Dynamics & Stochastics

Institute of Mathematical Statistics LECTURE NOTES–MONOGRAPH SERIES

Dynamics & Stochastics Festschrift in honor of M. S. Keane

Dee Denteneer, Frank den Hollander, Evgeny Verbitskiy, Editors

Volume 48

ISBN 0-940600-64-1

Institute of Mathematical Statistics LECTURE NOTES–MONOGRAPH SERIES Volume 48

Dynamics & Stochastics Festschrift in honor of M. S. Keane

Dee Denteneer, Frank den Hollander, Evgeny Verbitskiy, Editors

Institute of Mathematical Statistics Beachwood, Ohio, USA

Institute of Mathematical Statistics Lecture Notes–Monograph Series

Series Editor: Richard A. Vitale

The production of the Institute of Mathematical Statistics Lecture Notes–Monograph Series is managed by the IMS Office: Jiayang Sun, Treasurer and Elyse Gustafson, Executive Director.

Library of Congress Control Number: 2006924051 International Standard Book Number 0-940600-64-1 c 2006 Institute of Mathematical Statistics Copyright
All rights reserved Printed in the United States of America

Contents Preface Dee Denteneer, Frank den Hollander, Evgeny Verbitskiy . . . . . . . . . . . . . . . . .

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PERCOLATION AND INTERACTING PARTICLE SYSTEMS Polymer pinning in a random medium as influence percolation V. Beffara, V. Sidoravicius, H. Spohn and M. E. Vares . . . . . . . . . . . . . . . . .

1

Proof of a conjecture of N. Konno for the 1D contact process J. van den Berg, O. H¨ aggstr¨ om and J. Kahn . . . . . . . . . . . . . . . . . . . . . . .

16

Uniqueness and multiplicity of infinite clusters Geoffrey Grimmett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

A note on percolation in cocycle measures Ronald Meester . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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RANDOM WALKS On random walks in random scenery F. M. Dekking and P. Liardet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

Random walk in random scenery: A survey of some recent results Frank den Hollander and Jeffrey E. Steif . . . . . . . . . . . . . . . . . . . . . . . . . .

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Linearly edge-reinforced random walks Franz Merkl and Silke W. W. Rolles . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

Recurrence of cocycles and stationary random walks Klaus Schmidt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

RANDOM PROCESSES Heavy tail properties of stationary solutions of multidimensional stochastic recursions Yves Guivarc’h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

Attractiveness of the Haar measure for linear cellular automata on Markov subgroups Alejandro Maass, Servet Mart´ınez, Marcus Pivato and Reem Yassawi . . . . . . . . . 100 Weak stability and generalized weak convolution for random vectors and stochastic processes Jolanta K. Misiewicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Coverage of space in Boolean models Rahul Roy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 RANDOM FIELDS Strong invariance principle for dependent random fields Alexander Bulinski and Alexey Shashkin . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Incoherent boundary conditions and metastates Aernout C. D. van Enter, Karel Netoˇ cn´ y and Hendrikjan G. Schaap . . . . . . . . . . 144 Markovianity in space and time M. N. M. van Lieshout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

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Contents

Mixing and tight polyhedra Thomas Ward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 NUMBER THEORY AND SEQUENCES Entropy quotients and correct digits in number-theoretic expansions Wieb Bosma, Karma Dajani and Cor Kraaikamp . . . . . . . . . . . . . . . . . . . . . 176 Mixing property and pseudo random sequences Makoto Mori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Numeration systems as dynamical systems – introduction Teturo Kamae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 Hyperelliptic curves, continued fractions, and Somos sequences Alfred J. van der Poorten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 Old and new results on normality Martine Queff´ elec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Differentiable equivalence of fractional linear maps Fritz Schweiger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 ERGODIC THEORY Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem Michael Keane and Karl Petersen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 Purification of quantum trajectories Hans Maassen and Burkhard K¨ ummerer . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Finitary Codes, a short survey Jacek Serafin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 DYNAMICAL SYSTEMS Entropy of a bit-shift channel Stan Baggen, Vladimir Balakirsky, Dee Denteneer, Sebastian Egner, Henk Hollmann, Ludo Tolhuizen and Evgeny Verbitskiy . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 Nearly-integrable perturbations of the Lagrange top: applications of KAM-theory H. W. Broer, H. Hanßmann, J. Hoo and V. Naudot . . . . . . . . . . . . . . . . . . . 286 Every compact metric space that supports a positively expansive homeomorphism is finite Ethan M. Coven and Michael Keane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 On g -functions for subshifts Wolfgang Krieger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

Contributors to this volume Baggen, S., Philips Research Laboratories Eindhoven Balakirsky, V., Eindhoven University of Technology Beffara, V., CNRS–ENS-Lyon Bosma, W., Radboud University Nijmegen Broer, H. W., Groningen University Bulinski, A., Moscow State University Coven, E. M., Wesleyan University Dajani, K., Utrecht University Dekking, F. M., Thomas Stieljes Institute for Mathematics and Delft University of Technology den Hollander, F., Leiden University & EURANDOM Denteneer, D., Philips Research Laboratories Eindhoven Egner, S., Philips Research Laboratories Eindhoven Grimmett, G., University of Cambridge Guivarc’h, Y., Universit´e de Rennes 1 H¨aggstr¨ om, O., Chalmers University of Technology Hanßmann, H., RWTH Aachen, Universiteit Utrecht Hollmann, H., Philips Research Laboratories Eindhoven K¨ ummerer, B., Technische Universit¨ at Darmstadt Kahn, J., Rutgers University Kamae, T., Matsuyama University Keane, M., Wesleyan University Kraaikamp, C., University of Technology Delft Krieger, W., University of Heidelberg Liardet, P., Universit´e de Provence Maass, A., Universidad de Chile Maassen, H., Radboud University Nijmegen Mart´ınez, S., Universidad de Chile Meester, R., Vrije Universiteit Merkl, F., University of Munich Misiewicz, J. K., University of Zielona G´ ora Mori, M., Nihon University Naudot, V., Groningen University Netoˇcn´ y, K., Institute of Physics Petersen, K., University of North Carolina Pivato, M., Trent University v

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Contributors to this volume

Queff´elec, M., Universit´e Lille1 Rolles, S. W. W., University of Bielefeld Roy, R., Indian Statistical Institute Schaap, H. G., University of Groningen Schmidt, K., University of Vienna and Erwin Schr¨ odinger Institute Schweiger, F., University of Salzburg Serafin, J., Wroc
law University of Technology Shashkin, A., Moscow State University Sidoravicius, V., IMPA Spohn, H., TU-M¨ unchen Steif, J. E., Chalmers University of Technology Tolhuizen, L., Philips Research Laboratories Eindhoven den Berg, J., CWI and VUA der Poorten, A. J., Centre for Number Theory Research, Sydney Enter, A. C. D., University of Groningen Lieshout, M. N. M., Centre for Mathematics and Computer Science, Amsterdam Vares, M. E., CBPF Verbitskiy, E., Philips Research Laboratories Eindhoven

van van van van

Ward, T., University of East Anglia Yassawi, R., Trent University

Preface The present volume is a Festschrift for Mike Keane, on the occasion of his 65th birthday on January 2, 2005. It contains 29 contributions by Mike’s closest colleagues and friends, covering a broad range of topics in Dynamics and Stochastics. To celebrate Mike’s scientific achievements, a conference entitled “Dynamical Systems, Probability Theory and Statistical Mechanics” was organized in Eindhoven, The Netherlands, during the week of January 3–7, 2005. This conference was hosted jointly by EURANDOM and by Philips Research. It drew over 80 participants from 5 continents, which is a sign of the warm affection and high esteem for Mike felt by the international mathematics community. Mike is one of the founders of EURANDOM, and has seen this institute come to flourish since it opened its doors in 1998, bringing much extra buzz and liveliness to Dutch stochastics. Mike has also been a scientific consultant to Philips Research for some 20 years, and since 1998 spends one month per year with Philips Research Eindhoven in that capacity. It is therefore particularly nice that the conference could take place under the umbrella of these two institutions so close to Mike’s heart. Most people retire at 65. Not Mike, who continues to be enormously energetic and full of plans. After a highly active career, taking him to professorships in a number of countries (Germany 1962–1968, USA 1968–1970, France 1970–1980, The Netherlands 1981–2002), Mike is now with Wesleyan University in Middletown, Connecticut, USA, from where he continues his relentless search for fundamental questions, elegant solutions and powerful applications. All this takes place in an atmosphere of warm hospitality through the help of Mike’s spouse Mieke, who has welcomed more guests at her home than the American ambassador. A special event took place in the late afternoon of January 5, 2005, in the main auditorium on the Philips High Tech Campus. On behalf of Her Majesty the Queen, Mr. A. B. Sakkers, the mayor of Eindhoven, presented Mike with the decoration in the highest civilian order: “Knight in the Order of the Dutch Lion.” This decoration was bestowed upon Mike for his outstanding scientific contributions, his stimulating international role, as well as his service to industry.

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A number of organizations generously supported the conference: • NWO (Netherlands Organisation for Scientific Research). • KNAW (Royal Netherlands Academy of Arts and Sciences). • MRI (Mathematical Research Institute) and TSI (Thomas Stieltjes Institute), the two main collaborative research schools for mathematics in The Netherlands. • University of Amsterdam: Korteweg de Vries Institute. • Eindhoven University of Technology: Department of Mathematics and Computer Science. • Wesleyan University: Department of Mathematics. • ESF (European Science Foundation), through its scientific program “Random Dynamics of Spatially Extended Systems.” We are grateful to these organizations for their contribution. We are also grateful to the program committee (Rob van den Berg, Michel Dekking, Hans van Duijn, Chris Klaassen, Hans Maassen and Ronald Meester) for their help with putting together the conference program. We wish Mike and Mieke many healthy and active years to come, and trust that the reader will find in this Festschrift much that is to her/his interest and liking. Eindhoven, December 2005

Dee Denteneer Frank den Hollander Evgeny Verbitskiy

Mieke and Mike Keane January 5, 2005, Eindhoven, The Netherlands

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 1–15 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000022

Polymer pinning in a random medium as influence percolation V. Beffara1 , V. Sidoravicius2 , H. Spohn3 and M. E. Vares4 CNRS–ENS-Lyon, IMPA, TU-M¨ unchen and CBPF Abstract: In this article we discuss a set of geometric ideas which shed some light on the question of directed polymer pinning in the presence of bulk disorder. Differing from standard methods and techniques, we transform the problem to a particular dependent percolative system and relate the pinning transition to a percolation transition.

1. Introduction Motivating example. The totally asymmetric exclusion process (TASEP for short) is defined as follows on the set Z: At time 0, a (possibly random) configuration of particles is given, in such a way that each site contains at most one particle. To each edge of the lattice is associated a Poisson clock of intensity 1. Whenever this clock rings, and there is a particle at the left-end vertex of this edge and no particle at the right-end vertex, the particle moves to the right; otherwise the ring of the clock is ignored. The product over Z of Bernoulli measures of density ρ ∈ (0, 1) is invariant by this dynamics; in that case, the average number of particles passing through the origin up to time t is equal to ρ(1 − ρ)t, i.e. the flux through a given bond is exactly ρ(1 − ρ). The process is modified at the origin, by imposing that the Poisson clock associated with the bond e0 = 0, 1 is λ > 0. When λ > 1, one can still prove that the above expression for the flux holds asymptotically, although the Bernoulli measure is not an equilibrium measure anymore. From now on, we shall assume that 0 < λ ≤ 1. One of the fundamental questions in driven flow is to understand under which conditions such a static obstruction results in the formation of a “platoon” starting at the origin and propagating leftward. A convenient quantitative criterion for platoon formation is to start the TASEP with step initial conditions, i.e. all sites x ≤ 0 filled and all sites x ≥ 1 empty, and to consider the average current, j(λ), in the long time limit t → ∞. j(0) = 0, j is non-decreasing, and j(λ) = 1/4 for λ = 1. Thus the issue is to determine the critical intensity λc , which is defined as the supremum of all the λ for which j(λ) < 41 . Estimates for the value of λc were given in [10] and [5], and the full hydrodynamical picture is proved in [15]. 1 CNRS – UMPA – ENS Lyon, 46 All´ ee d’Italie, F-69364 Lyon Cedex 07, France, e-mail: [email protected] 2 IMPA, Estrada Dona Castorina, 110, Rio de Janeiro RJ 22460-320, Brasil, e-mail: [email protected] 3 Zentrum Mathematik, TU Muenchen, D-85478 Garching, Germany, e-mail: [email protected] 4 CBPF, Rua Dr. Xavier Sigaud, 150, Rio de Janeiro RJ 22290-180, Brasil, e-mail: [email protected] AMS 2000 subject classifications: primary 60G55, 60K35; secondary 60G17. Keywords and phrases: last-passage percolation, pinning, exclusion process.

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The blockage problem for the TASEP has been studied numerically and by exact enumerations. On the basis of these data, in [8] the value λc = 1 is conjectured. Recently this result has been challenged ([6]) and λc ∼ = 0.8 is claimed. One-dimensional driven lattice gases belong to the universality class of KardarParisi-Zhang (KPZ) type growth models. In particular the asymmetric simple exclusion process can be represented as the so-called body-centered solid-on-solid version of (1+1)-dimensional polynuclear growth model, or as directed polymer subject to a random potential. The TASEP also has a well known representation in terms of last-passage percolation (or maximal increasing subsequences, known as Ulam’s problem). In this article we choose to work in the setup of last passage percolation. The slow bond induces an extra line of defects relative to the disordered bulk. If λ < λc , the optimal path, i.e. the geodesic, is pinned to the line of defects. As λ → λc , the geodesic wanders further and further away from the line of defects and the density of intersections with the line of defects tends to zero. For λ > λc , the fluctuations of the geodesic are determined by the bulk, and the line of defects is irrelevant. In the present work we will not establish the actual value of λc (since we cannot) or settle the question as to whether it is equal to 1; the main goal of this paper is to describe a new way of looking at the problem which gives some insight about the precise behavior of the system. More precisely, we show how the problem can be studied through particular dependent percolative systems constructed in such a way that the pinning transition can be understood in terms of a percolation transition. 2. Interpretation as pinning in Ulam’s problem In this section we describe the representation of our initial problem in terms of Ulam’s problem or polynuclear growth. For more detailed explanations, see e.g. [1] or [11]. 2.1. The Model Let P (2) be the distribution of a Poisson point process of intensity λ(2) = 1 in the plane R2 , and let Ω(2) the set of all its possible configurations; for all n > 0, let (2) (2) Pn be the law of its restriction to the square Qn := [0, n] × [0, n] and Ωn be the configuration space of the restricted process. The maximal increasing subsequence problem, or Ulam’s problem, can be formulated in the following geometric way: Given a configuration ω (2) ∈ Ω(2) and (2) its restriction ωn to Qn , look for an oriented path π (moving only upward and (2) rightward) from (0, 0) to (n, n) collecting as many points from ωn as possible; as in the case of last-passage percolation described in the introduction, we shall call such a path a geodesic. Let Nn denote the number of collected points along such an optimal path (which need not be unique). It is a well known fact (see [1]) that E (2) Nn = 2. (2.1) n→+∞ n In a remarkable paper [9], K. Johansson showed that for this model the transversal fluctuations of the geodesics are of order n2/3 . A closely related problem is considered in [3]. We now modify the original model in the following way: Assume in addition, that on the main diagonal y = x there is an independent one-dimensional Poisson lim

Polymer pinning in a random medium as influence percolation

3

point process of intensity λ(1) = λ, and let P (1) be its distribution. We will denote (1) by Ω(1) (resp. Ωn ) the configuration space of this process (resp. of its restriction to Qn ). Finally if ω (1) is a realization of P (1) and ω (2) one of P (2) , let ω = ω (2) ∪ ω (1) be their union, and ωn the restriction of ω to Qn . What can now be said about an optimal directed path starting at (0, 0) and ending at (n, n)? If λ 1, clearly the geodesic will stay close to the diagonal and E (1) × E (2) Nn =: e(λ) > 2. n→∞ n lim

(2.2)

Instead of (2.2) in our context it will be more convenient to use a more geometric notion. We say that the directed polymer is pinned (with respect to the diagonal) if P (1) × P (2) -almost surely, the number of visits made by geodesics to {(u, u) : 0 ≤ u ≤ t} is of order γt, for some γ > 0, for all t large enough. One expects the “energy” notion (2.2) and geometric notion of pinning to be identical, but this is yet another point which remains to be proved. e(λ) is non-decreasing and there is a critical value λc , where e hits the value 2. The same arguments which predict λc = 1 in the case of the TASEP yield λc = 0 in the case of our model. Thus any extra Poisson points along the diagonal are expected to pin the directed polymer. Such a behavior is extremely delicate, and the answer depends on the nature and behavior of the geodesics in the initial, unperturbed system. Very little is known, even on a heuristic level, when the underlying measure governing the behavior of the polymer is not “nice” (with a kind of Markov property, for example a simple symmetric random walk). Our criteria presented below give partial but rigorous answers as to whether λc is strictly positive or not. In passing let us note that for a symmetric environment pinning can be proved, at least on the level of e(λ) [7]. Symmetric means that P (2) is concentrated on point configurations which are symmetric relative to the diagonal. In this case λc = 1, i.e. e(λ) = 2 for 0 ≤ λ ≤ 1 and e(λ) > 2 for λ > 1. Indeed in the symmetric case, the system is amenable to exact computations in terms of Fredholm determinants; a trace of the simplification can also be seen in our later discussion (see Section 2.6). 2.2. Construction of the broken-lines process Hammersley gave a representation of the longest increasing subsequence problem for a random permutation in terms of broken lines built from a Poisson point process in the positive quadrant (we describe the construction in some detail below). The length of the longest increasing subsequence can then be seen to be the number of lines which separate the points (0, 0) and (n, n) from one another. The purpose of that representation is to obtain a superadditivity property which easily implies the existence of the limit in (2.1) — but doesn’t specify its value. It is a very convenient formalism, which was used in [11, 13, 14] and [16, 17]. The broken line process ΓS in a finite domain S can be defined as the space-time trace of some particle system with birth, death and immigration. For convenience we rotate the whole picture by an angle of π/4 clockwise, so that the geodesic is restricted to never have a slope which is larger than 1 in absolute value. In what follows we will consistently use the letters t and x for the first and second coordinates, respectively, in the rotated picture; we will refer to t as “time” (the reason for that will become clear shortly). The geodesics can then be seen as curves of space-type (using the usual language of general relativity).

V. Beffara et al.

4

x

x

t

t

Fig 1. The construction of a Hammersley process.

Let S be the planar, bounded domain defined (cf. Figure 1) as S := {(t, x) : t0 < t < t1 , g− (t) < x < g+ (t)},

(2.3)

where t0 < t1 are given points and g− (t) < g+ (t), t0 < t < t1 are piecewise linear − continuous functions such that, for some t+ 01 (resp. t01 ) in the interval [t0 , t1 ], g+ (t) − (respectively g− (t)) increases (respectively, decreases) on (t0 , t+ 01 ) (resp. (t0 , t01 )) − and decreases (respectively, increases) on (t+ 01 , t1 ) (resp. (t01 , t1 )), always forming an angle of ±π/4 with the t-axis. Consider four independent Poisson processes Π0,+ , Π0,− , Π+ and Π− on the boundary of S. The processes Π0,± are supported on the leftmost vertical boundary component 0 S := {(t0 ,
x) : g0,− < x < g0,+ } of S, where g0,± := g± (t0 ), and they both have intensity λ(2) /2. The process Π+ is defined on the “northwest” boundary + S := {(t, g0,+ (t)) : t0 < t < t+ 01 }, and the process Π− is defined on the “southwest” boundary − S := {(t, g0,− (t)) : t0 < √ t < t− 01 }; their intensities (2) (with respect to the length element of ± S) are both λ . Finally, let Πin be a Poisson point process in S, with intensity λ(2) , and independent of the previous four. Following the general strategy for the definition of Markov polygonal fields of [2], we define a broken line process as follows. Each point of the Poisson process Πin is the point of birth of two particles which start moving in opposite directions, i.e. with velocities +1, −1. At each random point of Π0,+ , Π− a particle is born having velocity +1. Similarly at each random point of Π0,− , Π+ , a particle is born having velocity −1. All particles move with constant velocity until two of them collide, after which both colliding particles are annihilated (see Figure 1). The state space XS of the process ΓS is the set of all finite collections (γ1 , . . . , γk ) (including the empty one) of disjoint “broken lines” γj inside S. By a broken line in S we mean the graph γ = {(t, x) ∈ S : t = γ (x)} in S of a continuous and piecewise linear function γ , with slopes all ±1. Let PS denote the probability distribution on XS corresponding to the broken line process ΓS defined above.

Polymer pinning in a random medium as influence percolation

5

Let S  ⊂ S  be two bounded domains of the form of (2.3), and let PS
and PS

be the probability distributions of the broken line processes in S  and S  , respectively. The probability measure PS

on XS

induces a probability measure PS

|S
on XS
, which is the distribution of the restricted process ΓS

∩ S  . Then, by the choice we made of the boundary conditions, the following consistency property holds (see [2]): PS

|S
= PS
. (2.4) This guarantees the existence of the broken line process Γ ≡ ΓR2 on R2 , its distribution P on XR2 being such that for every S of the above shape, P|S = PS . Moreover P is invariant with respect to the translations of R2 . Remark. The same description holds for the polynuclear growth (PNG) model, which describes a crystal growing layer by layer on a one-dimensional substrate through the random deposition of particles. They nucleate on existing plateaus of the crystal, forming new islands. In an idealization these islands spread laterally with constant speed by steady condensation of further material at the edges of the islands. Adjacent island of the same level coalesce upon meeting and on the top of the new levels further islands emerge. Observe that a path π which can move only in the northeast-southeast cone (i.e. a path of space type) can collect at most one initial Poisson point from each broken line. In other words, the broken lines “factorize” the points of the configuration ω, in such a way that it tells us which points cannot be collected by the same path, and that the maximal number of points is bounded by the number of broken lines which lie in between start-point and end-point of the path π. In fact, the lines also provide an explicit construction of a geodesic, as follows: Start at point (t1 , x) on the right boundary and move leftward until you meet a line, then follow this line until you arrive at a point, which you can collect. Then start moving leftward again until you collect a point in the second-to-last broken line, and so on. The number of collected points is then essentially equal to the number of broken lines, though a little care needs to be taken as far as boundary conditions are concerned if this comparison is to be made completely formal. This observation led to a new proof of (2.1) in [2]. It is also the starting point of our argument. 2.3. Essential and non-essential points We now return to the question asked at the beginning of this section: How are extra added points affecting the initial system? We begin with a few purely deterministic observations and statements. We will need some extra notations: Given any configuration ω n (not necessarily sampled from a Poisson process) of points in Qn , let H( ωn ) be the number of broken lines produced by the above construction. For x = (x, t) and A = {x1 , . . . , xk } we will denote by ω n ∪ x or ω n ∪ A the configuration obtained from ω n by the addition of the points x1 , . . . , xk , and by Γn ( ωn ) the associated configuration of broken lines. Proposition 2.1 (Abelian property, see [17]). For any choice of ω n , x1 and x2 we have Γn ( ωn ∪ (x1 ∪ x2 )) = Γn (( ωn ∪ x1 ) ∪ x2 ) = Γn (( ω n ∪ x2 ) ∪ x1 ) .

(2.5)

Proposition 2.2 (Monotonicity, see [17]). For any choice of ω n and x1 we have ωn ∪ x1 ) ≤ H( ωn ) + 1. (2.6) H( ωn ) ≤ H(

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Definition 2.3. Given ω n and x we say that x is essential for ω n if H( ω n ∪ x1 ) = H( ωn ) + 1. Remark. The above definition is domain-dependent: If Sn ⊂ Sm are two domains in R2 , and ω n , ω m are the restrictions of a configuration ω  to Sn and Sm respectively, then an extra added point which is essential for ω n might not be essential for ω m . If an added point x is essential, its presence is felt on the boundary of the domain by the appearance of an extra broken line going outside of the area. If an added point x is not essential, its presence can be felt on the boundary or not, but in any case it will change the local geometry of existing broken lines. Speaking informally, the configuration ω n determines a partition of the domain into two (possibly disconnected) regions E and B, such that any additional point chosen in E will be essential for ω n , while if it is in B it will be not essential for ω n . It is easy to construct examples of configurations ω  for which E is empty (i.e., that are very insensitive to local changes). On the other hand B is never empty as soon as ω  is not empty. It is also easy to give examples of the following situations: • x1 is not essential for ω n and x2 is not essential for ω n , but x1 is essential for ω n ∪ x2 and x2 is essential for ω  n ∪ x1 ; • x1 is essential for ω n and x2 is essential for ω n , but x2 is not essential for ω  n ∪ x1 .

Remark. In order to simplify our explanations and make some concepts as well as the notations more transparent (and lighter), we will change from the consideration of point-to-point case to the point-to-hyperplane case, i.e. instead of looking for a geodesic connecting (0, 0) to (t, 0), we will be looking for an optimal path connecting (t, 0) to the line x = 0. This change is of purely “pedagogical” nature: All the ideas discussed above and below are easily transferred to the point-to-point case. Nevertheless we will not deny that it requires some amount of additional work due to boundary conditions. The reader should also not be surprised with our taking of starting point as (t, 0) and moving backward to the x-axis in the point-to-hyperplane case (see the remark at the end of Section 2.2). Since our broken lines were constructed by drawing the space-time trajectories of particles in “forward time”, the information provided by the broken lines about the underlying point configuration is useful in the backward direction, and thus it forces us to construct the geodesic this way. Conversely, in order to construct a forward geodesic, one could construct the broken lines on the same point configuration but backward in time. Due to that, the area to which we will be restricting our process will be the triangular area Sn enclosed by segments connecting points (0, −n), (0, n) and (n, 0). If confusion doesn’t arise, we will keep denoting related quantities by the same sub-index n as for the square case, for example ω n will stay from now on for the configuration of points in this triangular area. The next proposition is the crucial point in our construction. Proposition 2.4 (see [17]). If x is essential for ω n then the point-to-plane geodesic in configuration ω n ∪ x has to collect point x.

Again this is a purely deterministic statement, and does not depend on the choice of ω n or x. Further we will be considering only cases when extra points are added only along the t-axis.

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2.4. Propagation of influence Once an extra point x is added to the system, we need to update the configuration of broken lines. One way to do that is to redo the whole construction from scratch, i.e. to erase all the existing broken lines and redraw them using the algorithm we described previously, taking the new point into account. It is then natural to ask how much the new picture differs from the old one, which is not perturbed by addition of point x. It turns out that there is a very simple algorithm allowing us to trace all the places of the domain where the addition of x will be felt, i.e. where local modification will be done. n ∪ {x}. In order to see where Consider an augmented configuration ω n = ω and how the broken lines of Γn ( ωn ) will be modified, we look at a new interacting particle system, starting from the points of ω n , but with new interaction rules: 1. Particles starting from the points of ω n are following the same rules as before, i.e. they are annihilated at the first time when they collide with any other particle. These particles will be called “regular particles”; 2. The two particles which start from the newly added point x (also with velocities +1 and −1) will be called “superior particles”, and they obey different rules: (a) Superior particles annihilate if and only if they collide with each other; (b) If a superior particle collides with some regular particle, the velocity of the superior particle changes to that of the incoming regular particle, which is annihilated while the superior particle continues to move. − We will denote by p+ x (resp. px ) the superior particle which starts from x with − initial velocity +1 (resp. −1); The space-time trajectories of p+ x and px will be + − denoted by πx and πx , respectively. Observe that if any of these two trajectories leaves the triangular area Sn , it will never come back to it. (Notice that since superior particles can change their velocities during their evolution, both particles can leave the triangular area from the same side.) If πx+ and πx− intersect inside of Sn , then, according to rule 2a the corresponding superior particles are annihilated. The path πx+ (resp. πx− ) can be represented as an alternating sequence of concatenated segments πx+ (1, +), πx+ (1, −), πx+ (2, +), πx+ (2, −), . . . (resp. πx− (1, −), πx− (1, +), πx− (2, −), πx− (2, +), . . . ), where each segment corresponds to the time interval between two consecutive velocity changes of the superior particle p+ x (resp., − px ), and during which its velocity is equal to +1 or −1 according to the sign given as second argument in the notation. − The trajectories of p+ n ∪ {x}. Observe x and px are completely determined by ω + changes its velocity that each time px changes its velocity from +1 to −1 (resp. p− x from −1 to +1), it starts to move along a segment which also belongs to some broken line γi from Γn ( ωn ), and when it changes its velocity back to +1, it leaves this broken line, and moves until the next velocity flip, which happens exactly when the superior particle collides with the next broken line γi+1 in Γn ( ωn ). This gives  an extremely simple rule how to transform Γn ( ωn ) into Γn ( ωn ) (see Figure 2):

• Erase all the πx± (j, ∓) (i.e., all parts of the path of the superior particle which are contained in one of the original broken lines) to obtain an intermediate picture Γn ( ωn ); ωn ). • Add all the πx± (j, ±) thus obtaining the picture Γn (

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πx+

x

x

x

πx−

Fig 2. Propagation of influence.

It is not hard to conclude that Γn ( ωn ) = Γn ( ωn ). In other words the paths πx+ and πx− show how the “influence” of x spreads along the configuration Γn ( ωn ). If − both superior particles p+ and p collide inside the domain S , the trajectories πx+ n x x − and πx close into a loop, and outside of this loop the configuration Γn ( ωn ) was not modified, i.e. the presence of x was not felt at all. Definition 2.5. Given a configuration ω of the underlying Poisson process, and an added point x on the time axis, we will denote by τ (x; ω) (or simply τ (x) if there is no confusion possible) the self-annihilation time of the pair of particles created at x, i.e. the time at which the paths πx− and πx+ meet, if such a time exists; let τ (x) = +∞ otherwise. In the specific case we are looking at, τ (x) is almost surely finite if λ(2) > 0, but it need not be the case for other underlying point processes. 2.5. Interaction and Attractors Another important step in the analysis of the spread of influence, is to understand how the influence paths interact with each other if we add multiple points x1 , . . . , x
to the initial configuration. Proposition 2.1 implies that we can obtain the full picture by adding the points one by one; to simplify the notations, in our description of the procedure we will also use the fact that the additional points will be placed along t-axis, though this is not essential. Again, due to the presence of time orientation, the nature of the interaction between influence paths becomes exposed in a more transparent way if we proceed backward, i.e. if we begin to observe the modifications first when adding the rightmost point, and then continue progressively, adding the points one by one, moving leftward, each time checking the effect created by the newly added point. For notational convenience let us index the new points in the backward direction, i.e. xi = (ti , 0) with t1 > t2 > · · · . Applying the construction described in the previous subsection successively for each of the new points, we obtain the following rules for the updating a configuration (r) n ∪ with multiple points added: Take the initial configuration ω n and let ω n = ω {xi , 1 ≤ i ≤ r} be the modified configuration. In order to see where and how the broken lines of Γn ( ωn ) will be updated, consider a new particle representation built (r) on the configuration ω n , and obeying the following rules: 1. Regular particles, starting from the points of ω n , are annihilated as soon as they collide with any other particle;

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2. The particles starting from the (xi )1≤i≤r , with velocities +1 and −1 are de− noted by p+ xi and pxi respectively; again we shall call them superior particles. They behave as follows: (a) Whenever two superior particle of different types collide (“+−” collision), they annihilate and both disappear; (b) If two superior particles of the same type collide (“++” or “−−” collision), then they exchange their velocities (elastic interaction) and continue to move; (c) If a superior particle collides with a regular particle, the velocity of the superior particle changes that of the incoming regular particle; the regular particle is annihilated, while the superior particle survives and continues to move. + − Denote the space-time trajectories of superior particles p+ xi and pxi by πxi and + − respectively. As before, each pair of paths πxi and πxi can be represented as an alternating sequence of concatenated segments πx+i (1, +), πx+i (1, −), πx+i (2, +), πx+i (2, −), . . . or πx−i (1, −), πx−i (1, +), πx−i (2, −), πx−i (2, +), . . . , respectively, with the same convention for the sign of the velocities. We are now ready to complete the (r) set of rules which govern the transformation of Γn ( ωn ) in to Γn ( ωn ):

πx−i ,

• Erase each segment πx±i (j, ∓) from Γn ( ωn ), producing an intermediate picture Γn ( ωn ); (r) ωn ) = Γn ( ωn ). ωn ), thus producing Γn ( • Add the segments πx±i (j, ±) to Γn ( (Here as previously, two ± in the same formula are taken to be equal signs, while ± and ∓ in the same formula stand for opposite signs.) Recall that we are working in the bounded triangular domain Sn with the configuration ω n ∪{xi , 1 ≤ i ≤ r}, where r is the number of added points. By f+ i = (tf+ , xf+ ) i

i

and f− , xf− ) we shall denote the end-points of the influence paths πx+i and πx−i i = (tf− i i — they can be points where the corresponding paths exit the triangular domain, or points where a “+−” collision happens, in which case the two corresponding end-points are equal. Besides, let r+ , 0) and r− , 0) be the points of first return to the i = (tr+ i = (tr− i i + − t-axis of the paths πxi and πxi , respectively, and define  ti := min{tf+ , tf− , tr+ , tr− }. i

i

i

i

(2.7)

− + −   Last, let e+ i := πxi ∩ {(ti , x), x ∈ R} and ei := πxi ∩ {(ti , x), x ∈ R}, and denote x−i the parts of πx+i and πx−i lying between xi and e+ by π x+i , π i , and respectively − between xi and ei .

Definition 2.6. Let Ji be the (random) Jordan curve starting at xi , following the +  path π x−i until the point e− i , then the vertical line t = ti up to ei , and then the path π x+i backward until it comes back to xi . The domain bounded by Ji will be called the attractor of the point xi and denoted by Ai (see Figure 3). The part of its boundary which is contained in π x+i (resp. π x−i ) will be called the upper (resp. lower ) boundary of the attractor. It is important to understand how attractors are affected by one another, in order to give a convenient description of the whole augmented process. The key remark is the following: Informally speaking, a superior particle of a given type is

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πx+5 x5

πx+2 πx+1

πx+3

x3

x2

x1

x4 πx−5

πx−1 πx−2

Fig 3. Attractors (shaded).

not affected by an older one (i.e. one with larger index) of the same type. Indeed this is a consequence of the previous construction, and especially of the Abelian property (see Proposition 2.1). Of course this does not mean that the attractors Aj for j < j0 do not change when xj0 is added, since it remains possible that a “+−”-collision happens. In that case, we get some kind of a monotonicity property, for the statement of which some additional notation will be needed. Recall that τ (x; ω) stands for the selfannihilation time of a particle in the underlying scenery, i.e. ignoring the effect of the other new particles, both younger and older. We will denote by τ (x, y; ω) the annihilation time of the set of superior particles born from x and y, i.e. the last time at which any of the corresponding four superior particles is still alive. Such an annihilation can happen in one of four ways: • Flat: x is born, the two particles thus created annihilate, then y is born and its two particles collide; + • Embedded: x is born, then y appears between p− x and px , then the particles issued from y collide, then so do those issued from x; + • Parallel: x is born, then y appears outside of (p− x , px ), then the particles issued from y collide, then so do those issued from x; + • Crossed: x is born, then y (also outside of (p− x , px )), then one particle issued from x annihilates the particle of the other type coming from y, then the remaining two collide. The combinatorics become much more involved when more particles are added; nevertheless, it is possible (if a bit technical if a formal proof is needed, see [4]) to show the following: Proposition 2.7 (Monotonicity of the influence). For any two added points x and y, we have the following inequality: τ (x, y; ω) ≥ Max(τ (x, ω), τ (y; ω)); and more generally the annihilation time of the union of two finite families of added points is at least equal to the larger of the two annihilation times of the parts. In the flat, embedded and parallel cases, the monotonicity extends to the shapes of the attractors (the attractor of y in the presence of x contains the one without); there is true reinforcement in the embedded case, in that the inclusion is strict as

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soon as there is a “++”- or “−−”-collision. This is not always the case in crossed configurations (cf. Fig. 3, where the attractor of x3 is shortened by the addition of x4 ), which leads us to the following definition: Definition 2.8. We say that two attractors Ai and Aj , i > j are connected if there exists a sub-sequence j = i0 < i1 < i2 < · · · < ik = i such that xir ∈ Air+1 for all 0 ≤ r < k. We will call (ij )0≤j≤k a connecting subsequence between Ai and Aj . Observe that if i > j > k, if Ai is connected to Aj and Aj is connected to Ak , then Ai is connected to Ak . Nevertheless, due to the presence of orientation in the temporal direction, the above implication generally does not hold without the condition i > j > k. Our construction immediately implies the following: Proposition 2.9. If Ai is connected to Aj , i > j, then  ti ≥  tj .

− Corollary 2.10. If Ai is connected to Aj , i > j, and the end-points e+ j and ej belong respectively to the south-east and north-east boundaries of the triangular − domain Sn , then so do e+ i and ei .

Corollary 2.11. Assume that Ai is connected to Aj , i > j, with connecting subsequence (is )0≤s≤k : If xj is essential for the configuration ω n , then so are all the xis , 1 ≤ s ≤ k. 2.6. Pinning of the geodesics We now return to our original problem. Observe that if, for some fixed configuration ω n in Sn , we pick a realization of the points (xi )1≤i≤v in such a way that A1 and Av are connected (say), with connecting subsequence (ij )0≤j≤k , then all the xij must be essential for ω n , and therefore the point-to-plane geodesics for the configuration ω n ∪ {xi } has to visit all the xij . In the new formalism, the original question of whether, for any given density λ(1) > 0 of the one-dimensional Poisson point process, the limiting value in (2.1) is increased, becomes equivalent to the following: Is there a positive δ such that, with high probability as n goes to infinity, at least a fraction δ of the newly added points are essential for ω (2) ? This question is more complicated than simply whether there exists a chain of pairwise connected attractors spanning from the left to the right boundary of the domain: Indeed, such a chain does not necessarily have positive density. In the next Section we also mention some of the interesting mathematical questions that arise in the construction. It is not an easy task to understand how the attractors behave. The fact that the ωn ∪{xj , 1 ≤ j ≤ i}), x−i+1 depends on Γn ( structure of the influence paths π x+i+1 and π but not on the xj for j > i + 1, reduces the problem to checking whether none of x−i+1 hits the t-axis before xi , in which case Ai+1 is the influence paths π x+i+1 , π connected to Ai (see Figure 3). For a single point x added to the initial configuration, each influence path πx+ , πx− has the same statistical properties as what is known as a “second-class particle” in the framework of exclusion processes. Since in the definition of an attractor an important role is played by the (possible) return times of the influence paths to the t-axis, several things must be settled: 1. The first return time to the t-axis of a single influence path. It is believed (but remains a challenging open problem) that in the case of a one-dimensional

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exclusion process, a second-class particle behaves super-diffusively. Though some bounds are available, and we know the mean deviations of the second class particle [12], they do not provide good control on return times; 2. The joint behavior of the influence paths π x+ and π x− . Generally, it is a complicated question too, but for our purposes we need to have such control only up to the first times when one of π x+ , π x− returns to the t-axis. Before such a time, both paths stay apart from each other, and some good mixing properties of the system come into play; so the question reduces to how efficiently we control point 1. The fact that the influence lines of “younger” points (with smaller indices, i.e. sitting more to the right) repeal the influence lines of “older” points, leads to the following observation: Once the attractor Ai+m of an older point reaches the younger point xi , then it cannot end before the attractor Ai ends. If the attractor Ai ends before reaching the next point xi−1 , then Ai+m can still go forward, and possibly itself reach xi−1 . Observe that at the time Ai ends, the boundaries of Ai+m are necessarily at a positive distance from the t-axis. − That, together with the fact that the evolution of p+ xi and pxi in the slab (ti , ti−1 )×R depends only on Γn ( ωn ) brings some notion of week dependence to the system from one side, and the idea of a “re-start point” from another. This reduces the study of percolation of attractors to a more general problem of one-dimensional, long-range, dependent percolation which we formulate in the next section. There we also mention some of the interesting mathematical questions that have arisen. 3. Stick percolation In this section we introduce two “stick percolation” models, which will serve as toy models in the study of the propagation of influence in the broken-line model. In spite of their apparent simplicity, these models can be very useful studying effects of columnar defects and establishing (bounds for) critical values for asymptotic shape changes for some well known one-dimensional growth systems (see [4]). 3.1. Model 1: overlapping sticks Let (xi )i∈N be a Poisson point process of intensity λ > 0 on the positive real line. We call the points of this process “seeds”, and assume that they are ordered, x0 being the point closest to the origin. To each seed xi we associate a positive random variable Si (a “length”) and assume that the (Si )i∈N are i.i.d. with common distribution function F . The system we consider is then the following: For every i ∈ N, construct the segment Si = [xi , xi + Si ] (which we will call the i-th “stick”). We say that the sticks Si and Sj , i < j are connected if xj < xi + Si , i.e. if they have non-empty intersection; the system percolates if and only if there is an infinite chain of distinct, pairwise connected sticks, which (with probability 1) is equivalent to saying that the union of the sticks contains a half-line. It is easy to see that the system percolates with probability 0 or 1 (it is a tail event for the obvious filtration); and in fact there is a complete characterization of both cases: Proposition 3.1 ([4]). Let  xR(x) = P (S1 > x) be the tail of the stick length distribution, and let ϕ(x) = 0 R(u)du. Then the system percolates with probability

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1 if and only if



13

+∞

e−λϕ(x) dx < +∞.

(3.1)

0

This leads us to the following definition: Definition 3.2. The distribution F governing the stick process is said to be clusterstable if the system percolates for every positive value of λ. Example 1 (Return times of random walks). One natural distribution for the length of the sticks, in view of the previous construction, is the following: At time xi , start two random walks with no drift (the specifics, e.g. whether they are discrete or continuous time walks, will not matter at this point — for that matter we could also take two Brownian motions), one from +1 and the other from −1. Then, let xi + Si be the first time when these two walks meet. It is well known that, up to multiplicative constants, P (Si > t) behaves as t−1/2 for large t, as soon as the walks are irreducible and their step distribution have finite variance. It is easy to check that the obtained distribution is cluster-stable. Example 2. Change the previous example a little, as follows: For every value of i, start a two-dimensional Brownian motion (or random walk) starting at (xi,1 ), and let xi + Si be the first hitting point of the axis by this Brownian motion; but erase the stick if Si < 0. Then the distribution F is Cauchy restricted to be positive, and its tail is equivalent to P (Si > t) ∼ c/t as t goes to infinity. In that case, the stick length distribution is not cluster-stable for model 1, and there exists a critical value λc = 1/c, such that for λ > λc the system percolates while it does not if λ < λc . 3.2. Model 2: reinforced sticks There are several ways to mimic the interaction between “funnels”. One of the most simple possibilities is to add extra rules to Model 1 to account for the interaction. The basic idea behind this modification, is that if two or more sticks from Model 1 overlap, then there is a certain reinforcement of the system, which depends on how many sticks overlap, and then the whole connected component is enlarged correspondingly. One way to do that can be described informally as the following dynamic process. First, see each stick Si as the flight time of a particle πi born at time xi . We want to model the fact that if πi wants to land when a younger one (say πj ) is still flying, instead πi “bounces” on πj ; πj on the other hand should not be affected by πi , if the process is to look like the propagation of influence described in he previous section. Assign to each particle a “counter of chances” Ni which is set to 1 when the particle is born (formally it should be a function from R+ to N, and we let Ni (xi ) = 1). Then, two things can happen. If xi + Si < xi+1 , there is no interaction and πi dies at time xi + Si . If on the other hand xi + Si ≥ xi+1 , at time xi+1 the particle πi gets a “bonus”, so that Ni (xi+1 ) = 2; and similarly, it gets a bonus each time it passes above a seed point, so that Ni (xi + Si −) − 1 is the number of seed points in Si . Now when π lands, its counter is decreased by 1, but if it is still positive the particle bounces on the axis and restarts using an independent copy of S. If all the chances are exhausted before the closest seed is reached, πi gets killed. Again, we may ask a similar question: Given λ > 0, for which distribution functions F is the probability for a given particle to survive up to infinity positive? It is

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obvious that even with E(S) < +∞ we can obtain infinite trajectories for certain (large) values of λ. We will call F cluster-stable for the reinforced process if this happens for every positive λ. Example. The Cauchy distribution used in the previous example cluster-stable for the reinforced model. It is easy to see (e.g. by a coupling argument) that if the original system percolates, the reinforced version (for the same value of λ and the same length distribution F ) percolates too. In particular a cluster-stable distribution is cluster-stable for the reinforced problem. Nevertheless it is an interesting open problem to give full characterization of distributions which are cluster-stable for the reinforced models. Acknowledgments. The authors wish to thank K. Alexander, H. Kesten and D. Surgailis for many hours of fruitful and clarifying discussions scattered over the past four years. We also thank CBPF, IMPA, TU-M¨ unchen for hospitality and financial support. References [1] Aldous, D. and Diaconis, P. (1995). Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theory Related Fields 103, 2, 199–213. MR1355056 [2] Arak, T. and Surgailis, D. (1989). Markov fields with polygonal realizations. Probab. Theory Related Fields 80, 4, 543–579. MR980687 [3] Baik, J., Deift, P., and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12, 4, 1119–1178. MR1682248 [4] Beffara, V. and Sidoravicius, V. (2005) Effect of columnar defect on asymptotic shape of some growth processes. Preprint, in preparation. [5] Covert, P. and Rezakhanlou, F. (1997). Hydrodynamic limit for particle systems with non-constant speed parameter. J. Statist. Phys. 88, 1-2, 383–426. MR1468390 [6] Ha, M., Timonen, J., and den Nijs, M. (2003) Queuing transitions in the asymmetric simple exclusion process. Phys. Rev. E, 68:056122. [7] Sasamoto, T. and Imamura, T. (2004). Fluctuations of the one-dimensional polynuclear growth model in half-space. J. Statist. Phys. 115, 3-4, 749–803. MR2054161 [8] Janowsky, S. A. and Lebowitz, J. L. (1994). Exact results for the asymmetric simple exclusion process with a blockage. J. Statist. Phys. 77, 1-2, 35–51. MR1300527 [9] Johansson, K. (2000). Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Related Fields 116, 4, 445–456. MR1757595 [10] Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 324. Springer-Verlag, Berlin. MR1717346 ¨hofer, M. and Spohn, H. (2000). Statistical self-similarity of one[11] Pra dimensional growth processes. Phys. A 279, 1-4, 342–352. MR1797145 ¨hofer, M. and Spohn, H. (2002). Current fluctuations for the totally [12] Pra asymmetric simple exclusion process. In In and Out of Equilibrium (Mambu-

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caba, 2000). Progr. Probab., Vol. 51. Birkh¨ auser Boston, Boston, MA, 185–204. MR1901953 ¨hofer, M. and Spohn, H. (2002). Scale invariance of the PNG droplet Pra and the Airy process. J. Statist. Phys. 108, 5-6, 1071–1106. MR1933446 ¨hofer, M. and Spohn, H. (2004). Exact scaling functions for onePra dimensional stationary KPZ growth. J. Statist. Phys. 115, 1-2, 255–279. MR2070096 ¨ la ¨inen, T. (2001). Hydrodynamic profiles for the totally asymmetSeppa ric exclusion process with a slow bond. J. Statist. Phys. 102, 1–2, 69–96. MR1819699 Sidoravicius, V., Vares, M. E., and Surgailis, D. (1999). Poisson broken lines process and its application to Bernoulli first passage percolation. Acta Appl. Math. 58, 1–3, 311–325. MR1734758 Sidoravicius, V., Vares, M. E., and Surgailis, D. (2005). Discrete broken line process and applications to the first passage percolation models. Preprint, in preparation.

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 16–23 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000031

Proof of a conjecture of N. Konno for the 1D contact process J. van den Berg1,∗ , O. H¨ aggstr¨ om2 and J. Kahn3,† CWI and VUA, Chalmers University of Technology and Rutgers University Abstract: Consider the one-dimensional contact process. About ten years ago, N. Konno stated the conjecture that, for all positive integers n, m, the upper invariant measure has the following property: Conditioned on the event that O is infected, the events {All sites −n, . . . , −1 are healthy} and {All sites 1, . . . , m are healthy} are negatively correlated. We prove (a stronger version of) this conjecture, and explain that in some sense it is a dual version of the planar case of one of our results in [2].

1. Introduction and statement of the main result Consider the contact process on Z with infection rates λ(x, y), x, y ∈ Z, |x − y| = 1, and recovery rates δx , x ∈ Z. This model can, somewhat informally, be described as follows: Each site x ∈ Z has, at each time t ≥ 0, a value ηt (x) ∈ {0, 1}. Usually, 1 is interpreted as ‘infected’ (or ‘ill’) and 0 as ‘healthy’. When a site y is ill, it infects each neighbour x at rate λ(x, y). In other words, at time t each healthy site x becomes infected at rate λ(x, x + 1)ηt (x + 1) + λ(x, x − 1)ηt (x − 1). Further, when a site x is ill, it recovers (becomes healthy) at rate δx . We assume that the above-mentioned rates are bounded. In fact, the most commonly studied case is where all recovery rates are constant, say 1, and all infection rates are equal to some value λ > 0. See [6] and [7] for background and further references. Let νt be the law of (ηt (x), x ∈ Z). It is well known that if at time 0 all sites are infected, νt converges, as t → ∞, to a distribution called the upper invariant measure. We denote this limit distribution by ν. About ten years ago N. Konno proposed the following conjecture (see [4], Conjecture 4.5.2 or [5], Conjecture 2.3.2): Conjecture 1. Let ν be the upper invariant measure for the 1D contact process with infection rate λ and recovery rate 1. Let n, m be positive integers. Then ν(η(x) = 0, x = −n, . . . , −1, 1, . . . , m | η(0) = 1) ≤ ν(η(x) = 0, x = −n, . . . , −1 | η(0) = 1) × ν(η(x) = 0, x = 1, . . . , m | η(0) = 1). * Part

(1)

of JvdB’s research is financially supported by BRICKS project AFM2.2. in part by NSF grant DMS0200856. 1 CWI (Department PNA), Kruislaan 413, 1098 SJ Amsterdam, The Netherlands, e-mail: [email protected] 2 Chalmers University of Technology, S-412 96 G¨ oteborg, Sweden, e-mail: [email protected] 3 Department of Mathematics, Rutgers University, Piscataway NJ 07059, USA, e-mail: [email protected] AMS 2000 subject classifications: primary 60K35, 60J10; secondary 92D30. Keywords and phrases: contact process, correlation inequality. † Supported

16

Conjecture for the 1D contact process

17

Before we state our stronger version, we give some notation and terminology. A finite collection, say X1 , . . . , Xn , of 0 − 1 valued random variables is said to be positively associated if, for all functions f, g : {0, 1}n → R that are both increasing or both decreasing, E(f (X1 , . . . , Xn ) g(X1 , . . . , Xn )) ≥ E(f (X1 , . . . , Xn ))E(g(X1 , . . . , Xn )).

(2)

Equivalently, if f is increasing and g decreasing (or vice versa), E(f (X1 , . . . , Xn ) g(X1 , . . . , Xn )) ≤ E(f (X1 , . . . , Xn ))E(g(X1 , . . . , Xn )).

(3)

Further, a countable collection of random variables is said to be positively associated, if every finite subcollection is positively associated. We are now ready to state our main result, a stronger version of Konno’s conjecture. Theorem 2. Let ηt (x), x ∈ Z, t ≥ 0 be the 1D contact process with deterministic initial configuration, and bounded infection and recovery rates. For each t we have that, conditioned on the event that ηt (0) = 1, the collection of random variables {1 − ηt (x) : x < 0} ∪ {ηt (x) : x > 0} is positively associated. Remark. This theorem easily implies Conjecture 1: Start the contact process with all sites infected. Let t > 0 and let n, m be positive integers. Let A be the event {ηt (x) = 0, −n ≤ x ≤ −1}, and B the event {ηt (x) = 0, 1 ≤ x ≤ m}. The indicator function of A is an increasing function of the tuple (1 − ηt (x), −n ≤ x ≤ −1), and the indicator function of B is a decreasing function of the tuple (ηt (x), 1 ≤ x ≤ m). Hence, by Theorem 2, conditioned on the event {ηt (0) = 1}, the events A and B are negatively correlated. This holds for each t. The conjecture follows by letting t → ∞. 2. Slight extension of an earlier inequality In this section we present a slight extension of a result in [2]. In Section 3 we will prove (for certain graphs) a dual version of this extension. Let G be a finite, or countably infinite, mixed graph. The word ‘mixed’ means that we allow that some of the edges are oriented and others non-oriented. A nonoriented edge between vertices x and y is denoted by {x, y}, and an oriented edge from x to y by (x, y). Let V = V (G) and E = E(G) be the vertex and edge sets of G. Let p(e), e ∈ E, be values in [0, 1]. Consider the percolation model on G where each edge e, independently of the others, is open with probability pe and closed with probability 1 − pe . When we speak of a path in G, we assume that it respects the orientation of its edges. As usual, an open path is a path every edge of which is open. For S, T ⊂ V (G), the event that there is an open path from (some vertex in) S to (some vertex in) T will be denoted by {S → T }, the complement of this event by {S → T }, and the indicators of these events by IS→T and IS→T respectively. Theorem 3. Let S and T be disjoint subsets of V (G). Let, for each edge e, Xe and Ye be the indicators of the events {e belongs to an open path beginning in S} and {e belongs to an open path ending in T } respectively. Then, conditioned on the event {S → T }, the collection {Xe : e ∈ E(G)} ∪ {1 − Ye : e ∈ E(G)} is positively associated.

18

J. van den Berg, O. H¨ aggstr¨ om and J. Kahn

This is an oriented generalization of Theorem 1.5 of [2] and follows from a straightforward modification of the arguments in that paper. (See Section 3 of [2] for other generalizations). 3. A planar dual version of Theorem 3 When the graph G in Section 2 is embeddable in the plane, one can obtain a ‘planar dual version’ of Theorem 3. This (for the case of finite G, which for our purpose is sufficient) is Theorem 4 below. In Section 4 we will apply Theorem 4 to a special graph, which can be regarded as a discrete-time version of the ususal space–time diagram of the contact process. This application will yield Theorem 2. Theorem 4. Let G = (V, E) be a finite, planar, mixed graph. Let C be a subset of E which, when one disregards edge orientations, forms a face-bounding cycle in some planar embedding of G. Let u1 , . . . , uk , a1 , . . . , am , w1 , . . . , wl and b1 , . . . , bn denote (in some cyclic order) the vertices of C. Let U ⊂ {u1 , . . . , uk } and W ⊂ {w1 , . . . , wl }. Consider bond percolation on G with parameters pe , e ∈ E. Then, conditioned on the event {U → W }, the collection of random variables {IU →ai : 1 ≤ i ≤ m} ∪ {IU →bj : 1 ≤ j ≤ n} is positively associated. Proof. We assume, without loss of generality that C bounds the outer face in the given embedding of G, that the vertices of C are given above in clockwise order, and that k =  = 1. (To justify the last assertion, add vertices u, w outside C and undirected edges joining u to the ui ’s in U and w to the wi ’s in W , and let these new edges be open with probability 1.) So we may simply take U = {u} and W = {w}, and condition on {u → w}. If, for some x, y ∈ V (G), we have (x, y) ∈ E(G) but (y, x) ∈ E(G), we can just add (y, x) to E(G) and take p(y,x) = 0 without essentially changing anything. So (again, w.l.o.g.) we assume that (x, y) ∈ E(G) iff (y, x) ∈ E(G). Finally, if {x, y} is an undirected edge, which is open with probability p, we replace this edge by two directed edges which are independently open with probability p. It is well-known and easy to check that this does not change the distribution of the collection (Iu→v , v ∈ V (G)), and hence it does not affect the assertion of Theorem 4. Therefore, we assume w.l.o.g. that all edges of G are directed. Next, for convenience, we slightly vary the usual definition of the undirected graph, G, underlying G. The graph G has the same vertices as G. All edges of G are undirected, and {x, y} is an edge of G iff (x, y) (and hence, by one of the assumptions above, also (y, x)) is an edge of G. It is clear that the alternative form of Theorem 4 obtained by replacing G by G in the second line, is equivalent to the original form. From now on we will refer to that alternative form. The following conventions also turn out to be convenient: we consider a “drawing” (not, strictly speaking, embedding) of G which coincides with the given embedding of G, in the sense that (x, y), (y, x) ∈ E(G) are both drawn as orientations of the curve representing the corresponding edge in the embedding of G. These conventions will also apply to the dual-like graph H defined below. The dual e∗ of an edge e will always be oriented to cross e from left to right (as these sides are understood when one follows the direction of e).

Conjecture for the 1D contact process

19

For x, y ∈ V (C) we use [x, y] for the set of edges of G whose underlying edges (in G) belong to the path obtained by following C clockwise from x to y. We form a graph H, a variant of the planar dual of G, as follows. Start with vertices corresponding to the bounded faces of (our drawing of) G, joining them by dual edges as usual (oriented according to the preceding convention, and again taking (x, y)∗ and (y, x)∗ to be represented by the same curve). Then, for each e ∈ E(G) with underlying edge belonging to C, add a dual edge e∗ joining the vertex of H corresponding to the inner face containing e in its boundary to a new vertex se lying in the outer face. The se ’s are distinct except that s(x,y) = s(y,x) . (To avoid introducing unwanted crossings, take se to be drawn just outside e.) For x ∈ V (C) \ {u}, let Sx = {se : e ∈ [u, x]}, and Tx = {se : e ∈ [x, u]}, and set Sw = S, Tw = T . We couple percolation on H with that on G in the natural way, by declaring an edge e∗ of H to be open (closed) if the corresponding edge e of G is closed (open). Let V ∗ and E ∗ denote the vertex set and the edge set of H respectively. For connection events in H we use similar notation as for G, with the symbol ‘∗’ added to indicate that we consider the dual. For instance, if s and t are vertices of H, ∗

∗

s → t denotes the event that there is an open path in H from s to t, and s → t the complement of that event. We will apply Theorem 3 to the graph H. Observations ∗ (i) For each x ∈ V (C) \ {u} , u → x iff Sx →  Tx . (This is an analog of standard ∗

duality properties of planar percolation). In particular, u → w iff S → T . (ii) Let, for each edge e∗ of H, Xe∗ and Ye∗ be as in Theorem 3 (i.e. Xe∗ is the indicator of the event that e∗ belongs to an an open path in H beginning in S, and Ye∗ is the indicator of the event that e∗ belongs to an an open path in H ending in T ). By observation (i), for each i, 1 ≤ i ≤ m, Iu→ai is a decreasing function of the collection (Xe∗ : e∗ ∈ E ∗ ), and for each j, 1 ≤ j ≤ n, Iu→bj is a decreasing function of the collection (Ye∗ : e∗ ∈ E ∗ ). Now let f and g be increasing functions of the collection {Iu→ai : 1 ≤ i ≤ m} ∪ {Iu→bj : 1 ≤ j ≤ n}. By observation (ii), f and g are decreasing functions of the collection {Xe∗ : e∗ ∈ E ∗ } ∪ {1 − Ye∗ : e∗ ∈ E ∗ }. We get: ∗

E(f g | u → w)

= E(f g | S → T ) ∗

(4) ∗

≥ E(f | S → T ) E(g | S → T ) = E(f | u → w) E(g | u → w), where the equalities follow from observation (i), and the inequality follows from Theorem 3. This completes the proof of Theorem 4. 3.1. An alternative proof of Theorem 4 We think, and we believe this is also part of Mike’s philosophy, that a problem is best understood by approaching it in several ways. This subsection gives a sketch of a self-contained proof of Theorem 4. Instead of using explicit results from [2], it uses ideas similar to those which play a key role in some of the proofs in that paper.

J. van den Berg, O. H¨ aggstr¨ om and J. Kahn

20

Let A = {a1 , · · · am } and B = {b1 , · · · , bn }, with the a’s and b’s as in Theorem 4. Again we assume w.l.o.g. that, in the statement of Theorem 4, the circuit C bounds the outer face in the given embedding of G, that the vertices of C are given in clockwise order, and that U = {u}, W = {w}. In this proof we will not use the notion of G and that of a drawing of G. In the description below we always have in mind the embedding of G given in the statement of the theorem, with the abovementioned assumptions. Each path π from u to w partitions the set E(G) into three subsets: the set of edges of π itself, the edges in the part of G to the left of π (when we follow π in the direction of w), and the edges in the part of G to the right of π. We denote these sets by E(π), EL (π) and ER (π) respectively. For each configuration ω ∈ {0, 1}E(G) which has an open path from u to w, we will consider the left-most self-avoiding path from u to w (this is similar to the wellknown notion of lowest crossing in, e.g., bond percolation on a box in the square lattice). Analogously we will consider the right-most self-avoiding open path. For brevity we will drop the word ‘self-avoiding’. Let P denote the measure on {0, 1}E(G) corresponding to the given bond percolation model; that is, the product measure with parameters pe , e ∈ E(G). Let π be a path from u to w. Observation. Conditioned on the event that π is the leftmost open path from u to w, each edge e in ER (π) is, independently of the others, open with probability pe . An analogous observation holds when we replace ‘leftmost’ by ‘rightmost’ and ER (π) by EL (π). It is easy to check that similar properties hold for the distribution µ obtained from P by conditioning on having an open path from u to w: µ(ωe = ·, e ∈ ER (π) | π is the leftmost open path from u to w) is the product distribution on {0, 1}ER (π) with parameters pe , e ∈ ER (π), and similarly if we replace leftmost by rightmost, and ER by EL . Let Γ be the set of all configurations ω that have an open path from u to w. We will construct a Markov chain ωn , n = 0, 1, · · · , with state space Γ and stationary distribution µ. To do this, we first introduce auxiliary 0−1 valued random variables ln (e), rn (e), e ∈ E(G), n = 0, 1, · · · . These random variables are independent and, for each e and n, P (ln (e) = 1) = P (rn (e) = 1) = pe . As initial state of the Markov chain we take ω0 = α, for some α ∈ Γ; the precise choice does not matter. The transition from time n to time n + 1 of this Markov chain consists of two substeps, (i) and (ii) below: Substep (i): Denote by πn the leftmost open path from u to w in the configuration ωn . Using the rn variables introduced above, we update all edges in ER (πn ). This gives a new configuration, which we denote by ωn . More precisely, we define ωn (e) = rn (e), if e ∈ ER (πn ), and ωn (e) otherwise.

Conjecture for the 1D contact process

21

Substep (ii): To ωn we apply, informally speaking, the same action as in substep (i), but now with ‘left’ and ‘right’ exchanged. The resulting configuration is ωn+1 . More precisely, with πn denoting the rightmost open path from u to w in ωn , we define ωn+1 (e) = ln (e), if e ∈ EL (πn ), and ωn (e) otherwise. Let µn denote the distribution of ωn . From the above construction it is clear that ωn , n = 0, 1, · · · is indeed a Markov chain with state space Γ. Moreover, it is clear from the construction and the above-mentioned Observation that µ is invariant under the above dynamics. (In fact, it is invariant under substep (i) as well as under substep (ii)). It is also easy to see that this Markov chain is aperiodic and irreducible. Hence, µn converges to µ. Let, for each vertex x, ηn (x) be the indicator of the event that the configuration ωn has an open path from u to x. By the above arguments, it is sufficient to show that, for each n, the collection of random variables {ηn (x), x ∈ A} ∪ {1 − ηn (y), y ∈ B} is positively associated. This, in turn, follows from the following Claim and the well-known Harris-FKG theorem that independent random variables are positively associated: Claim. Fix the initial configuration α of the Markov chain. For each n and for each x ∈ A, ηn (x) is then a function of the variables lk (e) and rk (e), e ∈ E(G), 0 ≤ k ≤ n − 1. Moreover, it is increasing in the l variables and decreasing in the r variables. An analogous statement, but with l and r interchanged, holds for x ∈ B. We give a brief sketch of the proof of this claim: Consider the following partial order, called ‘more leftish than’, on Γ. First let, for ω ∈ Γ, πL (ω) and πR (ω) denote the leftmost and the rightmost open paths from u to w respectively. If ω, ω ˆ ∈ Γ, we say that ω ˆ is more leftish than ω iff each of the following ((a) and (b)) holds: (a) The leftmost and the rightmost open path of ω ˆ are located to the left of the corresponding paths of ω. More precisely, E(πL (ˆ ω )) ⊂ EL (πL (ω)) ∪ E(πL (ω)), and E(πR (ω)) ⊂ ER (πR (ˆ ω )) ∪ E(πR (ˆ ω )). (b) ω ˆ (e) ≥ ω(e), e ∈ EL (πL (ˆ ω )), and ω ˆ (e) ≤ ω(e), e ∈ ER (πR (ω)). With fixed l’s and r’s this order is preserved under substep (i) as well as under substep (ii). Moreover, it is easy to check that if we apply substep (i) or substep (ii) to some configuration ω, and increase some of the l variables or decrease some of the r variables involved in that substep, the configuration resulting from that substep will become more leftish. These arguments, together with the fact that if ω ˆ is more leftish than ω, then I{u→x in

ω ˆ}

≥ I{u→x in

ω} ,

x ∈ A,

J. van den Berg, O. H¨ aggstr¨ om and J. Kahn

22

and I{u→x in

ω ˆ}

≤ I{u→x in

ω} ,

x ∈ B,

imply the above Claim. 4. Proof of Theorem 2 Consider the contact process in the statement of Theorem 2. A useful and wellknown way to describe this process is by a so-called space-time diagram, or graphical representation (see e.g. figure 1 in part I of [7]): We represent each site x as the point (x, 0) in the plane and we assign to it a vertical line (time axis) lx = {(x, t) : t > 0}. On lx we consider three independent Poisson point processes: one with density δx , corresponding to recovery attempts; one with density λx−1,x corresponding to attempts to infect site x−1; and one with density λx+1,x corresponding to attempts to infect site x + 1. At each point in the first point process we draw a symbol ∗ on lx ; from each point in the second point process we draw a horizontal arrow to lx−1 ; similarly, from each point in the third point process we draw a horizontal arrow to lx+1 . By an allowable path we mean a continuous trajectory along the lx ’s and the arrows specified above, which satisfies the following conditions: along the lx ’s it goes only upward, and is not allowed to cross a ∗; when it goes along an arrow it must respect the direction of that arrow. A site y is infected at time t iff for some site x that is infected at time 0, there is an allowable path from (x, 0) to (y, t) in the space-time diagram; that is, ηt (y) =I{∃x s.t.

x

is infected at time

0

and

there is an allowable path from

(x,0)

to

(y,t)}

.

We will apply Theorem 4 to a discrete-time approximation of this process. Similar discretization arguments for contact processes (and many other interacting particle systems) are quite common (see e.g. pages 11 and 65 of [7]). Let N be a positive integer. Consider bond percolation on the following graph, G. The vertex set of G is {(x, k/N ) : x ∈ Z, k = 0, 1, · · · }. Each vertex (x, k/N ) is the starting point of three oriented edges: one to (x + 1, k/N ), one to (x − 1, k/N ), and one to (x, (k + 1)/N ). We take these edges to be open with probabilities λ(x + 1, x)/N , λ(x − 1, x)/N and 1 − δx /N respectively. Let U = {(u, 0) : u is infected at time 0}. Now fix a t > 0 and a positive integer n. Let tˆ be the smallest multiple of 1/N that is larger than or equal to t. Let Gn,N be the (finite) subgraph induced by Vn,N = {(x, j/N ) : |x| ≤ n, j ≤ N tˆ}. (n,N ) Let, for each integer x, ηt (x) denote the indicator of the event {U → (x, tˆ) in Gn,N }. (It would be more correct here to write U ∩ ([−n, n] × 0) in place of U .) Fix a positive integer m. It is quite standard that the joint distribution of the (n,N ) random variables ηt (x), −m ≤ x ≤ m, converges to that of ηt (x), −m ≤ x ≤ m, when we let N → ∞ and then n → ∞. Moreover, to each graph Gn,N we can apply Theorem 4, which tells us that, conditioned on the event {U → (0, tˆ) in Gn,N }, the collection of random variables (n,N )

{1 − ηt

(n,N )

(x) : −m ≤ x ≤ −1} ∪ {ηt

(x) : 1 ≤ x ≤ m}

is positively associated. This, combined with the above-mentioned limit considerations, gives us that, conditioned on the event that ηt (0) = 1, the collection {1 − ηt (x) : −m ≤ x ≤ −1} ∪ {ηt (x) : 1 ≤ x ≤ m} is positively associated. Since this holds for all m, Theorem 2 follows.

Conjecture for the 1D contact process

23

Acknowledgment We thank N. Konno for bringing his conjecture to our attention. References [1] Belitsky, V., Ferrari, P. A., Konno, N., and Liggett, T. M. (1997). A strong correlation inequality for contact processes and oriented percolation. Stochastic Process. Appl. 67, 2, 213–225. MR1449832 ¨ggstro ¨ m, O., Kahn, J. (2005). Some conditional cor[2] van den Berg, J., Ha relation inequalities for percolation and related processes, to appear in Random Structures and Algorithms. [3] Diestel, R. (2000). Graph Theory. Graduate Texts in Mathematics, Vol. 173. Springer-Verlag, New York. MR1743598 [4] Konno, N. (1994). Phase Transitions of Interacting Particle Systems, World Scientific, Singapore. [5] Konno, N. (1997). Lecture Notes on Interacting Particle Systems, Rokko Lectures in Mathematics 3, Kobe University. [6] Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 276. Springer-Verlag, New York. MR776231 [7] Liggett, T. M. (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 324. Springer-Verlag, Berlin. MR1717346

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 24–36 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000040

Uniqueness and multiplicity of infinite clusters Geoffrey Grimmett1 University of Cambridge Abstract: The Burton–Keane theorem for the almost-sure uniqueness of infinite clusters is a landmark of stochastic geometry. Let µ be a translationinvariant probability measure with the finite-energy property on the edge-set of a d-dimensional lattice. The theorem states that the number I of infinite components satisfies µ(I ∈ {0, 1}) = 1. The proof is an elegant and minimalist combination of zero–one arguments in the presence of amenability. The method may be extended (not without difficulty) to other problems including rigidity and entanglement percolation, as well as to the Gibbs theory of random-cluster measures, and to the central limit theorem for random walks in random reflecting labyrinths. It is a key assumption on the underlying graph that the boundary/volume ratio tends to zero for large boxes, and the picture for non-amenable graphs is quite different.

1. Introduction The Burton–Keane proof of the uniqueness of infinite clusters is a landmark in percolation theory and stochastic geometry. The general issue is as follows. Let ω be a random subset of Zd with law µ, and let I = I(ω) be the number of unbounded components of ω. Under what reasonable conditions on µ is it the case that: either µ(I = 0) = 1, or µ(I = 1) = 1? This question arose first in percolation theory with µ = µp , where µp denotes product measure (on either the vertex-set or the edge-set of Zd ) with density p. It was proved in [2] that µp (I = 1) = 1 for any value of p for which µp (I ≥ 1) > 0, and this proof was simplified in [9]. Each of these two proofs utilized a combination of geometrical arguments together with a large-deviation estimate. The true structure of the problem emerged only in the paper of Robert Burton and Michael Keane [5]. Their method is elegant and beautiful, and rests on the assumptions that the underlying measure µ is translation-invariant with a certain ‘finite-energy property’, and that the underlying graph is amenable (in that the boundary/volume ratio tends to zero in the limit for large boxes). The Burton– Keane method is canonical of its type, and is the first port of call in any situation where such a uniqueness result is needed. It has found applications in several areas beyond connectivity percolation, and the purpose of this paper is to summarize the method, and to indicate some connections to other problems in the theory of disordered media. Michael Keane’s contributions to the issue of uniqueness are not confined to [5]. The results of that paper are extended in [10] to long-range models (see also [33]), and to models on half-spaces. In a further paper, [6], he explored the geometrical 1 Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom, e-mail: [email protected] AMS 2000 subject classifications: 60K35, 60D05, 82B20, 82B43. Keywords and phrases: percolation, stochastic geometry, rigidity, entanglement, randomcluster model, random labyrinth.

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Uniqueness and multiplicity of infinite clusters

25

properties of infinite clusters in two dimensions, and in [11] the existence of circuits. He wrote in the earlier paper [21] of uniqueness in long-range percolation. The Burton–Keane approach to uniqueness is sketched in Section 2 in the context of percolation. Its applications to rigidity percolation and to entanglement percolation are summarized in Sections 3 and 4. An application to the random-cluster model is described in Section 5, and another to random walks in random reflecting labyrinths in Section 6. The reader is reminded in Section 7 that infinite clusters may be far from unique when the underlying graph is non-amenable. We shall make periodic references to lattices, but no formal definition is given here. 2. Uniqueness of infinite percolation clusters The Burton–Keane argument is easiest described in the context of percolation, and we begin therefore with a description of the bond percolation model. Let G = (V, E) be a countably infinite connected graph with finite vertex-degrees. The configuration space of the model is the set Ω = {0, 1}E of all 0/1-vectors ω = (ω(e) : e ∈ E). An edge e is called open (respectively, closed ) in the configuration ω if ω(e) = 1 (respectively, ω(e) = 0). The product space Ω is endowed with the σ-field F generated by the finite-dimensional cylinder sets. For p ∈ [0, 1], we write µp for product measure with density p on (Ω, F). The percolation model is central to the study of disordered geometrical systems, and a reasonably full account may be found in [16]. Let ω ∈ Ω, write η(ω) = {e ∈ E : ω(e) = 1} for the set of open edges of ω, and consider the open subgraph Gω = (V, η(ω)) of G. For x, y ∈ V , we write x ↔ y if x and y lie in the same component of Gω . We write x ↔ ∞ if the component of Gω containing x is infinite, and we let θx (p) = µp (x ↔ ∞). The number of infinite components of Gω is denoted by I = I(ω). It is standard that, for any given p ∈ [0, 1], for all x, y ∈ V, and that

θx (p) = 0 if and only if θy (p) = 0,


= 0 if p < pc (G), θx (p) > 0 if p > pc (G),

(1)

(2)

where the critical probability pc (G) is given by pc (G) = sup{p : µp (I = 0) = 1}.

(3)

The event {I ≥ 1} is independent of the states of any finite collection of edges. Since the underlying measure is product measure, it follows by the Kolmogorov zero–one law that µp (I ≥ 1) ∈ {0, 1}, and hence
= 0 if p < pc (G), µp (I ≥ 1) = 1 if p > pc (G). It is a famous open problem to determine for which graphs it is the case that µpc (I ≥ 1) = 0, see Chapters 8–10 of [16]. We concentrate here on the case when G is the d-dimensional hypercubic lattice. Let Z = {. . . , −1, 0, −1, . . . } be the integers, and Zd the set of all d-vectors

G. Grimmett

26

x = (x1 , x2 , . . . , xd ) of integers. We turn Zd into a graph by placing an edge between any two vertices x, y with |x − y| = 1, where |z| =

d 

|zi |,

z ∈ Zd .

i=1

We write E for the set of such edges, and Ld = (Zd , E) for the ensuing graph. Henceforth, we let d ≥ 2 and we consider bond percolation on the graph Ld . Similar results are valid for any lattice in two or more dimensions, and for site percolation. d d A box Λ is a subset of Z of the form i=1 [xi , yi ] for some x, y ∈ Zd . The boundary ∂S of the set S of vertices is the set of all vertices in S which are incident to some vertex not in S. A great deal of progress was made on percolation during the 1980s. Considerable effort was spent on understanding the subcritical phase (when p < pc ) and the supercritical phase (when p > pc ). It was a key discovery that, for any p with µp (I ≥ 1) = 1, we have that µp (I = 1) = 1; that is, the infinite cluster is (almost surely) unique whenever it exists. Theorem 1 ([2]). For any p ∈ [0, 1], either µp (I = 0) = 1 or µp (I = 1) = 1. This was first proved in [2], and with an improved proof in [9]. The definitive proof is that of Burton and Keane, [5], and we sketch this later in this section. Examination of the proof reveals that it relies on two properties of the product measure µp , namely translation-invariance and finite-energy. The first of these is standard, the second is as follows. A probability measure µ on (Ω, F) is said to have the finite-energy property if, for all e ∈ E, 0 < µ(e is open | Te ) < 1

µ-almost-surely,

where Te denotes the σ-field generated by the states of edges other than e. The following generalization of Theorem 1 may be found in [5]. Theorem 2 ([5]). Let µ be a translation-invariant probability measure on (Ω, F) with the finite-energy property. Then µ(I ∈ {0, 1}) = 1. If, in addition, µ is ergodic, then I is µ-almost-surely constant, and hence: either µ(I = 0) = 1 or µ(I = 1) = 1. A minor complication arises for translation-invariant non-ergodic measures, and this is clarified in [6] and [12], page 42. Proof of Theorem 1. The claim is trivial if p = 0, 1, and we assume henceforth that 0 < p < 1. There are three steps. Since I is a translation-invariant function and µp is ergodic, I is µp -almost-surely constant. That is, there exists ip ∈ {1, 2, . . . }∪{∞} such that µp (I = ip ) = 1. (4) Secondly, let us assume that 2 ≤ ip < ∞. There exists a box Λ such that µp (Λ intersects ip infinite clusters) > 0. By replacing the state of every edge in Λ by 1, we deduce by finite-energy that µp (I = 1) > 0, in contradiction of (4). Therefore, ip ∈ {0, 1, ∞}. In the third step we prove that ip = ∞. Suppose on the contrary that ip = ∞. We will derive a contradiction by a geometrical argument. A vertex x is called a trifurcation if:

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(i) x lies in an infinite open cluster, (ii) there exist exactly three open edges incident to x, and (iii) the deletion of x and its three incident open edges splits this infinite cluster into exactly three disjoint infinite clusters and no finite clusters. We write Tx for the event that x is a trifurcation. By translation-invariance, the probability of Tx does not depend on the choice of x, and thus we set τ = µp (Tx ). Since ip = ∞ by assumption, there exists a box Λ such that µp (Λ intersects three or more infinite clusters) > 0. On this event, we may alter the configuration inside Λ in order to obtain the event T0 . We deduce by the finite-energy property of µp that τ > 0. The mean number of trifurcations inside Λ is τ |Λ|. This implies a contradiction, as indicated by the following rough argument. Select a trifurcation (t1 , say) of Λ, and choose some vertex y1 (∈ ∂Λ) which satisfies t1 ↔ y1 in Λ. We now select a new trifurcation t2 ∈ Λ. By the definition of the term ‘trifurcation’, there exists y2 ∈ ∂Λ such that y1 = y2 and t2 ↔ y2 in Λ. We continue similarly, at each stage picking a new trifurcation tk ∈ Λ and a new vertex yk ∈ ∂Λ. If there exist N trifurcations in Λ, then we obtain N distinct vertices yk lying in ∂Λ. Therefore |∂Λ| ≥ N . We take expectations to find that |∂Λ| ≥ τ |Λ|, which is impossible with τ > 0 for large Λ. We deduce by this contradiction that ip = ∞. The necessary rigour may be found in [5, 16]. 3. Rigidity percolation Theorems 1 and 2 assert the almost-sure uniqueness of the infinite connected component. In certain other physical situations, one is interested in topological properties of subgraphs of Ld other than connectivity, of which two such properties are ‘rigidity’ and ‘entanglement’. The first of these properties may be formulated as follows. Let G = (V, E) be a finite graph and let d ≥ 2. An embedding of G into Rd is an injection f : V → Rd . A framework (G, f ) is a graph G together with an embedding f . A motion of a framework (G, f ) is a differentiable family f = (ft : 0 ≤ t ≤ 1) of embeddings of G, containing f , which preserves all edge-lengths. That is to say, we require that f = fT for some T , and that ft (u) − ft (v) = f0 (u) − f0 (v)

(5)

for all edges u, v ∈ E, where · is the Euclidean norm on Rd . We call the motion f rigid if (5) holds for all pairs u, v ∈ V rather than adjacent pairs only. A framework is called rigid if all its motions are rigid motions. The above definition depends on the value of d and on the initial embedding f . For given d, the property of rigidity is ‘generic’ with respect to f , in the sense that there exists a natural measure π (generated from Lebesgue measure) on the set of embeddings of G such that: either (G, f ) is rigid for π-almost-every embedding f , or (G, f ) is not rigid for π-almost-every embedding. We call G rigid if the former holds. Further details concerning this definition may be found in [13, 14, 27]. Note that rigid graphs are necessarily connected, but that there exist connected graphs which are not rigid.

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We turn now to the rigidity of infinite graphs. Let G be a countably infinite graph with finite vertex-degrees. The graph G is called rigid if every finite subgraph of G is contained in some finite rigid subgraph of G. Next we introduce probability. Let L be a lattice in d dimensions, and consider bond percolation on L having density p. The case L = Ld is of no interest in the context of rigidity, since the lattice Ld is not itself rigid. Let R be the event that the origin belongs to some infinite rigid subgraph of L all of whose edges are open. The rigidity probability is defined by θrig (p) = µp (R). Since R is an increasing event, θrig is a non-decreasing function, whence
= 0 if p < prig c (L), θrig (p) rig > 0 if p > pc (L), where the rigidity critical probability prig c (L) is given by rig prig c (L) = sup{p : θ (p) = 0}.

The study of the rigidity of percolation clusters was initiated by Jacobs and Thorpe, see [30, 31]. Since rigid graphs are connected, we have that θrig (p) ≤ θ(p), implying that prig c (L) ≥ pc (L). The following is basic. Theorem 3 ([16, 27]). Let L be a d-dimensional lattice, where d ≥ 2. (i) We have that pc (L) < prig c (L). rig (ii) pc (L) < 1 if and only if L is rigid. How many (maximal) infinite rigid components may exist in a lattice L? Let J be the number of such components. By the Kolmogorov zero–one law, for any given value of p, J is µp -almost-surely constant. It may be conjectured that µp (J = 1) = 1 whenever µp (J ≥ 1) > 0. The mathematical study of rigidity percolation was initiated by Holroyd in [27], where it was shown amongst other things that, for the triangular lattice T in two dimensions, µp (J = 1) = 1 for almost every p ∈ (prig c (T), 1]. The proof was a highly non-trivial development of the Burton–Keane method. The main extra difficulty lies in the non-local nature of the property of rigidity. See also [29]. Considerably more general results have been obtained since by H¨ aggstr¨ om. Holroyd’s result for almost every p was extended in [23] to for every p, by using the two-dimensional uniqueness arguments of Keane and co-authors to be found in [11]. More recently, H¨ aggstr¨ om has found an adaptation of the Burton–Keane argument which (almost) settles the problem for general rigid lattices in d ≥ 2 dimensions. Theorem 4 ([25]). Let d ≥ 2 and let L be a rigid d-dimensional lattice. We have that µp (J = 1) = 1 whenever p > prig c (L). There remains the lacuna of deciding what happens when p = prig c , that is, of rig (J = 1) = 1. (J = 0) = 1 or that µ proving either that µprig pc c 4. Entanglement in percolation In addition to connectivity and rigidity, there is the notion of ‘entanglement’. The simplest example of a graph which is entangled but not connected comprises two

Uniqueness and multiplicity of infinite clusters

29

disjoint circuits which cannot be separated without one of them being broken. Such entanglement is intrinsically a three-dimensional affair, and therefore we restrict ourselves here to subgraphs of L3 = (Z3 , E) viewed as graphs embedded in a natural way in R3 . We begin with some terminology. For E ⊆ E, we denote by [E] the union of all unit line-segments of R3 corresponding to edges in E. The term ‘sphere’ is used to mean a subset of R3 which is homeomorphic to the 2-sphere {x ∈ R3 : x = 1}. The complement of any sphere S has two connected components; we refer to the bounded component as the inside of S, written ins(S), and to the unbounded component as the outside of S, written out(S). There is a natural definition of the term ‘entanglement’ when applied to finite sets of edges of the lattice L3 , namely the following. We call the finite edge-set E entangled if, for any sphere S not intersecting [E], either [E] ⊆ ins(S) or [E] ⊆ out(S). Thus entanglement is a property of edge-sets rather than of graphs. However, with any edge-set E we may associate the graph GE having edge-set E together with all incident vertices. Graphs GE arising in this way have no isolated vertices. We call GE entangled if E is entangled, and we note that GE is entangled whenever it is connected. There are several possible ways of extending the notion of entanglement to infinite subgraphs of L3 , and these ways are not equivalent. For the sake of being definite, we adopt here a definition similar to that used for rigidity. Let E be an infinite subset of E. We call E entangled if, for any finite subset F (⊆ E), there exists a finite entangled subset F
of E such that F ⊆ F
. We call the infinite graph GE , defined as above, entangled if E is entangled, and we note that GE is entangled whenever it is connected. A further discussion of the notion of entanglement may be found in [20]. Turning to percolation, we declare each edge of L3 to be open with probability p. We say that the origin 0 lies in an infinite open entanglement if there exists an infinite entangled set E of open edges at least one of which has 0 as an endvertex. We concentrate on the event N = {0 lies in an infinite open entanglement}, and the entanglement probability θent (p) = µp (N ). Since N is an increasing event, θent is a non-decreasing function, whence
= 0 if p < pent c , θent (p) > 0 if p > pent c , where the entanglement critical probability pent is given by c = sup{p : θent (p) = 0}. pent c

(6)

Since every connected graph is entangled, it is immediate that θ(p) ≤ θent (p), ≤ pc . whence 0 ≤ pent c Theorem 5 ([1, 28]). The following strict inequalities are valid: 0 < pent < pc . c

(7)

G. Grimmett

30

Entanglements in percolation appear to have been studied first in [32], where it was proposed that pc − pent

1.8 × 10−7 , c implying the strict inequality pent < pc . It is a curious fact that we have no rigorous c insight into the numerical value of pent c . For example, we are unable to decide on the basis of mathematics whether pent is numerically close to either 0 or pc . c ent Suppose that p is such that θ (p) > 0. By the zero–one law, the number K of (maximal) infinite entangled open edge-sets satisfies µp (K ≥ 1) = 1. The almostsure uniqueness of the infinite entanglement has been explored in [20, 29], and the situation is similar to that for rigidity percolation. H¨ aggstr¨ om’s proof of the following theorem uses a non-trivial application of the Burton–Keane method. Theorem 6 ([24]). We have that µp (K = 1) = 1 whenever p > pent c . As is the case with rigidity, there remains the open problem of proving either that µpent (K = 0) = 1 or that µpent (K = 1) = 1. c c There are several other open problems concerning entangled graphs, of which we mention a combinatorial question. Let n ≥ 1, and let En be the set of all subsets E of E with cardinality n such that: (i) some member of e is incident to the origin, and (ii) E is entangled. Since every connected graph is entangled, En is at least as large as the family of all connected sets of n edges touching the origin. Therefore, |En | grows at least exponentially in n. It may be conjectured that there exists κ such that |En | ≤ eκn

for all n ≥ 1.

The best inequality known currently is of the form |En | ≤ exp{κn log n}. See [20]. 5. The random-cluster model The random-cluster model on a finite graph G = (V, E) is a certain parametric family of probability measures φp,q indexed by two parameters p ∈ [0, 1] and q ∈ (0, ∞). When q = 1, the measure is product measure with density p; when q = 2, 3, . . . , the corresponding random-cluster measures correspond to the Ising and qstate Potts models on G. The random-cluster model provides a geometrical setting for the correlation functions of the ferromagnetic Ising and Potts models, and it has proved extremely useful in studying these models. Recent accounts of the theory, and of its impact on Ising/Potts models, may be found in [18, 19]. The configuration space is the set Ω = {0, 1}E of 0/1-vectors indexed by the edge-set E. The probability measure φp,q on Ω is given by   1  ω(e) 1−ω(e) p (1 − p) q k(ω) , ω ∈ Ω, (8) φp,q (ω) = Z e∈E

where k(ω) is the number of connected components (or ‘open clusters’) of the graph Gω = (V, η(ω)). When G is finite, every φp,q -probability is a smooth function of the parameters p and q. The situation is more interesting when G is infinite, since infinite graphs may display phase transitions. For simplicity, we restrict the present discussion to

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31

the graph Ld = (Zd , E) where d ≥ 2. We introduce next the concept of boundary conditions. Let Λ be a finite box, and write EΛ for the set of edges joining pairs of members of Λ. We write TΛ for the σ-field generated by the states of edges in E \ EΛ . For ξ ∈ Ω, we write ΩξΛ for the (finite) subset of Ω containing all configurations ω satisfying ω(e) = ξ(e) for e ∈ Ed \ EΛ ; these are the configurations which ‘agree with ξ off Λ’. Let ξ ∈ Ω, and write φξΛ,p,q for the random-cluster measure on the finite graph Λ ‘with boundary condition ξ’. That is to say, φΛ,p,q is given as in (8) subject to ω ∈ ΩξΛ , and with k(ω) replaced by the number of open clusters of Ld that intersect Λ. A probability measure φ on (Ω, F) is called a random-cluster measure with parameters p and q if for all A ∈ F and all finite boxes Λ, φ(A | TΛ )(ξ) = φξΛ,p,q (A)

for φ-a.e. ξ.

The set of such measures is denoted Rp,q . The reader is referred to [18, 19] for accounts of the existence and properties of random-cluster measures. One may construct infinite-volume measures by taking weak limits. A probability measure φ on (Ω, F) is called a limit random-cluster measure with parameters p and q if there exist ξ ∈ Ω and a sequence Λ = (Λn : n ≥ 1) of boxes satisfying Λn → Zd as n → ∞ such that φξΛn ,p,q ⇒ φ

as n → ∞.

The two ‘extremal’ boundary conditions are the configurations ‘all 0’ and ‘all 1’, denoted by 0 and 1, respectively. It is a standard application of positive association that the weak limits φbp,q = lim φbΛ,p,q Λ↑Zd

exist for b = 0, 1 and q ≥ 1. It is an important fact that these limits belong to Rp,q . Theorem 7 ([15]). Let p ∈ [0, 1] and q ∈ [1, ∞). The limit random-cluster measures φbp,q , b = 0, 1, belong to Rp,q . The proof hinges on the following fact. Let φ be a limit random-cluster measure with parameters p, q such that the number I of infinite open clusters satisfies φ(I ∈ {0, 1}) = 1.

(9)

It may then be deduced that φ ∈ Rp,q . The uniqueness theorem, Theorem 2, is used to establish (9) for the measures φ = φbp,q , b = 0, 1. Let q ≥ 1. The random-cluster model has a phase transition defined as follows. For b = 0, 1, let θb (p, q) = φbp,q (0 ↔ ∞), and define the critical points pbc (q) = sup{p : θb (p, q) = 0}. It is standard that φ0p,q = φ1p,q for almost every p. It follows that p0c (q) = p1c (q), and we write pc (q) for the common critical value. It is known that φ0p,q = φ1p,q when p < pc (q), and it is an important open problem to prove that φ0p,q = φ1p,q

if p > pc (q).

See [18, 19] for further discussion of the uniqueness of random-cluster measures.

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6. Random walks in random labyrinths Suppose that a ball is propelled through a random environment of obstacles, off which it rebounds with perfect reflection. We ask for information about the trajectory of the ball. This classical problem is often termed the ‘Lorentz problem’, and it has received much attention in both the mathematics and physics literature. If the obstacles are distributed at random in Rd then, conditional on their placements, the motion of the ball is deterministic. It is a significant problem of probability theory to develop a rigorous analysis of such a situation. Two natural questions spring to mind. (i) Non-localization. What is the probability that the trajectory of the ball is unbounded? (ii) Diffusivity. Suppose the trajectory is unbounded with a strictly positive probability. Conditional on this event, is there a central limit theorem for the ball’s position after a large time t. These questions seem to be difficult, especially when the obstacles are distributed aperiodically. The problem is much easier when the environment of obstacles is ‘lubricated’ by a positive density of points at which the ball behaves as a random walk. We consider a lattice model of the following type. The obstacles are distributed around the vertex-set of the d-dimensional hypercubic lattice Ld , and they are designed in such a way that the ball traverses the edges of the lattice. Some of the associated mathematics has been surveyed in [4, 17], to which the reader is referred for an account of the literature. The main result of [4] is that, subject to certain conditions on the density of obstacles, the ball’s trajectory satisfies a functional central limit theorem. The Burton–Keane method plays a crucial role in the proof. We make this more concrete as follows. Our model involves a random environment of reflecting bodies distributed around the vertices of Ld . Each vertex is designated either a ‘reflector’ (of a randomly chosen type) or a ‘random walk point’. The interpretation of the term ‘random walk point’ is as follows: when the ball hits such a point, it departs in a direction chosen randomly from the 2d available directions, this direction being chosen independently of everything else. The defining properties of a reflector ρ are that: (i) to each incoming direction u there is assigned a unique outgoing direction ρ(u), and (ii) the ball will retrace its path if its direction is reversed. Let I = {e1 , e2 , . . . , ed } be the set of positive unit vectors of Zd , and let I ± = {αej : α = ±, 1 ≤ j ≤ d}. A reflector is defined to be a map ρ : I ± → I ± with the property that ρ(−ρ(u)) = −u for all u ∈ I ± . We write R for the set of all reflectors. One particular reflector is special, namely the identity map satisfying ρ(u) = u for all u ∈ I ± ; we call this the crossing, and we denote it by +. Crossings do not deflect the ball. The following random environment will be termed a random labyrinth. Let prw and p+ be non-negative reals such that prw + p+ ≤ 1, and let π be a probability mass function on  the set R \ {+} of ‘non-trivial’ reflectors (that is, π(ρ) ≥ 0 for d ρ ∈ R\{+} and ρ∈R\{+} π(ρ) = 1). Let Z = (Zx : x ∈ Z ) be a family of

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independent random variables, taking values in R ∪ {∅}, with probabilities   if β = ∅, prw P(Zx = β) = p+ if β = +,   (1 − prw − p+ )π(ρ) if β = ρ ∈ R\{+}.

A vertex x is called a crossing if Zx = +, and a random walk (rw) point if Zx = ∅. We now study admissible paths in the labyrinth Z. Consider a path in Ld which visits (in order) the vertices x0 , x1 , . . . , xn ; we allow the path to revisit a given vertex more than once, and to traverse a given edge more than once. This path is called admissible if it conforms to the reflectors that it meets, which is to say that xj+1 − xj = Zxj (xj − xj−1 )

for all j such that Zxj = ∅.

Remarkably little is known about random labyrinths when prw = 0. One notorious open problem concerns the existence (or not) of infinite admissible paths in L2 when prw = 0. The problem is substantially easier when prw > 0, and we assume this henceforth. We explain next how the labyrinth Z generates a random walk therein. Let x be a rw point. A walker starts at x, and flips a fair 2d-sided coin in order to determine the direction of its first step. Henceforth, it is required to traverse admissible paths only, and it flips the coin to determine its exit direction from any rw point encountered. We write PxZ for the law of the random walk in the labyrinth Z, starting from a rw point x. There is a natural equivalence relation on the set R of rw points of Zd , namely x ↔ y if there exists an admissible path with endpoints x and y. Let Cx be the equivalence class containing the rw point x. We may follow the progress of a random walk starting at x by writing down (in order) the rw points which it visits, say X0 (= x), X1 , X2 , . . .. Given the labyrinth Z, the sequence X = (Xn : n ≥ 0) is an irreducible Markov chain on the countable state space Cx . Furthermore, this chain is reversible with respect to the measure µ given by µ(y) = 1 for y ∈ Cx . We say that x is Z-localized if |Cx | < ∞, and Z-non-localized otherwise. We call Z localized if all rw points are Z-localized, and we call Z non-localized otherwise. By a zero–one law, we have that P(Z is localized) equals either 0 or 1. Suppose that the origin 0 is a rw point. As before, we consider the sequence X0 (= 0), X1 , X2 , . . . of rw points visited in sequence by a random walk in Z beginning at the origin 0. For  > 0, we let X  (t) = X−2 t

for t ≥ 0,

and we are interested in the behaviour of the process X  (·) in the limit as  ↓ 0. We study X  under the probability measure P0 , defined as the measure P conditional on the event {0 is a rw point, and |C0 | = ∞}. for the critical probability of site percolation on Zd . We write psite c Theorem 8 ([4]). Let d ≥ 2 and prw > 0. There exists a strictly positive constant A = A(prw , d) such that the following holds whenever either 1 − prw − p+ < A or prw > psite c : (i) P(0 is a rw point, and |C0 | = ∞) > 0, and √ (ii) as  ↓ 0, the re-scaled process X  (·) converges P0 -dp to δW , where W is a standard Brownian motion in Rd and δ is a strictly positive constant.

G. Grimmett

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With E denoting expectation, the convergence ‘P0 -dp’ is to be interpreted as P0Z (f (X  )) → E(f (W ))

in P0 -probability,

for all bounded continuous functions f on the Skorohod path-space D([0, ∞), Rd ). The proof of Theorem 8 utilizes the Kipnis–Varadhan central limit theorem, [34], together with its application to percolation, see [7, 8]. A key step in the proof is to show that, under the conditions of the theorem, there exists a unique infinite equivalence class, and this is where the Burton–Keane method is key. 7. Non-uniqueness for non-amenable graphs Let G = (V, E) be an infinite, connected graph with finite vertex-degrees. We call G amenable if its ‘isoperimetric constant’   |∂W | χ(G) = inf : W ⊆ V, 0 < |W | < ∞ (10) |W | satisfies χ(G) = 0, where the infimum in (10) is over all non-empty finite subsets W of V . The graph is called non-amenable if χ(G) > 0. We have so far concentrated on situations where infinite clusters are (almost surely) unique, as is commonly the case for an amenable graph. The situation is quite different when the graph is non-amenable, and a systematic study of percolation on such graphs was proposed in [3]. The best known example is bond percolation on the infinite binary tree, for which there exist infinitely many infinite clusters whenever the edge-density p satisfies 12 < p < 1. Let G = (V, E) be an infinite graph and let p ∈ (0, 1). For ω ∈ Ω = {0, 1}E , let I = I(ω) be the number of infinite clusters of ω. It has been known since [22] that there exist graphs having three non-trivial phases characterized respectively by I = 0, I = 1, I = ∞. One of the most interesting results in this area is the existence of a critical point for the event {I = 1}. This is striking because the event {I = 1} is not increasing. Prior to stating this theorem, we introduce some jargon. The infinite connected graph G = (V, E) is called transitive if, for all x, y ∈ V , there exists an automorphism τ of G such that y = τ (x). The graph G is called quasi-transitive if there exists a finite set V0 of vertices such that, for all y ∈ V , there exists x ∈ V0 and an automorphism τ such that y = τ (x). The following result was obtained by H¨ aggstr¨ om and Peres under a further condition, subseqently lifted by Schonmann. Theorem 9 ([26, 35]). Let G be an infinite, connected, quasi-transitive graph. There exist pc , pu ∈ [0, 1] satisfying 0 ≤ pc ≤ pu ≤ 1 such that: µp (I = 0) = 1 µp (I = ∞) = 1 µp (I = 1) = 1

if if if

0 ≤ p < pc , pc < p < pu , pu < p ≤ 1.

(11) (12) (13)

Here are some examples. 1. For an amenable graph, we have by the Burton–Keane argument that pc = pu . 2. For the binary tree, we have pc = 21 and pu = 1. 3. For the direct product of the binary tree and a line, we have that 0 < pc < pu < 1, see [22].

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References [1] Aizenman, M. and Grimmett, G. (1991). Strict monotonicity for critical points in percolation and ferromagnetic models. J. Statist. Phys. 63, 5–6, 817–835. MR1116036 [2] Aizenman, M., Kesten, H., and Newman, C. M. (1987). Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Comm. Math. Phys. 111, 4, 505–531. MR901151 [3] Benjamini, I. and Schramm, O. (1996). Percolation beyond Zd , many questions and a few answers. Electron. Comm. Probab. 1, no. 8, 71–82 (electronic). MR1423907 [4] Bezuidenhout, C. and Grimmett, G. (1999). A central limit theorem for random walks in random labyrinths. Ann. Inst. H. Poincar´e Probab. Statist. 35, 5, 631–683. MR1705683 [5] Burton, R. M., and Keane, M. S. (1989). Density and uniqueness in percolation. Comm. Math. Phys. 121, 3, 501–505. MR990777 [6] Burton, R. M., and Keane, M. S. (1991). Topological and metric properties of infinite clusters in stationary two-dimensional site percolation. Israel J. Math. 76, 3, 299–316. MR1177347 [7] De Masi, A., Ferrari, P. A., Goldstein, S., and Wick, W. D. (1985). Invariance principle for reversible Markov processes with application to diffusion in the percolation regime. In Particle Systems, Random Media and Large Deviations (Brunswick, Maine, 1984). Contemp. Math., Vol. 41. Amer. Math. Soc., Providence, RI, 71–85. MR814703 [8] De Masi, A., Ferrari, P. A., Goldstein, S., and Wick, W. D. (1989). An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Statist. Phys. 55, 3-4, 787–855. MR1003538 [9] Gandolfi, A., Grimmett, G., and Russo, L. (1988). On the uniqueness of the infinite cluster in the percolation model. Comm. Math. Phys. 114, 4, 549–552. MR929129 [10] Gandolfi, A., Keane, M. S., and Newman, C. M. (1992). Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses. Probab. Theory Related Fields 92, 4, 511–527. MR1169017 [11] Gandolfi, A., Keane, M., and Russo, L. (1988). On the uniqueness of the infinite occupied cluster in dependent two-dimensional site percolation. Ann. Probab. 16, 3, 1147–1157. MR942759 ¨ggstro ¨ m, O., and Maes, C. (2001). The random geom[12] Georgii, H.-O., Ha etry of equilibrium phases. In Phase Transitions and Critical Phenomena, Vol. 18. Academic Press, San Diego, CA, 1–142. MR2014387 [13] Gluck, H. (1975). Almost all simply connected closed surfaces are rigid. In Geometric Topology (Proc. Conf., Park City, Utah, 1974). Springer, Berlin, 225–239. Lecture Notes in Math., Vol. 438. MR400239 [14] Graver, J., Servatius, B., and Servatius, H. (1987). Combinatorial Rigidity. Mem. Amer. Math. Soc. 381. MR1251062 [15] Grimmett, G. (1995). The stochastic random-cluster process and the uniqueness of random-cluster measures. Ann. Probab. 23, 4, 1461–1510. MR1379156 [16] Grimmett, G. (1999). Percolation. Grundlehren der Mathematischen Wissenschaften, Vol. 321. Springer-Verlag, Berlin. MR1707339 [17] Grimmett, G. R. (1999). Stochastic pin-ball. In Random Walks and Discrete Potential Theory (Cortona, 1997). Sympos. Math., XXXIX. Cambridge Univ.

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IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 37–46 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000059

A note on percolation in cocycle measures Ronald Meester1 Vrije Universiteit, Amsterdam Abstract: We describe infinite clusters which arise in nearest-neighbour percolation for so-called cocycle measures on the square lattice. These measures arise naturally in the study of random transformations. We show that infinite clusters have a very specific form and direction. In concrete situations, this leads to a quick decision whether or not a certain cocycle measure percolates. We illustrate this with two examples which are interesting in their own right.

1. Introduction Much of Mike’s work in probability and percolation theory has been inspired by his background in ergodic theory. His ergodic-theoretical viewpoint of spatial stochastic models turned out to be very fruitful, both for answering long standing open questions as for generating new problems. Among many other things, Mike taught me how to think ‘ergodically’, and I have enjoyed the interplay between probability and ergodic theory ever since. In this note, we will further illustrate this interplay in a concrete situation; we discuss some percolation properties of a particular class of random colourings of the nearest neighbour edges of the square lattice Z2 . The (probability) measures in this note are related to measure-preserving random transformations, and for reasons that will become clear we shall call it the class of cocycle measures. We consider colourings of the edges of Z2 with two colours, red and blue, with the cocycle property. This property can be reformulated as follows. Consider four nearest-neighbour edges forming a square. When you travel in two steps from southwest to north-east along this square, the number of blue and red edges you see along the way does not depend on the route you take. A second way of defining this class of colourings is as follows. Take any two vertices x and y ∈ Z2 , and consider a vertexself-avoiding path π between x and y, i.e. a sequence of distinct edges ei = (ui , vi ) i = 1, . . . , k such that u1 = x, vk = y, vi = ui+1 for i = 1, . . . , k −1. When travelling along π from x to y, we travel edges horizontally to the right, vertically upwards, horizontally to the left or vertically downwards. We collect the first two types of edges in a set π + , and the last two in a set π − . Now consider the number of red edges in π + , minus the number of red edges in π − and call this number f1 (π). Similarly, f2 (π) is defined as the number of blue edges in π + minus the number of blue edges in π − . The requirement we impose on the configurations is that (f1 (π), f2 (π)) is the same for all paths π from x to y. Motivation for this type of measures can for instance be found in Burton, Dajani and Meester (1998). Indeed, the last characterisation above is in fact a formulation of a so called cocycle-identity, but this will play no role in the present note. We give some examples of cocycle measures in the last section of this note. 1 Deptartment

of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands, e-mail: [email protected] AMS 2000 subject classifications: primary 60K35; secondary 82B20. Keywords and phrases: percolation, cocycles, cocycle measure. 37

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38

Let µ be a stationary, ergodic (with respect to the group of all translations of Z ) cocycle measure. We are interested in percolation properties of µ, i.e. we are interested in the question whether or not infinite red or blue clusters exist, and if so, how many. It is easy to come up with examples in which both blue and red edges percolate; for instance take the measure µ which makes all horizontal edges blue and all vertical edges red. On the other hand, it is just as easy to find an example where neither the blue nor the red edges percolate; just colour the four edges of every second square blue and the remaining edges red, and choose the origin randomly as to get something stationary. We leave it to the reader to find an easy example of a measure for which exactly one colour percolates. The goal of the present note is to discuss some general percolation properties for this type of measures, which in concrete examples lead to a quick decision whether or not a given measure actually percolates. For reasons that will become clear soon, we will no longer speak about red or blue edges, but about edges labelled 0 or 1, and from now on this refers to a number, not a colour. We shall concentrate on percolation of edges labelled 0. Of course, this is in some sense arbitrary, but 0’s really seem to have advantages over 1’s, as we shall see. The next section gives some general background on cocycle measures. Section 3 deals with general facts about percolation in cocycle measures, and the last section is devoted to a number of examples. The first example in the last section was the motivation to study percolation properties of cocycle measures; in this example we answer a question which was asked by T. Hamachi. 2

2. General background We start with some notation. The expection of the label of a horizontal edge is denoted by h, the expection of the label of a vertical edges by v. To avoid trivial situations, we assume that 0 < h, v < 1. We write f (z) = f (z1 , z2 ) = f (z1 , z2 , ω) for the sum of the labels in π + minus the sum of the labels in π − , where π is an arbitrary self-avoiding path from 0 to z = (z1 , z2 ). L1 distance is denoted by  · . Note that our weak assumption on ergodicity of µ does not imply that the right or up shift are individually ergodic. However, since by the defining property of cocycle measures we have that |(f (n, m) − f (0, m)) − (f (n, m + 1) − f (0, m + 1))| ≤ 2, the limits limn→∞ (f (n, m) − f (0, m))/n, which exist by stationarity, are invariant under both horizontal and vertical translations and therefore a.s. constant. This constant has to be h then. A similar remark is valid for vertical limits. The cone y {(x, y) ∈ Z2 : α −  ≤ ≤ α + } x is denoted by C(α, ). Throughout, µ denotes a cocycle measure. The following lemma is taken from Dajani and Meester (2003). Lemma 2.1. Let {(kn , mn )} be a sequence of vectors in Z2 . (i) Suppose that (kn , mn ) → (c1 · ∞, c2 · ∞) for some c1 , c2 ∈ {1, −1} and in n addition that m kn → α ∈ [−∞, ∞]. Then c1 c2 |α| f (kn , mn ) → h+ v |kn | + |mn | 1 + |α| 1 + |α|

A note on percolation in cocycle measures

39

1 is to be interpreted as 0 and in µ-probability as n → ∞. (The quotient 1+∞ ∞ 1+∞ as 1.) (ii) Suppose that {kn } is bounded and mn → c3 · ∞ for some c3 ∈ {1, −1}. Then

f (kn , mn ) →c3 v |kn | + |mn | in µ-probability as n → ∞. (iii) Suppose that {mn } is bounded and kn → c4 · ∞ for some c4 ∈ {1, −1}. Then f (kn , mn ) →c4 h |kn | + |mn | in µ-probability as n → ∞. This leads to Lemma 2.2. Let α ∈ (−∞, ∞). Then for any  > 0, there a.s. exist N
> 0 and δ
> 0 such that whenever |mn |, |kn | > N
and (kn , mn ) ∈ C(α, δ
), then



f (kn , mn ) c c |α| 1 2

(1)
|kn | + |mn | − 1 + |α| h − 1 + |α| v
< ,

for appropriate c1 and c2 . When α = ±∞, δ
should be replaced by a constant M
and the condition (kn , mn ) ∈ C(α, δ
) should be replaced by |mn /kn | > M
. Moreover, similar statements are valid for all other cases of Lemma 2.1. Proof. Draw a uniform (0, 1) distributed random variable U and consider the line y = αX + U . Let yn be the (random) point on the vertical line {(x, y) : x = n} closest to this line. It is not hard to see that (f (y0 ), f (y1 ), . . .) forms a random walk with (dependent) stationary increments. We write xn = (kn , mn ) and write yj(n) for the (or a) vertex among (y0 , y1 , . . .) which is closest to xn . We then have   f (yj(n) ) f (xn ) − f (yj(n) ) yj(n)  f (xn ) = + . xn  yj(n)  yj(n)  xn  Since (f (y0 ), f (y1 ), . . .) has stationary increments, the ergodic theorem tells us that f (yn )/yn  converges a.s., and it then follows from the corresponding convergence in probability in Lemma 2.1 that this a.s. limit must be the same limit as in Lemma 2.1. Therefore, if kn and mn are large enough and |mn /kn −α| is small enough, then j(n) is large and therefore f (yj(n) )/yj(n)  is close to the correct limit in Lemma 2.1. At the same time, the term yj(n) /xn  is close to 1 by construction. Finally, the norm of the vector (f (xn ) − f (yj(n) ))/yj(n)  is bounded above by xn − yj(n)  . yj(n)  This last expression is close to 0 when kn and mn are large and |mn /kn − α| is small. For α = α0 := − hv , the limit in Lemma 2.1(i) is equal to 0, and we write C0 () for C(α0 , ). Lemma 2.3. Let  > 0. With µ-probability one, only finitely many points z outside C0 () have f (z) = 0.

R. Meester

40

Proof. The proof is by contradiction. Suppose infinitely many such z exist. Look at the set B of directions β outside C0 () such that for every δ > 0, the cone C(β, δ) contains infinitely many z with f (z) = 0. The set B is closed and invariant under translations and therefore β¯ := sup{β : β ∈ B} is well defined and an a.s. constant. According to Lemma 2.2, for every γ > 0, we now have infinitely many z with f (z) = 0 for which

¯ β| ¯

f (z) c1 β¯ c2 β|


z − 1 + |β| ¯ h − 1 + |β| ¯ v
< γ. But since f (z) = 0 and γ is arbitrary, this implies that ¯ β| ¯ c1 β¯ c2 β| h + ¯ ¯ v = 0, 1 + |β| 1 + |β| which implies that β¯ = α0 , a contradiction. 3. Percolation As mentioned in the introduction, we will concentrate on percolation of edges labelled with 0. The cluster C(z) of the vertex z is the set of vertices that can be reached from z by travelling over 0-labelled edges only. We are interested in the question whether or not infinite clusters exist and if so, how many. A subset S of Z2 is said to have density r if for each sequence R1 ⊆ R2 ⊆ · · · of rectangles in Z2 with ∪n Rn = Z2 , it is the case that #(S ∩ Rn ) = r, n→∞ #(Rn ) lim

where #(·) denotes cardinality. Burton and Keane (1991) showed that for every stationary percolation process Z2 , a.s. all clusters have density. In addition, they showed that either all infinite clusters have positive density a.s., or all infinite clusters have zero density a.s. Lemma 3.1. For any cocycle measure µ, all clusters have zero density a.s. Proof. If infinite clusters exist with positive probability, then the probability that the cluster of the origin is infinite must be positive. All elements z in this cluster have f (z) = 0. But according to Lemma 2.3, this implies that for all  > 0, up to a finite number of vertices, the whole cluster is contained in C0 (). Since the density of C0 () goes to zero with  tending to zero, we find that the cluster of the origin has density zero a.s. The result now follows from the result of Burton and Keane just mentioned. Lemma 3.2. If infinite clusters exist with positive µ-probability (and hence with probability one according to the ergodicity of µ), then there are infinitely many infinite clusters µ-a.s. Proof. There are at least two quick proofs of this fact. (1): If infinite clusters exists, then the set of vertices z for which C(z) is infinite has positive density a.s. Since each single cluster has zero density, the only conclusion is that there are infinitely many clusters. (Note that density is not countably additive!) (2): If there are infinite clusters a.s., then there are infinitely many points z on the x-axis for which C(z)

A note on percolation in cocycle measures

41

is infinite. It is clear that f (z) takes infinitely many values among these points z. However, when f (z) = f (z  ), then C(z)∩C(z  ) = ∅, since f (z) is obviously constant on a cluster. Hence infinitely many infinite clusters must exist. The following result could be stated in higher generality. Strictly speaking, it follows from a general result like Lemma 2.3 in H¨ aggstr¨ om and Meester (1996), but for this particular situation an independent simple proof is possible. See also Meester (1999) for related results. Lemma 3.3. For any cocycle measure µ, with probability one every infinite cluster C satisfies sup{y ; (x, y) ∈ C} = ∞, and inf{y ; (x, y) ∈ C} = −∞. Similar statements are true for the horizontal direction. Proof. Since C is contained (up to finitely many points) in every C0 (), the assumption that C is infinite implies that C is unbounded in at least one of the two vertical directions. We now assume (wlog) that with positive probability (and therefore with probability 1) a cluster C exists for which sup{y ; (x, y) ∈ C} = ∞ and inf{y ; (x, y) ∈ C} > −∞. According to Lemma 2.3, the intersecion of C with a horizontal line contains at most finitely many points, and therefore there is a leftmost point C (n) at every level y = n, for n large enough. The collection {C (n)}, where C ranges over all upwards unbounded clusters of the halfspace {y ≥ n}, has a well defined (one-dimensional horizontal) density dn . It is clear from the construction that dn+1 ≥ dn , since the restriction to {y ≥ n + 1} of every upwards unbounded cluster of {y ≥ n} contains at least one upwards unbounded cluster in {y ≥ n + 1}. At the same time, (dn ) forms a stationary sequence. We conclude that dn is constant a.s. On the other hand, note that our assumption implies that any given vertex z is the left-lowest point of an upwards unbounded cluster with positive probability. Therefore the line y = n + 1 contains a positive density of such points. Clearly, these points are ‘new’ in the sense that they are not in previous clusters. From this we see that dn+1 > dn , the required contradiction. Next, we show that percolation occurs in a directed sense: Lemma 3.4. Every infinite cluster C contains a strictly northwest-southeast directed bi-infinite path. More precisely, the left boundary of C forms such a path. Proof. Define, for every n, the leftmost vertical edge in C between {y = n} and {y = n + 1} by en . Connect, for all n, the upper endpoint of en with the lower endpoint of en+1 through the horizontal edges in between them (if necessary). I claim that the union of these vertical and horizontal edges is the required path. To see this, note that the upper endpoint of en is connected by a path of 0-edges to the upper endpoint of en+1 since they belong to the same cluster C. Since we can also travel from the upper endpoint of en to the upper endpoint of en+1 by first travelling horizontally to the lower endpoint of en+1 and the last step via en+1 , all these last travelled horizontal edges must have zero labels. Finally, en+1 cannot be strictly at the right of en , since then the path constructed by traveling vertically from the upper endpoint of en and then horizontally to the upper endpoint of en+1 would consist of zero labels only, which contradicts the definition of en+1 . Finally, we show that ‘dead ends’ are impossible:

42

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Lemma 3.5. Consider the following event: there is a directed path from the origin going down-right, which is completely labelled 0, and the two edges ((−1, 0), (0, 0)) and ((0, 1), (0, 0)) are both labelled 1. This event has probability 0. Proof. According to the previous lemma, the left boundary of the 0-cluster of the origin forms a bi-infinite directed path π. This bi-infinite path crosses the x-axis to the left of the origin. The cluster of the origin must contain a connection between π and the directed path going down from the origin. Now label all edges which are forced to be zero, given this connection. It is easy to see that these 0’s run into conflict with one of the two designated edges having label 1. By now we have a fairly precise and specific description of the geometry of infinite clusters in cocycle measures, if they exist: there are in that case infinitely many such clusters, essentially contained in a cone in the α0 -direction, and bounded at the left by a bi-infinite directed path. This description is so specific that it makes it easy in many case to rule out percolation almost immediately. On the other hand, one might suspect that this specific description makes it almost impossible for ‘natural’ cocycle measures to percolate, and that a percolating cocycle measure must be more or less constructed for that purpose. It seems hard to formulate a general property which excludes percolation. One is tempted to try to connect certain ergodic-theoretical mixing properties with percolation here, since the above description of percolation clusters seems highly non-mixing. However, we will see that mixing cocycle measures that percolate can be constructed; they can even have trivial full tail. 4. An example The following example is discussed to some extent in Burton, Dajani and Meester (1998). Here we shall discuss the example in detail. Choose 0 < p < 1 and let q = 1 − p. Label all edges of the x-axis 0 with probability q and 1 with probability p, independently of each other. For the y-axis we do the same with interchanged probabilities. Now denote the square [n, n + 1] × [0, 1] by Wn , and denote the lower and upper edge of Wn by en and fn respectively. The labelling procedure is at follows: first label the remaining edges of W1 ; if there are two possibilities for doing this we choose one of them with equal probabilities. At this point, the lower and left edge of W2 are labelled, and we next complete the labelling of W2 , noting again that if there are two ways to do this, we choose one of them with equal probabilities. This procedure is continued and gives all labels in the strip [0, ∞) × [0, 1]. Then we move one unit upwards, and complete in a similar fashion the labels in the strip [0, ∞) × [1, 2]. (Of course, if you want to carry out this labelling, you never actually finish any strip. Instead, you start at some moment with the second strip which can be labelled as far as the current labelling of the first strip allows, etc.) This procedure yields a random labelling of all edges in the first quadrant. Using for instance Kolmogorov’s consistency condition, we can extend this to a cocycle measure on the labels in the whole plane. Lemma 4.1. The procedure described above yields a stationary and mixing measure µ. In particular, the labelling is ergodic. Proof. If we can show that the labelling of the edges fn has the same distribution as the labelling of the edges en , then we have shown that the labelling in the quadrant [0, ∞) × [1, ∞) has the same distribution as the labelling in [0, ∞) × [0, ∞) and we

A note on percolation in cocycle measures

43

can use a similar argument for vertical lines plus induction to finish the argument. Therefore we only need to show that the labelling of the edges fn is i.i.d. with the correct marginals. To do this properly, consider the labels of the edges of Wn . There are six possible labellings of the edges of Wn . Four of these are such that en and fn have the same label. The exceptional labellings are (starting at the lower left vertex and moving clockwise) 0110 and 1001. Denote the labelling of the edges of Wn by Ln . Then it is not hard to see that Ln is a Markov chain on the state space {0110, 1001, 1010, 0101, 1111, 0000}. Take the transition matrix P of Ln , interchange the rows and the columns corresponding to 0110 and 1001 to obtain P  , and consider the backward Markov chain corresponding to P  , denoted by Mn . An easy calculation then shows that Ln and Mn have the same transition matrix and that they are both in stationarity. But now note that Mn represents the right-to-left labelling of the strip [0, ∞) × [0, 1]. This means that the distribution of the random vector (fk1 , fk2 , . . . , fkn ) is the same as the distribution of (ekn , ekn−1 , . . . , ek1 ). The last vector has independent marginals, hence so has the first, and we are done. Next we show that µ is mixing. For this, consider finite-dimensional cylinder events A and B, i.e. A and B only depend on edges in the box Bn = [0, n]2 . Denote by Mk the labelling of all the edges in the box Bn + (k, 0). It is easy to check that Mk is a mixing Markov chain. This implies that for the events A and B, we have −1 µ(A ∩ T(k,0) (B)) → µ(A)µ(B),

where Tz denotes translation over the vector z. This shows that µ is mixing in the horizontal direction. For the vertical direction, we consider the Markov chain associated with the labellings of the boxes Bn + (0, k), k = 0, 1, . . ., and repeat the argument. Theorem 4.2. The measure µ described in this subsection does not percolate a.s. Before we give a proof, we need to look at the construction of µ. The above definition of µ is simple, but has the disadvantage that we need to appeal to Kolmogorov’s consistency theorem to define it on the whole plane. In this sense, the definition is not constructive. There is an alternative way of defining µ that is constructive, and that will be quite useful. The first step towards this construction is the following lemma. Lemma 4.3. Let π be a bi-infinite path (. . . , z−1 , z0 , z1 , . . .), where zi = (zi1 , zi2 ), and with the property that zk1 is non-decreasing in k, and zk2 is non-increasing in k. Denote the edge (zi , zi+1 ) by ei . Then the labels (. . . , c(e−1 ), c(e0 ), c(e1 ), . . .) form an independent sequence. Proof. Independence is defined in terms of finite collection of edges, so by stationarity we need only look at finite paths with these monotonicity properties which travel from the y-axis to the x-axis. That is, we only work in the first quadrant for now. Denote such a path by (z0 , . . . , zk ), where z0 is on the y-axis, and zk is on the xaxis. We may assume that z0 and zk are the only points of the path on the coordinate axes. Again denoting the edge (zi , zi+1 ) by ei , We claim that the last edge c(ek−1 ) is independent of the collection {c(e0 ), . . . , c(ek−2 )}. To see this, first assume that ek−2 is a horizontal edge. (Note that ek1 is always vertical by assumption.) Then by considering the reversed Markov chain in the proof of Lemma 4.1, it follows immediately that c(ek−1 ) is independent of c(ek−2 ) and is also independent of the labels of all horizontal edges to the left of ek−2 . But all labels {c(e1 ), . . . , c(ek−2 )}

44

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are measurable with respect to these last labels together with some independent labels on the y-axis and some independent choices when appropriate. This proves the claim. If ek−2 is vertical, we walk back along the path until the first horizontal edge, and repeat the argument with the Markov chain corresponding to the strip with the appropriate width. The lemma now follows with induction in the obvious way. The last lemma tells us how to construct a labelling of the whole plane in a constructive manner. We first take a bi-infinite northwest-southest directed path π that is unbounded in all directions. Label the edges of this path in an independent fashion, with the correct one-dimensional marginals. Next start ‘filling the plane’ in a way similar to the original construction. Above the path, we can do what we did before, and label strips from left to right; below the path we use the backwards Markov chain mentioned in the proof of Lemma 4.1, and we label strips from right to left. It is clear that the labelling obtained this way has the correct distribution: just note that all finite-dimensional distributions are correct. Here we need Lemma 4.3 of course, to make sure we start off with the correct distribution on our path π. Note that the labellings above and below π are conditionally independent, given the labelling of π itself. Also note that the sigma fields generated by E 1 (π) := {e : e is both below π and to the right of the line x = 0 (inclusive)} and E 2 := {e : e is both above π and to the left of the line x = 0 (inclusive)} are independent. Proof of Theorem 4.2. We assume µ does percolate, and show that this leads to a contradiction. Choose a northwest-southeast directed path π = (. . . , z−1 , z0 , z1 , . . .) as follows (recall the definition of h and v): z0 is the origin, and for some (possibly large) M , z−M , . . . , zM are all on the x-axis. The vertices zM +1 , zM +2 , . . . can now be chosen in such a way that they are all above the line through zM with direction −h/2v; the vertices z−M −1 , z−M −2 , . . . can now be chosen in such a way that they are all below the line through z−M with direction −h/2v. Label the edges of π independently (with the correct marginals). The point of this choice is that according to our geometrical picture obtained in the previous section, for some M , there must be a positive probability that the origin is contained in a 0-labelled directed infinite path going down-right, all whose edges are strictly below π. We call this event E1 . It is clear that E1 is measurable with respect to the sigma field generated by E 1 (π). On the other hand, there is a positive probability that the four edges between the origin, (0, 1), (−1, 0) and (−1, −1) are all labelled 1. This event, let’s call it E2 , is measurable with respect to the sigma field generated by E 2 (π). We noted above that these two sigma fields are independent and it follows that E1 and E2 are independent. Hence P (E1 ∩ E2 ) > 0, but on this event, the origin is a dead end in the sense of Lemma 3.5, a contradiction. It is interesting to compare the last theorem with a result of Kesten (1982). He showed that if we label all edges independently, with vertical edges being 0 with probability p and horizontal edges with probability 1 − p, then the system does not percolate. In fact, the system is critical in the sense that if one increases the probability for either the horizontal or vertical edges by a positive amount, the sytem does in fact percolate. 5. A percolating cocycle with trivial full tail The geometrical picture of infinite clusters looks highly non-mixing. After all, if we look at the realisation below the horizontal line y = n, we have a lot of information

A note on percolation in cocycle measures

45

about the infinite clusters, an this should tell a lot about the realisation in the halfplane y ≥ n + m, for m large. So it seems that a percolating cocycle measure has long distance dependencies and therefore weak mixing properties. But we shall now see that a percolating cocycle measure can be constructed which has trivial full tail, which is much stronger than being mixing. The construction is based on an exclusion process introduced in Yaguchi (1986) and studied by Hoffman (1998). We first describe Yaguchi’s construction. We shall initially work in the half plane x ≥ 0, but the measure can of course be extended to a measure on the full plane. Consider the y-axis. Each vertex is either blue, red or not coloured (probability comes in later). We next colour the line x = 1 as follows. Each coloured vertex z (on the y-axis) decides independently with a certain (fixed and constant) probability if it wants to move down one unit. It also checks whether or not the vertex below is not coloured. If both the vertex wants to move, and the vertex below is not coloured, then we colour the vertex z +(1, −1) with the same colour as z. Otherwise we colour z + (1, 0) with the same colour as z. This procedure is repeated when we go from x = 1 to x = 2, etc. Yaguchi (1986) characterised the stationary measures of the associated Z2 action, in particular he showed such measures exist. Hoffman (1999) showed that these measures have trivial full tail; in particular they are mixing. How does this relate to cocycle measures? We shall make a few minor modifications. First, we look at all red points on the y-axis that are between two given blue points (with no other blue points in between). Take the top vertex among these red vertices and change its colour into green. When a coloured vertex z causes z + (1, 0) to be coloured with the same colour as z we also colour the edge between these two vertices with the same colour. When z causes z + (1, −1) to be coloured, we colour the two edges (z, z + (0, −1)) and (z + (0, −1), z + (1, −1)) with the same colour. Finally we keep the green edges and ‘uncolour’ all other edges. A configuration now consists of infinitely many disjoint, bi-infinite strictly northwest-southeast directed green paths. We can transform a realisation to a labelling of the edges that satisfy the cocycle identity as follows: All green edges are labelled 0. All edges that have one endpoint in common with a green edge are label 1. All remaining edges are labelled 0. It is easy to prove that the realisation obtained this way satisfies our cocycle identity. It is obtained as a ergodic-theoretical factor of a mixing process, and therefore also mixing. On the other hand, it percolates along the edges that are coloured green. References [1] Burton, R., Dajani, K., and Meester, R. (1998). Entropy for random group actions. Ergodic Theory Dynam. Systems 18, 1, 109–124. MR1609487 [2] Burton, R. M. and Keane, M. (1991). Topological and metric properties of infinite clusters in stationary two-dimensional site percolation. Israel J. Math. 76, 3, 299–316. MR1177347 [3] Dajani, K. and Meester, R. (2003). Random entropy and recurrence. Int. J. Math. Math. Sci. 47, 2977–2988. MR2010744 ¨ggstro ¨ m, O. and Meester, R. (1996). Nearest neighbor and hard sphere [4] Ha models in continuum percolation. Random Structures Algorithms 9, 3, 295– 315. MR1606845 [5] Hoffman, C. (1999). A Markov random field which is K but not Bernoulli. Israel J. Math. 112, 249–269. MR1714990 [6] Kesten, H. (1982). Percolation Theory for Mathematicians. Progress in Probability and Statistics, Vol. 2. Birkh¨ auser Boston, MA. MR692943

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[7] Meester, R. (1999). Extremal points of infinite clusters in stationary percolation. Statist. Probab. Lett. 42, 4, 361–365. MR1707181 [8] Yaguchi, H. (1986). Stationary measures for an exclusion process on onedimensional lattices with infinitely many hopping sites. Hiroshima Math. J. 16, 3, 449–475. MR867575

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 47–52 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000068

On random walks in random scenery F. M. Dekking1 and P. Liardet2 Delft University of Technology and Universit´ e de Provence Abstract: This paper considers 1-dimensional generalized random walks in random scenery. That is, the steps of the walk are generated by an arbitrary stationary process, and also the scenery is a priori arbitrary stationary. Under an ergodicity condition—which is satisfied in the classical case—a simple proof of the distinguishability of periodic sceneries is given.

1. Introduction Random walks in random scenery have been studied by Mike Keane for quite some time (see [2] for his most recent work). In fact, he and Frank den Hollander were pioneers in this exciting area. Around 1985 they formulated a conjecture about “recovery of the scene” by a simple random walker. A weaker form of this: “distinguishability of two scenes”, was proven by Benjamini and Kesten ([1]). Since then there has been a lot of action in this field, especially by Matzinger and co-workers. We just mention the recent paper [5]. In the following we will introduce generalized random walks in random scenery, and analyse them from a dynamical point of view. This gives us in Section 2 a general scenery recovery result on the level of measures, from which we deduce in a simple way in Section 3 a proof for the distinguishability of periodic sceneries. We consider a random walker on the integers. The integers are coloured by colours from an alphabet C. This is the scenery. At time n the walker records the scenery at his position, this yields rn from C. To formalize somewhat more, let the random walk be described by a measure µ on the Borel sets of Ω, where Ω = {ω = (ωn )n∈Z : ωn ∈ J for all n}. Here the set J of the possible steps of the walk will simply be {−1, +1}, or somewhat more general {−1, 0, +1}. Although often a single scenery x = (xk )k∈Z is considered, it is useful to consider x as an element of the shift space X = C Z with shift map T : X → X, equipped with some ergodic T -invariant measure λ, which we will call the scenery measure. We then consider x picked according to the measure λ. The colour record ϕx of x can be written as a map ϕx : Ω → X: ϕx (ω) = (rn (ω, x))n∈Z , where in line with the description above, one has for n ≥ 1 rn (ω, x) = (T ω0 +···+ωn−1 x)0 . 1 Thomas

Stieljes Institute for Mathematics and Delft University of Technology Faculty EEMCS, Mekelweg 4, 2628 CD Delft, The Nethrlands, e-mail: [email protected] 2 Universit´ e de Provence, Centre de Math´ematiques et Informatique (CMI), 39 rue Joliot-Curie, F-13453 Marseille cedex 13, France, e-mail: [email protected] AMS 2000 subject classifications: primary 28D05. Keywords and phrases: random walk, random scenery, colour record, skew product transformation. 47

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48

This definition is completed by putting r0 (ω, x) = x0 , and for n < 0: rn (ω, x) = (T −ω−1 −ω−2 ···−ωn x)0 . The dynamics of the whole process is well described by a skew product transformation TΩ×X on the product space Ω × X defined by TΩ×X (ω, x) = (σω, T ω0 x), where σ denotes the shift map on Ω. Let us now look at the colour records of all x; we define the global recording map Φ : Ω × X → X by Φ(ω, x) = (rn (ω, x))n∈Z . Lemma 1. The map Φ is equivariant, that is, Φ ◦ TΩ×X = T ◦ Φ. Proof. One way: Φ ◦ TΩ×X (ω, x) = Φ((σω, T ω0 x)) = (rn (σω, T ω0 x))n∈Z = ((T ω1 +···+ωn T ω0 x)0 ) = (rn+1 (ω, x))n∈Z . The other way: T ◦ Φ(ω, x) = T ((rn (ω, x))n∈Z ) = (rn+1 (ω, x))n∈Z . Clearly product measure µ × λ is preserved by TΩ×X . We will be particularly interested in the image of µ×λ under the global recording map Φ, which we denote ρ: ρ = (µ×λ) ◦ Φ−1 . We call ρ the global record measure. It follows from Lemma 1 that ρ is invariant for T . Moreover, ρ will be ergodic when TΩ×X is ergodic for µ×λ. In the classical case were µ is product measure this is guaranteed by Kakutani’s random ergodic theorem. In this case, when λ and λ are two scenery measures, and ρ = (µ×λ)◦Φ−1 and ρ = (µ×λ ) ◦ Φ−1 are the corresponding global record measures, then either ρ = ρ or ρ ⊥ ρ . The colour record ϕx of a scenery x induces the record measure ρx defined by ρx = µ ◦ ϕ−1 x . Following [4] we call the two sceneries x and y distinguishable if ρx ⊥ρy . The following lemma shows that global distinguishability carries over to local distinguishability. Lemma 2. Let λ and λ be two scenery measures with corresponding global record measures ρ and ρ . Then ρ⊥ρ implies that ρx ⊥ρy for λ×λ almost all (x,y). Proof. By Fubini’s theorem, and recalling that Φ(ω, x) = ϕx (ω),

ρ(E) = 1E ◦ Φ(ω, x) dµ(ω) dλ(x) X Ω
= µ(ϕ−1 x E) dλ(x). X



So ρ = ρx dλ(x). Hence if E is a Borel set with the property that ρ(E) = 1 and ρ (E c ) = 1, then ρx (E) = 1 for λ-almost x, and ρy (E c ) = 1 for λ -almost y.

On random walks in random scenery

49

2. Reconstructing the scenery measure Here we consider the case of a generalized random walk with steps J = {−1, 0, +1} given by an ergodic stationary measure µ on Ω = J Z . For ease of notation we rename J to {l, h, r}. To simplify the exposition we will assume that there is no holding (µ([h]) = 0), and show at the appropriate moment that this restriction can trivially be removed. There is a basic difference between symmetric walks and asymmetric walks in the reconstruction of the scenery. We call µ symmetric if for each word w µ[w] = µ[w]. Here w denote the mirror image of w, that is, the word obtained from w by replacing r by l and l by r. Since he does not know left from right, a symmetric walker can only reconstruct a scenery x up to a reflection. This will result in two theorems, one for the asymmetric, and one for the symmetric case. Let λ be a scenery measure. We give a few examples of calculation of the ρprobabilities of cylinder sets. Let Wn be the set of all words w = w1 . . . wn of length n over the (colour) alphabet {0, 1}. For w ∈ Wn we let [w] denote the cylinder [w] = {x ∈ X : x0 . . . xn−1 = w}, and we will abbreviate ρ([w]) to ρ[w]. We will use the same type of notations and conventions for λ and µ. It is clear, using the stationarity of λ, that for instance ρ[001] = µ[rr]λ[001] + µ[ll]λ[100], and slightly more involved ρ[000] = (µ[rl] + µ[lr])λ[00] + (µ[rr] + µ[ll])λ[000], and ρ[0001] = µ[rll]λ[100] + µ[lrr]λ[001] + µ[rrr]λ[0001] + µ[lll]λ[1000]. In the sequel we shall denote the word r . . . r, N times repeated, as rN . Note ← how with each appearance of a word w on the right side also the reversed word w ← ← appears, defined by w = wn . . . w1 if w = w1 . . . wn . Words w that satisfy w = w are called palindromes. Now let us put all the words w from ∪1≤k≤n Wk in some fixed order, taking care that their lengths are non-decreasing and that for a fixed k we ← first take all palindromes, and then all non-palindromes in pairs (w, w). Let Vn (ρ) and Vn (λ) denote the vectors of length (2n+1 − 2) containing the real numbers ρ[w] respectively λ[w] in the chosen order. For example, V2T (ρ) = (ρ[0], ρ[1], ρ[00], ρ[11], ρ[01], ρ[10]). In general, if w is a word of length N + 1, then ρ[w] is obtained as a sum of products µ[u]λ[v], where the length of v is at most N + 1, and length N + 1 only occurs when the walker makes no turns, i.e., when u = rN or u = lN . Moreover, if w is a palindrome, then there is one maximal length term (µ[rN ] + µ[lN ])λ[w], and if w is not a palindrome, then there are two maximal length terms µ[rN ]λ[w], ← and µ[lN ]λ[w]. This observation shows that there exists an almost lower triangular (2n+1 − 2) × (2n+1 − 2) matrix An such that Vn (ρ) = An Vn (λ).

50

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Here ‘almost lower triangular’ means 
0  ∗
  ∗ ∗   ∗ ∗   ∗ ∗ ∗ ∗

that An has the form  0 0 0 0 0 0 0 0  
0 0 0  , ∗ ... 0 0   ∗ ∗
0  ∗ ∗ ∗


where (at palindrome entries)
is a 1 × 1 matrix µ[rN ] + µ[lN ], and (at nonpalindrome pairs)
is a 2 × 2 matrix of the form

µ[rN ] µ[lN ] . µ[lN ] µ[rN ] With simple linear algebra we find that An is non-singular if and only if µ[r] = µ[l], . . . , µ[rN ] = µ[lN ], . . . Let us call a generalized random walk given by µ strongly asymmetric if all these inequalities hold. For instance, if µ is a stationary Markov chain given by a 2 × 2 transition matrix (ps,s ), then µ is strongly asymmetric if and only if pr,r = pl,l . Note that when µ[h] > 0, then only some sub-diagonal elements of An will change from 0 to a positive value. We therefore obtained the following result. Theorem 1. For strongly asymmetric generalized random walk with holding the scenery measure λ can be reconstructed from ρ. What remains is the symmetric walker case. Then in general λ can not be reconstructed from ρ. However, often we can reconstruct the reversal symmetrized ˇ defined for each word w by measure λ 1 ← ˇ λ[w] = λ[w] + λ[w] . 2 For symmetric µ the equation for, e.g., ρ[0001] becomes ˇ ˇ ρ[0001] = 2µ[lrr]λ[001] + 2µ[rrr]λ[0001]. In general, if w is a word of length N +1, then ρ[w] is obtained as a sum of products ˇ µ[u]λ[v], where the length of v is at most N + 1, and length N + 1 only occurs when N ˇ u = r or u = lN . Moreover, now there is for all words w one term 2µ[rN ]λ[w] for the v = w with maximal length. So this time we obtain the existence of a (2n+1 − 2) × (2n+1 − 2) lower triangular matrix An such that ˇ Vn (ρ) = An Vn (λ). Let us call µ straightforward if arbitrary long words of r’s have positive probability to appear. Then the diagonal elements of An are positive for each n, and we obtain the following. Theorem 2. For straightforward symmetric generalized random walk with holding ˇ can be reconstructed from ρ. the reversal symmetrized scenery measure λ

On random walks in random scenery

51

3. Distinguishing periodic sceneries In this section we shall derive more general results with more simple proofs than in [3]. It is shown there that for asymmetric simple random walk with holding any two periodic sceneries x and y which are not translates of each other can be distinguished by their scenery records, i.e., ρx ⊥ρy . Our result is Theorem 3. Any strongly asymmetric generalized random walk with holding can distinguish two periodic sceneries that are not translates of each other, provided that their global record measures are ergodic. t

Proof. Let us write x ∼ y if x and y are translates of each other, i.e., for some k one has y = T k x. Let Per(x) be the period of x, i.e. p =Per(x) is the smallest natural number such that T p x = x. Let λ be the scenery measure generated by x, i.e., denoting point measure in z by δz , 1 λ= Per(x)

Per(x)−1

δT k x .

k=0

The scenery measure generated by y is denoted as λ . Now suppose that ρx is not orthogonal to ρy . Then, since λ and λ are discrete, it follows from Lemma 2 that also ρ and ρ are not orthogonal. But since these measures are ergodic, they must be equal. From Theorem 1 it then follows that also λ = λ . This implies that t x ∼ y, by the discreteness of λ and λ . Indeed, equality of these measures yields t that δT k x = δT j y for some k and j, and hence that x ∼ y. For a symmetric (generalized) random walk it is impossible to distinguish a ← ← sequence x from its reflection x , defined by x k = x−k . So let us call x and y equivalent, and we denote x ∼ y, if y can be obtained from x by translation and/or reflection. Theorem 4. Any straightforward symmetric generalized random walk with holding can distinguish two periodic sceneries that are not equivalent, provided that their global record measures are ergodic. Proof. The proof follows the same path as the proof of Theorem 3, using Theorem ˇ is a 2 instead of Theorem 1. The only other difference now is that the measure λ k j ←  ˇ ˇ mixture of point measures in T x and in T x . But then equality of λ and λ implies that y must be a translate, or the reflection of a translate of x, i.e., x ∼ y. Acknowledgment We are grateful, in chronological order, to Jeff Steif and Frank den Hollander for their useful comments on earlier versions of this paper. References [1] Benjamini, I. and Kesten, H. (1996). Distinguishing sceneries by observing the scenery along a random walk path. J. Anal. Math. 69, 97–135. MR1428097 [2] den Hollander, F., Keane, M. S., Serafin, J., and Steif, J. E. (2003). Weak Bernoullicity of random walk in random scenery. Japan. J. Math. (N.S.) 29, 2, 389–406. MR2036267

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[3] Howard, C. D. (1996). Detecting defects in periodic scenery by random walks on Z. Random Structures Algorithms 8, 1, 59–74. MR1368850 [4] Lindenstrauss, E. (1999). Indistinguishable sceneries. Random Structures Algorithms 14, 1, 71–86. MR1662199 [5] Matzinger, H. (2005). Reconstructing a two-color scenery by observing it along a simple random walk path. Ann. Appl. Probab. 15, 1B, 778–819. MR2114990

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 53–65 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000077

Random walk in random scenery: A survey of some recent results Frank den Hollander1,2,∗ and Jeffrey E. Steif 3,† Leiden University & EURANDOM and Chalmers University of Technology Abstract. In this paper we give a survey of some recent results for random walk in random scenery (RWRS). On Zd , d ≥ 1, we are given a random walk with i.i.d. increments and a random scenery with i.i.d. components. The walk and the scenery are assumed to be independent. RWRS is the random process where time is indexed by Z, and at each unit of time both the step taken by the walk and the scenery value at the site that is visited are registered. We collect various results that classify the ergodic behavior of RWRS in terms of the characteristics of the underlying random walk (and discuss extensions to stationary walk increments and stationary scenery components as well). We describe a number of results for scenery reconstruction and close by listing some open questions.

1. Introduction Random walk in random scenery is a family of stationary random processes exhibiting amazingly rich behavior. We will survey some of the results that have been obtained in recent years and list some open questions. Mike Keane has made fundamental contributions to this topic. As close colleagues it has been a great pleasure to work with him. We begin by defining the object of our study. Fix an integer d ≥ 1. Let X = (Xn )n∈Z be a sequence of i.i.d. random variables taking values in a possibly infinite set F ⊂ Zd according to a common distribution mF having full support on F . Let S = (Sn )n∈Z be the corresponding two-sided random walk on Zd , defined by S0 = 0

and

Sn − Sn−1 = Xn , n ∈ Z,

i.e., Xn is the step at time n and Sn is the position at time n. To make S into an irreducible random walk, we will assume that F generates Zd , i.e., for all x ∈ Zd there exist n ∈ N and x1 , . . . , xn ∈ F such that x1 + · · · + xn = x. The simple 1 for e ∈ F . random walk is the case where F = {e ∈ Zd : |e| = 1} and mF (e) = 2d * The

research of FdH was supported by the Deutsche Forschungsgemeinschaft and the Netherlands Organisation for Scientific Research through the Dutch-German Bilateral Research Group on “Mathematics of Random Spatial Models from Physics and Biology”. † The research of JES was supported by the Swedish National Science Research Council and by the G¨ oran Gustafsson Foundation (KVA). 1 Mathematical Institute, Leiden University, P.O. Box 9512, 2330 RA Leiden, The Netherlands, e-mail: [email protected] 2 EURANDOM, P.O.Box 513, 5600 MB Eindhoven, The Netherlands. 3 Department of Mathematics, Chalmers University of Technology, Gothenburg, Sweden, e-mail: [email protected] AMS 2000 subject classifications: primary 60G10; secondary 82B20. Keywords and phrases: random walk in random scenery, K-automorphism, Bernoulli, weak Bernoulli, finitary coding, conditional probability distribution, bad configuration, scenery reconstruction. 53

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54

Next, let C = (Cz )z∈Zd be i.i.d. random variables taking values in a finite set G according to a common distribution mG on G with full support. Unless stated otherwise, we will restrict to the case where G = {−1, +1} and mG (−1) = mG (+1) = 12 , although most results in this paper hold in general. We will refer to C as the random scenery, i.e., Cx is the scenery value at site x. In what follows, X and C will be taken to be independent. Let Y = (Yn )n∈Z

with

Yn = (C ◦ S)n = CSn

be the sequence of scenery values observed along the random walk. We will refer to Y as the scenery record. The joint process Z = (Zn )n∈Z

with

Zn = (Xn , Yn )

is called random walk in random scenery (RWRS). This process registers both the step taken by the walk and the scenery value at the site that is visited. Note that, while X is simple, Y is complicated (because it is a composition of two random processes). The interplay between X and Y will be important. We will assume that the reader is familiar with a number of key concepts from ergodic theory, namely, K-automorphism, Bernoulli, weak Bernoulli, and finitary factor of an i.i.d. process. For definitions we refer to Walters [41]. For reasons of exposition, we give loose definitions here. A K-automorphism is a random process with a trivial future tail σ–field. A Bernoulli process is one that can be coded (in an invertible and time-invariant manner) from an i.i.d. process. A weak Bernoulli process is one where the past and the far distant future have a joint distribution that is close in total variation norm to the distribution of the past and the far distant future put together independently. A finitary factor of an i.i.d. process is a Bernoulli process for which the coding is such that, in order to determine one of the output bits, one need only look at a finite (but random) number of bits in the i.i.d. process. In ergodic theory, when d = 1 and the random walk is simple, RWRS is referred to as the T, T −1 -process. The reason is that if T is the shift on the scenery sequence, then with each step the walker sees the scenery sequence shifted either by T or by T −1 depending on whether the step is to the right or to the left. The interest in RWRS originally came from the fact that, for simple random walk in d = 1, Z was conjectured to be a natural example of a K-automorphism that is not Bernoulli. As the history given below reveals, this conjecture turned out to be true. In our opinion it is by far the simplest such example (in terms of the description of the process, though not in terms of the proof). The outline of this paper is as follows. In Section 2 we focus on the ergodic properties of RWRS and present a history of results that have been obtained so far, organized in Sections 2.1-2.6. In Section 3 we describe a number of results for scenery reconstruction. In Section 4 we close by listing some open questions. Inevitably, what follows is our selection of highlights and omits certain works that could also have been included. 2. Ergodic properties In Sections 2.1 and 2.2 we list the main theorems that determine when RWRS is a K-automorphism, is Bernoulli, is weak Bernoulli, or is a finitary factor of an i.i.d. process. In Section 2.3 we have a look at when these properties hold for

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reduced RWRS, the second component of RWRS, i.e., the scenery record alone. In Section 2.4 we make a brief excursion into random walks with stationary increments and random sceneries with stationary components, in order to see what properties survive when we relax the i.i.d. assumptions. In Section 2.5 we consider induced RWRS, which is obtained by observing the scenery only when a +1 is visited. Finally, in Section 2.6 we investigate the continuity properties of the conditional probability distribution for RWRS at time zero given the configuration at all other times. 2.1. K-automorphism, Bernoulli and weak Bernoulli RWRS is clearly stationary. Since the walk may return to sites visited before, it is not i.i.d. By Kakutani’s random ergodic theorem, it is ergodic. Our starting point is the following general result. Theorem 2.1 (Meilijson [33]). RWRS associated with an arbitrary random walk is a K-automorphism (i.e., has a trivial future tail σ–field ). The intuition behind this result is that the distribution of the walk spreads out for large times, so that the walk sees an ergodic average of the scenery. Although Theorem 2.1 was proved only for d = 1 (in the more general setting of so-called skew-products), the argument easily extends to arbitrary d ≥ 1. See Rudolph [36] for related results. We point out that for the case of simple random walk in d = 1, the result in Theorem 2.1 was “known” prior to [33]. On the other hand, Meilijson actually proved his result for any totally ergodic random scenery (See Section 2.4 below). It was known early on that being Bernoulli implies being a K-automorphism, because the latter is isomorphism invariant (i.e., invariant under coding) and an i.i.d. random process has a trivial future tail σ-field. In 1971, Adler Ornstein and Weiss conjectured that RWRS associated with simple random walk in d = 1 is not Bernoulli (see [42] and [16]). If true, then this would provide a beautiful and natural example of a K-automorphism that is not Bernoulli. (At some earlier stage, it was an open question whether every K-automorphism was Bernoulli. Counterexamples were constructed by Ornstein and later by Ornstein and Shields, but these were much less natural.) The conjecture was settled in a deep paper by Kalikow. Theorem 2.2 (Kalikow [16]). RWRS associated with simple random walk in d = 1 is not Bernoulli. (In fact, Kalikow actually proves the stronger result that the process is not “loosely Bernoulli”, a notion we will not consider.) Theorem 2.2 was later extended to cover an almost arbitrary recurrent random walk. Theorem
2.3 (den Hollander and Steif [13]). If the random walk is recurrent with x∈Zd |x|δ mF (x) < ∞ for some δ > 0 and satisfies a certain technical condition, then the associated RWRS is not Bernoulli. We will not explain the “technical condition”, because it is extremely weak and is in fact conjectured in [13] to hold for an arbitrary random walk (!). In [13] it is proved that if mF has one component that is in the domain of attraction of a stable law with index ≥ 1, then the technical condition is already fulfilled. Theorem 2.3 shows that recurrence of the random walk essentially implies that the associated RWRS is not Bernoulli. The next result tells us that if the random

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walk is transient, then the associated RWRS is Bernoulli, providing us with a nice dichotomy. Theorem 2.4 (den Hollander and Steif [13]). If the random walk is transient, then the associated RWRS is Bernoulli. The concept of weak Bernoulli was studied in the early days of ergodic theory. It was introduced by Kolmogorov under the name absolutely regular, and is called β– mixing in some circles of probabilists. Weak Bernoulli was known early on to imply Bernoulli. The fact that the two are not equivalent was proved by Smorodinsky [38]. The characterization of when RWRS is or is not weak Bernoulli turns out to be very interesting. Rather than describing the results in complete generality, we restrict to special classes of random walk. Theorem 2.5 (den Hollander and Steif [13]). (i) For simple random walk on Zd , the associated RWRS is weak Bernoulli if and only if d ≥ 5. (ii) For random walk with bounded step size and nonzero mean, the associated RWRS is weak Bernoulli for any d ≥ 1. The proof of Theorem 2.5 is based on a coupling argument where, given two independent pasts of RWRS, the futures are coupled so as to make them agree far out. Loosely speaking, weak Bernoulli is equivalent to such a coupling being possible. The phase transition in the behavior of RWRS as d increases from 4 to 5 is due to a fundamental difference between the behavior of simple random walk in 4 and 5 dimensions, a difference that is less well known than the fundamental recurrence/transience dichotomy between 2 and 3 dimensions. If we take two independent simple random walks and look at the intersection of their two trajectories, then this intersection almost surely is infinite for 1 ≤ d ≤ 4 but finite for d ≥ 5. This result is described in detail in Lawler [19], Section 3. The general necessary and sufficient condition for weak Bernoulli is that |S[0, ∞) ∩ S(−∞, 0]| < ∞ almost surely, i.e., the future and the past trajectories of the walk have a finite intersection, providing us with another nice dichotomy. For simple random walk, the latter condition is equivalent to d ≥ 5, as we just described, and explains Theorem 2.5(i). This part in fact extends to a random walk with zero mean and finite variance. If, on the other hand, the random walk has a drift, then the latter condition holds because in positive time the walk is moving in the opposite direction of where it is moving in negative time. This explains Theorem 2.5(ii). 2.2. Finitary factor We now move on to discussing when RWRS is a “finitary factor of an i.i.d. process”. The following serves as a crude definition. First, if we have an i.i.d. process W = (Wn )n∈Z , taking values in a finite set A, and a map f from AZ to B Z , with B another finite set, such that f is translation invariant, then we say that the stationary random process f (W ) is a factor of W . If, in addition, any fixed coordinate of the image process can be determined by knowing a sufficiently large but finite (in general random) number of coordinates of the domain process relative to the position of that coordinate, then the factor map is called finitary.

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It is well known and quite elementary to show that if a process is a finitary factor of an i.i.d. process and the so-called expected coding length (which, loosely speaking, is the expected number of bits in the domain process one needs to look at to determine a single bit of the image process) is finite, then the process is weak Bernoulli. Without the finite expected coding length assumption, this is not true. An example can be found for example in Burton and Steif [4]. As to whether being weak Bernoulli implies being a finitary factor of an i.i.d. process, del Junco and Rahe [15] constructed a counterexample. Interestingly, RWRS provides a natural counterexample. For the sake of exposition we restrict to simple random walk, but the result holds in much greater generality. Theorem 2.6 (Steif [39]). For simple random walk on Zd , the associated RWRS is not a finitary factor of an i.i.d. process for any d ≥ 1. Observe that for d = 1 this result follows from Theorem 2.2. However, the proof of Theorem 2.6 is much simpler than the proof of Theorem 2.2. Theorems 2.5 and 2.6 together tell us that RWRS associated with simple random walk on Zd , d ≥ 5, provides us with a natural example of a random process that is weak Bernoulli but not a finitary factor of an i.i.d. process. Two key facts are used in the proof of Theorem 2.6: (i) a finitary factor of an i.i.d. process must satisfy “standard large deviation behavior”, meaning that the mean ergodic theorem holds at an exponential rate (see Marton and Shields [28]); (ii) for simple random walk on Zd , d ≥ 1, the number of sites visited satisfies “nonstandard large deviation behavior” (see Donsker and Varadhan [5]), and hence so does the scenery record. It turns out that adding a drift to the random walk (which we saw causes the associated RWRS to become weak Bernoulli) results in it being a finitary factor of an i.i.d. process. Theorem 2.7 (Keane and Steif [18]). For nearest-neighbor random walk on Z with nonzero mean, the associated RWRS is a finitary factor of an i.i.d. process. We point out that this result rests on deep results in Rudolph [34], [35]. In a more direct approach, without the use of these latter papers it is possible to prove the weaker result that RWRS associated with nearest-neighbor random walk on Z with nonzero mean is a finitary factor of a countable state Markov chain that has “exponentially decaying return probabilities”. The proof of Theorem 2.7 exploits a certain type of “regenerative structure” that is present in a random walk with positive drift. This regenerative structure is seen when observing the random walk between the time it visits z for the last time and the time it visits z + 1 for the last time. It is indicated in [18] how to generalize Theorem 2.7 to a nearest-neighbor random walk on Zd with nonzero drift. We summarize the results presented so far. For simple random walk on Zd , the associated RWRS, while K is not Bernoulli for d = 1 and 2. When we move to d = 3 and 4, it becomes Bernoulli, but not yet weak Bernoulli. When we move to d ≥ 5, it becomes weak Bernoulli, but not yet a finitary factor of an i.i.d. process. Finally, when a drift is added, it becomes a finitary factor of an i.i.d. process. 2.3. Reduced RWRS Which of the results in Sections 2.1 and 2.2 survive when we look at the second coordinate of RWRS alone, i.e., the scenery record without the steps of the walk. It

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is evident that all “positive results” survive the reduction. However, it is not clear which “negative results” do. The following are two negative results that do survive the reduction, generalizing Theorems 2.2 and 2.6. Theorem 2.8 (Hoffman [10]). For simple random walk on Z, the associated reduced RWRS is not Bernoulli. (Hoffman actually proves that the process is not even “loosely Bernoulli” as Kalikow did.) Theorem 2.9 (Steif [39]). For simple random walk on Zd , the associated reduced RWRS is not a finitary factor of an i.i.d. process for any d ≥ 1. The proof of Theorem 2.8 consists of the following ingredients: (1) Kalikow’s result that RWRS is not (loosely) Bernoulli; (2) Matzinger’s result that “scenery reconstruction” is possible; (3) Thouvenot’s “relative isomorphism theory”; (4) Rudolph’s result that a “two-point weakly mixing extension” of a Bernoulli process is a Bernoulli process. Scenery reconstruction will be described in Section 3. Ingredient (2) is crucial because it makes up for the loss of the first coordinate when going from RWRS to reduced RWRS. Indeed, for recurrent random walk the combination of the walk and the scenery record in RWRS allows us to retrieve the full scenery from a single realization of RWRS. 2.4. Non-i.i.d. random scenery or random walk If the i.i.d. random scenery is replaced by a stationary random field, then the situation becomes more subtle. In Section 2.1, we already mentioned that Meilijson proved Theorem 2.1 under the much weaker assumption that the random scenery is totally ergodic. In den Hollander [11], it was pointed out that Theorem 2.1 holds if and only if the random scenery is ergodic w.r.t. the subgroup of translations generated by F − F = {z − z  : z, z  ∈ F }. Beyond this, results are so far limited. One early result is the following. Suppose that the random scenery is obtained by taking an irrational rotation of the circle and putting a +1 each time the top of the circle is hit and a −1 when the bottom half is hit. Suppose that the random walk stands still with probability 21 and moves one unit to the right with probability 1 2 . Then, as was shown by Adler and Shields [1], [2] using geometric arguments, the associated RWRS is Bernoulli. In Shields [37] a combinatorial proof was given, and it was proved that the associated RWRS is not weak Bernoulli. The following is a generalization of Theorem 2.5. Theorem 2.10 (den Hollander, Keane, Serafin and Steif [12]). (i) If the random scenery is non-atomic, then for simple random walk on Zd , 1 ≤ d ≤ 4, the associated RWRS is not weak Bernoulli. (ii) If the random scenery is the plus state of the low temperature Ising model, then for simple random walk on Zd , d ≥ 5, the associated RWRS is weak Bernoulli. To obtain Theorem 2.10(ii), one needs to be able to control the dependencies in the random field on pairs of infinite sets that are far away from each other. In the case of the low temperature Ising model, the relevant methods were developed in Burton and Steif [3] with the help of techniques from percolation. These methods can be carried over to a more general class of Markov random fields, leading to

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extensions of Theorem 2.10(ii) (see [12]). Van der Wal [40] has further extended Theorem 2.10(ii) to a class of random fields that are “sufficiently rapidly mixing”. This class includes the two-dimensional Ising model at arbitrary supercritical temperatures, as well as all d-dimensional Gibbs measures at sufficiently high temperatures. He also generalized Theorem 2.3 to random sceneries that are “exponentially mixing”, which includes these same sets of examples. Theorem 2.11 (van der Wal [40]). Under the same conditions as in Theorem 2.3, if the random scenery is exponentially mixing, then the associated RWRS is not Bernoulli. Kalikow [16] states that, once Theorem 2.2 is obtained, one can argue (using abstract ergodic theory) that the same result holds for a random scenery with positive entropy. As far as replacing the steps of the random walk by a stationary random process is concerned, results are again limited. We mention one key result, which generalizes Theorem 2.3. Theorem 2.12 (Steif [39]). Let X = (Xn )n∈Z be a stationary random process taking values in Zd such that S = (Sn )n∈Z is transient. If X is Bernoulli and the random scenery is i.i.d., then the associated RWRS is Bernoulli. The technique for proving Theorem 2.12 is very different from that of proving Theorem 2.4. The former is obtained by constructing an explicit factor map from an i.i.d. process to the RWRS, while the latter is obtained by verifying the coupling property that characterizes Bernoulli for RWRS: the so-called “very weak Bernoulli” property (see [13]). Much remains to be done in further relaxing the mixing conditions on scenery and walk. 2.5. Induced RWRS Consider reduced RWRS. Suppose that we condition on the scenery at the origin being +1, consider only those times at which a +1 appears in the scenery record, and report the scenery seen by the walk at such times, i.e.,

where

 = (C k )k∈Z C

with

k = (Cz+S )z∈Zd , C Tk

T0 = 0, Tk = inf{n > Tk−1 : Yn = +1}, k ∈ N, Tk = sup{n < Tk+1 : Yn = +1}, k ∈ −N.

 is called the induced RWRS (with {Y0 = +1} the induction set). The process C  is stationary. By Kakutani’s random ergodic theorem, if C is ergodic, Clearly, C  Mixing properties, however, are in general not inherited under inducthen so is C. tion. The following is a positive result. Here, the σ-field at infinity consists of those events that do not depend on Cz for z in any finite subset of Zd . Theorem 2.13 (den Hollander [11]). Suppose that C has a trivial σ-field at infinity. If Y1 is not constant almost surely, then induced RWRS is strongly mixing. The strong mixing property was first proved by Keane and den Hollander [17] for the case where C is i.i.d. and the random walk is transient. Their proof uses

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a specific coupling technique, which was extended in [11] to cover the general case stated in the theorem. The coupling is delicate especially for recurrent random walk. Later Georgii [9] used a stronger form of coupling, called orbit coupling, weakened the condition on C to it being ergodic w.r.t. the subgroup of translations generated  is by F − F = {z − z  : z, z  ∈ F }, and proved that under this weaker condition C a K-automorphism. 2.6. Conditional probabilities Let us return to i.i.d. scenery and walk. We next investigate continuity properties of conditional probabilities for RWRS. Given a general stationary random process W = (Wn )n∈Z taking values in {−1, +1}Z , we may ask whether there is a version V ( · | η) of the conditional probability distribution (P is the law of W ) P (W0 ∈ · | W = η on Z\{0}) ,

η ∈ {−1, +1}Z\{0} ,

such that the map η → V ( · | η) is continuous. If there is not, then we may ask whether there is a version that is continuous almost everywhere. These types of questions are of interest in probability theory and in statistical physics. Indeed, it turns out that many natural transformations acting on the set of stationary random sequences are capable of turning random sequences for which the first statement holds into random sequences for which not even the second statement holds. The history and recent developments of this research area are highlighted in the proceedings of a workshop held at EURANDOM in December 2003 (see van Enter, Le Ny and Redig [7]). To tackle the question about the existence of “nice” conditional probabilities, a key concept is the following. For n ∈ N, let Λn = [−n, n] ∩ Z. A configuration (Wn )n=0 is said to be a bad configuration for W0 if there is an  > 0 such that for all n ∈ N there are m ∈ N with m ≥ n and δ ∈ {−1, +1}Z\{0} with δ = η on Λn \{0} such that  P (W0 ∈ · | W = η on Λm \{0})  − P (W0 ∈ · | W = δ on Λm \{0})  ≥ ,

where · denotes the total variation norm. In words, by tampering with the configuration outside any given finite box, the conditional distribution of the coordinate at the origin can be nontrivially affected. A configuration that is not bad is called good. The importance of these notions is described in Maes, Redig and Van Moffaert [27], where it is shown that every version of the conditional probability distribution is discontinuous at all the bad configurations, while there exists a version that is continuous at all the good configurations. More details can be found in den Hollander, Steif and van der Wal [14]. Returning to RWRS, the following results show that interesting behavior occurs for the conditional probability distribution of Y0 (the scenery value at time 0) given (Zn )n=0 . Once more we restrict to special classes of random walks, though the results hold in much greater generality. Theorem 2.14 (den Hollander, Steif and van der Wal [14]). (i) For arbitrary random walk, the set of bad configurations (Zn )n=0 for Y0 is nonempty.

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(ii) For simple random walk on Zd , there is a version of the conditional probability distribution of Y0 given (Zn )n=0 that is continuous almost everywhere if and only if d = 1 or 2. If instead we consider the conditional probability distribution of X0 (the step at time 0) given (Zn )n=0 , then the answer is different. Theorem 2.15 (den Hollander, Steif and van der Wal [14]). (i) For arbitrary random walk, the set of bad configurations (Zn )n=0 for X0 is nonempty. (ii) For simple random walk on Zd , there is a version of the conditional probability distribution of X0 given (Zn )n=0 that is continuous almost everywhere if and only if 1 ≤ d ≤ 4. Theorems 2.14 and 2.15 rely on a full classification of the bad configurations. What drives these results are the same intersection properties of the random walk that are behind Theorem 2.5. We point out that, in Theorems 2.14 and 2.15, for the good configurations the random variable at the origin is determined by a sufficiently large piece of the good configuration. For example, any realization where the walker eventually returns to the origin allows us to read off the scenery value at the origin. Since the latter cannot change with any further information on the process after the return, this explains Theorem 2.14(ii). If we consider the conditional probability distribution of Y0 given (Yn )n=0 instead, i.e., we pose the continuity problem for reduced RWRS, then we believe that very different behavior occurs. Conjecture 2.16. Consider the random walk on Z with F = {−1, +1}, mF (+1) = p and mF (−1) = 1 − p, where p ∈ [ 12 , 1]. For p ∈ [ 12 , 54 ) every configuration (Yn )n=0 is bad for Y0 , while for p ∈ ( 45 , 1] every configuration (Yn )n=0 is good for Y0 . We are presently attempting to prove this conjecture. If true, it would provide us with a remarkable example where the continuity problem has a phase transition in the drift parameter. 3. Scenery reconstruction Scenery reconstruction is the problem of recovering C given only Y+ = (Yn )n∈N0 . In words, given a single forward realization of the scenery record (i.e., the scenery as seen through the eyes of the walker at nonnegative times), is it possible to reconstruct the full scenery without knowing the walk? Remarkably, the answer is “sometimes yes”. In this section we mention a number of results that have been obtained in past years. A detailed overview, including a description of the main techniques, is given in Lember and Matzinger [22], which also contains a full bibliography. The scenery reconstruction problem was raised by den Hollander and Keane and by Benjamini and Kesten in the mid 1980’s. Most of the progress on this difficult problem has been achieved only since the late 1990’s. A precise formulation of the scenery reconstruction problem requires that we make some fair restrictions: 1. Scenery reconstruction is not possible when the random walk is transient, because then almost surely there are sites that the walker never visits. 2. For simple random walk, any scenery that puts +1’s in the interval Λk , k ∈ N, cannot be distinguished from the scenery that is obtained by shifting it l units

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to the left or to the right with l ≤ k even. Similarly, if the random walk is reflection invariant, then any two sceneries that are reflections of each other cannot be distinguished. 3. Lindenstrauss [23] has constructed a countably infinite collection of onedimensional sceneries (all different under translation and reflection) that cannot be distinguished with simple random walk. The sceneries in this collection have a certain “self-similar structure” and therefore have measure zero under i.i.d. scenery processes. Thus, scenery reconstruction is at best possible for recurrent random walk, up to translation and reflection (in general), for almost every scenery and almost surely w.r.t. the walk. It turns out that these restrictions are enough, and so the problem can now be formalized, as follows. Recall that F and G are the sets of possible values of the walk increments, respectively, the scenery components. Two sceneries C and C  are said to be equivalent, written C ∼ C  , if they can be obtained from each other by a translation or reflection. Then scenery reconstruction is said to be possible when there exists a d measurable function A : GN0 → GZ , called a reconstruction algorithm, such that P (A(Y+ ) ∼ C) = 1. Several methods have been developed to deal with scenery reconstruction in different cases. These methods vary substantially with modifications of F and G. Roughly speaking, the larger F is, the harder the scenery reconstruction, while the larger G is, the easier the scenery reconstruction. See Lember and Matzinger [22] for insight into why. Most results so far are restricted to i.i.d. random walk and i.i.d. random scenery. We mention three key results, all for d = 1 and for mF , mG the uniform distribution on F, G. Theorem 3.1 (Matzinger [29], [30]). Scenery reconstruction is possible when F = {−1, +1} and |G| = 3, or F = {−1, 0, +1} and |G| = 2. Theorem 3.2 (L¨ owe, Matzinger and Merkl [26]). Scenery reconstruction is possible when |F | < |G|. Theorem 3.3 (Lember and Matzinger [20], [21]). Scenery reconstruction is possible when F
{−1, 0, +1} and |G| = 2. (The uniformity restrictions on mF and mG may be relaxed.) Matzinger and Rolles [32] have shown that a finite piece of scenery can be reconstructed in a time that is polynomial in the length of the piece. In addition, Matzinger and Rolles [31] have shown that scenery reconstruction is robust against errors: if the scenery record Y+ is perturbed by randomly changing each bit with a probability  > 0, then the scenery can still be reconstructed from the perturbed scenery record, provided  is small enough. Scenery reconstruction is particularly challenging in two dimensions. This is because recurrent random walk on Z2 returns to sites extremely slowly. Very little is known so far, the most notable result being due to L¨ owe and Matzinger [24], stating that scenery reconstruction is possible for simple random walk on Z2 when |G| is sufficiently large. The proof requires |G| to be very large. Very little progress has been made so far on scenery reconstruction for non-i.i.d. sceneries. One result can be found in L¨ owe and Matzinger [25]. Partial progress is

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underway for one-dimensional Gibbs sceneries by Lember and Matzinger (private communication). 4. Open questions We close by formulating a number of open questions: 1. Does the technical condition in Theorem 2.3 hold for arbitrary random walk, as conjectured in [13]? 2. Is there an analogue of Theorem 2.4 for transient random walk and non-i.i.d. random scenery similar in spirit to Theorem 2.11, which generalizes Theorem 2.3 to recurrent random walk and exponentially mixing random sceneries? 3. How far can Theorem 2.10(ii) be extended within the class of Gibbs random sceneries, in particular, in the non-uniqueness regime? 4. Let X = (Xn )n∈Z be a stationary random process taking values in Z with zero mean, implying that the random walk S = (Sn )n∈Z is recurrent (Durrett [6], Section 6.3). For i.i.d. scenery, under what conditions on X is the associated RWRS not Bernoulli? This would generalize both Theorem 2.2 and Theorem 2.12? Can it be Bernoulli? 5. When is the induced RWRS Bernoulli or weak Bernoulli? This would generalize Theorem 2.13 and its subsequent extension in [9]. 6. Is scenery reconstruction possible as soon as the entropy of mF is strictly smaller than that of mG , as conjectured in [26]? This would generalize Theorem 3.2. 7. Is scenery reconstruction possible for an arbitrary recurrent random walk with i.i.d. increments and an arbitrary i.i.d. random scenery? 8. To what extent can scenery reconstruction be carried through for non-i.i.d. random sceneries, e.g. Gibbs random sceneries? A final reference on RWRS of interest is Gantert, K¨ onig and Shi [8], where the small, moderate and large deviation behavior of sums of scenery values seen along the walk are reviewed. This paper includes many references to the relevant literature. References [1] Adler, R. L. and Shields, P. C. (1972). Skew products of Bernoulli shifts with rotations. Israel J. Math. 12, 215–222. MR315090 [2] Adler, R. L. and Shields, P. C. (1974). Skew products of Bernoulli shifts with rotations. II. Israel J. Math. 19, 228–236. MR377013 [3] Burton, R. M. and Steif, J. E. (1995). Quite weak Bernoulli with exponential rate and percolation for random fields. Stochastic Process. Appl. 58, 1, 35–55. MR1341553 [4] Burton, R. M. and Steif, J. E. (1997). Coupling surfaces and weak Bernoulli in one and higher dimensions. Adv. Math. 132, 1, 1–23. MR1488237 [5] Donsker, M. D. and Varadhan, S. R. S. (1979). On the number of distinct sites visited by a random walk. Comm. Pure Appl. Math. 32, 6, 721–747. MR539157 [6] Durrett, R. (1996). Probability: Theory and Examples. Duxbury Press, Belmont, CA. MR1609153

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[7] van Enter, A. C. D., Redig, F., and Le Ny, A. (2004). Foreword: [Special issue containing proceedings of the Workshop “Gibbs vs. non-Gibbs” in Statistical Mechanics and Related Fields]. Markov Process. Related Fields 10, 3, 377–379. MR2097862 ¨ nig, W., and Shi, Z. (2004). Annealed deviations of ran[8] Gantert, N., Ko dom walk in random scenery. Preprint. [9] Georgii, H.-O. (1997). Mixing properties of induced random transformations. Ergodic Theory Dynam. Systems 17, 4, 839–847. MR1468103 [10] Hoffman, C. (2003). The scenery factor of the [T, T −1 ] transformation is not loosely Bernoulli. Proc. Amer. Math. Soc. 131, 12, 3731–3735 (electronic). MR1998180 [11] den Hollander, W. T. F. (1988). Mixing properties for random walk in random scenery. Ann. Probab. 16, 4, 1788–1802. MR958216 [12] den Hollander, F., Keane, M. S., Serafin, J., and Steif, J. E. (2003). Weak Bernoullicity of random walk in random scenery. Japan. J. Math. (N.S.) 29, 2, 389–406. MR2036267 [13] den Hollander, F. and Steif, J. E. (1997). Mixing properties of the generalized T, T −1 -process. J. Anal. Math. 72, 165–202. MR1482994 [14] den Hollander, F., Steif, J. E., and van der Wal, P. (2005). Bad configurations for random walk in random scenery and related subshifts, Stochastic Process. Appl. 115, no. 7, 1209–1232. MR2147247 [15] del Junco, A. and Rahe, M. (1979). Finitary codings and weak Bernoulli partitions. Proc. Amer. Math. Soc. 75, 2, 259–264. MR532147 [16] Kalikow, S. A. (1982). T, T −1 transformation is not loosely Bernoulli. Ann. of Math. (2) 115, 2, 393–409. MR647812 [17] Keane, M. and den Hollander, W. T. F. (1986). Ergodic properties of color records. Phys. A 138, 1-2, 183–193. MR865242 [18] Keane, M. and Steif, J. E. (2003). Finitary coding for the one-dimensional T, T −1 process with drift. Ann. Probab. 31, 4, 1979–1985. MR2016608 [19] Lawler, G. F. (1991). Intersections of Random Walks. Probability and its Applications. Birkh¨ auser Boston Inc., Boston, MA. MR1117680 [20] Lember, J. and Matzinger H. (2002). A localization test for observation, EURANDOM Report 2002-014. [21] Lember, J. and Matzinger H. (2002). Reconstructing a piece of 2-color scenery, EURANDOM Report 2002-042. [22] Lember, J. and Matzinger H. (2005). Scenery reconstruction: an overview, manuscript in preparation. [23] Lindenstrauss, E. (1999). Indistinguishable sceneries. Random Structures Algorithms 14, 1, 71–86. MR1662199 ¨ we, M. and Matzinger, III, H. (2002). Scenery reconstruction in two di[24] Lo mensions with many colors. Ann. Appl. Probab. 12, 4, 1322–1347. MR1936595 ¨ we, M. and Matzinger, III, H. (2003). Reconstruction of sceneries with [25] Lo correlated colors. Stochastic Process. Appl. 105, 2, 175–210. MR1978654 ¨ we, M., Matzinger, H., and Merkl, F. (2004). Reconstructing a mul[26] Lo ticolor random scenery seen along a random walk path with bounded jumps. Electron. J. Probab. 9, no. 15, 436–507 (electronic). MR2080606 [27] Maes, C., Redig, F., and Van Moffaert, A. (1999). Almost Gibbsian versus weakly Gibbsian measures. Stochastic Process. Appl. 79, 1, 1–15. MR1666839 [28] Marton, K. and Shields, P. C. (1994). The positive-divergence and blowing-up properties. Israel J. Math. 86, 1–3, 331–348. MR1276142

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[29] Matzinger H. (1999), Reconstructing a 2-color scenery by observing it along a simple random walk. PhD thesis, Cornell University. [30] Matzinger, H. (1999). Reconstructing a three-color scenery by observing it along a simple random walk path. Random Structures Algorithms 15, 2, 196–207. MR1704344 [31] Matzinger, H. and Rolles, S. W. W. (2003). Reconstructing a random scenery observed with random errors along a random walk path. Probab. Theory Related Fields 125, 4, 539–577. MR1974414 [32] Matzinger, H. and Rolles, S. W. W. (2003). Reconstructing a piece of scenery with polynomially many observations. Stochastic Process. Appl. 107, 2, 289–300. MR1999792 [33] Meilijson, I. (1974). Mixing properties of a class of skew-products. Israel J. Math. 19, 266–270. MR372158 [34] Rudolph, D. J. (1981). A characterization of those processes finitarily isomorphic to a Bernoulli shift. In Ergodic Theory and Dynamical Systems, I (College Park, Md., 1979–80). Progr. Math., Vol. 10. Birkh¨ auser Boston, MA., 1–64. MR633760 [35] Rudolph, D. J. (1982). A mixing Markov chain with exponentially decaying return times is finitarily Bernoulli. Ergodic Theory Dynamical Systems 2, 1, 85–97. MR684246 [36] Rudolph, D. J. (1986). Zn and Rn cocycle extensions and complementary algebras. Ergodic Theory Dynam. Systems 6, 4, 583–599. MR873434 [37] Shields, P. C. (1977). Weak and very weak Bernoulli partitions. Monatsh. Math. 84, 2, 133–142. MR476997 [38] Smorodinsky, M. (1971). A partition on a Bernoulli shift which is not weakly Bernoulli. Math. Systems Theory 5, 201–203. MR297971 [39] Steif, J. E. (2001). The T, T −1 process, finitary codings and weak Bernoulli. Israel J. Math. 125, 29–43. MR1853803 [40] van der Wal, P. (2003). (Weak) Bernoullicity of random walk in exponentially mixing random scenery, EURANDOM Report 2003–032, to appear in Ann. Prob. [41] Walters, P. (1975). Ergodic Theory — Introductory Lectures. SpringerVerlag, Berlin. MR480949 [42] Weiss, B. (1972). The isomorphism problem in ergodic theory. Bull. Amer. Math. Soc. 78, 668–684. MR304616

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 66–77 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000103

Linearly edge-reinforced random walks Franz Merkl1 and Silke W. W. Rolles2 University of Munich and University of Bielefeld Abstract: We review results on linearly edge-reinforced random walks. On finite graphs, the process has the same distribution as a mixture of reversible Markov chains. This has applications in Bayesian statistics and it has been used in studying the random walk on infinite graphs. On trees, one has a representation as a random walk in an independent random environment. We review recent results for the random walk on ladders: recurrence, a representation as a random walk in a random environment, and estimates for the position of the random walker.

1. Introduction Consider a locally finite graph G = (V, E) with undirected edges. All the edges are assigned weights which change in time. At each discrete time step, the edgereinforced random walker jumps to a neighboring vertex with probability proportional to the weight of the traversed edge. As soon as an edge is traversed, its weight is increased by one. The model was introduced by Diaconis in 1986 (see [11] and [3]). Mike Keane made the model popular in The Netherlands. In particular, Mike Keane introduced Silke Rolles to the intriguing questions raised by this model. One fundamental question, asked by Diaconis, concerns recurrence: Do almost all paths of the edgereinforced random walk visit all vertices infinitely often? By a Borel-Cantelli argument, this is equivalent to the following question: Does the edge-reinforced random walker return to the starting point infinitely often with probability one? For all dimensions d ≥ 2, it is an open problem to prove or disprove recurrence on Zd . Only recently, recurrence of the edge-reinforced random walk on ladders has been proven. The present article focuses on linear edge-reinforcement as described above. In the past two decades, many different reinforcement schemes have been studied. We briefly mention some of them: Vertex-reinforced random walk was introduced by Pemantle [25] and has been further analyzed by Bena¨ım [1], Dai ([5], [6]), Pemantle and Volkov ([27], [38]), and Tarr`es [35]. Sellke [30] and Vervoort [37] proved recurrence results for once-reinforced random walks on ladders; Durrett, Kesten, and Limic [13] analyzed the process on regular trees. Recurrence questions for reinforced random walks of sequence type were studied by Davis ([7], [8]) and Sellke [31]. Takeshima ([33], [34]) studied hitting times and recurrence for reinforced random walk of matrix type. Limic [18] proved a localization theorem for superlinear reinforcement. Weakly reinforced random walks in one dimension were studied by 1 Mathematical

Institute, University of Munich, Theresienstr. 39, D-80333 Munich, Germany, e-mail: [email protected] 2 Department of Mathematics, University of Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany, e-mail: [email protected] AMS 2000 subject classifications: primary 82B41; secondary 60K35, 60K37. Keywords and phrases: reinforced random walk, random environment, recurrence, Bayesian statistics. 66

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T´oth [36]. Directionally reinforced random walks are the subject of Mauldin, Monticino, and von Weizs¨ acker [20] and Horv´ ath and Shao [14]. Othmer and Stevens [23] suggested linearly edge-reinforced random walks as a simple model for the gliding of myxobacteria. These bacteria produce a slime and prefer to move on the slime trail produced earlier. Overviews with a different emphasis than the present article have been written by Davis [9] and Pemantle [26]. The article is organized as follows: In Section 2, we give a formal definition of the edge-reinforced random walk. On finite graphs, the edge-reinforced random walk has the same distribution as a random walk in a dependent random environment. This representation together with related limit theorems is presented in Section 3. The edge-reinforced random walk on finite graphs gives rise to a family of prior distributions for reversible Markov chains. This application to statistics is the content of Section 4. A characterization of the process (which is also applied in the Bayesian context) is given in Section 5. In Section 6, we state results for the process on acyclic graphs. Recently, progress has been made in understanding the edge-reinforced random walk on graphs of the form Z × {1, . . . , d} and, more generally, on Z × T when T is a finite tree. Section 7 reviews these recent results and presents some simulations. 2. Formal description of the model Let G = (V, E) be a locally finite undirected graph. Every edge is assumed to have two different endpoints; thus there are no direct loops. We identify an edge with the set of its endpoints. Formally, the edge-reinforced random walk on G is defined as follows: Let Xt denote the random location of the random walker at time t. Let ae > 0, e ∈ E. For t ∈ N0 , we define wt (e), the weight of edge e at time t, recursively as follows: w0 (e) :=ae for all e ∈ E,
wt (e) + 1 for e = {Xt , Xt+1 } ∈ E, wt+1 (e) := wt (e) for e ∈ E \ {{Xt , Xt+1 }}.

(2.1) (2.2)

Let Pv0 ,a denote the distribution of the edge-reinforced random walk on G starting in v0 with initial edge weights equal to a = (ae )e∈E . The distribution Pv0 ,a is a probability measure on V N0 , specified by the following requirements: X0 = v0

Pv0 ,a -a.s.,

Pv0 ,a (Xt+1

   wt ({Xt , v}) = v|Xi , i = 0, 1, . . . , t) = {e∈E:Xt ∈e} wt (e)  0

(2.3) if {Xt , v} ∈ E, otherwise. (2.4)

3. Reinforced random walk on finite graphs Throughout this section, we assume the graph G to be finite. For t ∈ N and e ∈ E, set kt (e) := wt (e) − ae

and

αt (e) :=

kt (e) . t

(3.1)

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In particular, αt (e) denotes the proportion of crossings of the edge e up to time t. The random vector αt := (αt (e))e∈E takes values in the simplex    E ∆ := (xe )e∈E ∈ (0, ∞) : xe = 1 . (3.2) e∈E

For x ∈ (0, ∞)E and v ∈ V , we define xv :=



xe .

(3.3)

e∈E:v∈e

Now view the graph G as an electric network, and consider the space H1 of all electric current distributions (ye )e∈E on the graph, such that Kirchhoff’s vertex rule holds at all vertices: For every vertex, the sum of all ingoing currents should equal the sum of all outgoing currents. More formally, we proceed as follows: We assign a counting direction to every edge, encoded in a signed incidence matrix s = (sve )v∈V,e∈E ∈ {−1, 0, 1}V ×E . Here, sve = +1 means that e is an ingoing edge into the vertex v; sve = −1 means that e is an outgoing edge from vertex v, and sve = 0 means that the vertex v is not incident to the edge e. Then, for a given current distribution y = (ye )e∈E ∈ RE , the current balance at any vertex v ∈ V equals  sve ye , (3.4) (sy)v = e∈E

and the space of all current distributions satisfying Kirchhoff’s vertex rule equals H1 = kernel(s) = {y ∈ RE : sy = 0}    = (ye )e∈E ∈ RE : sve ye = 0 for all v ∈ V .

(3.5)

e∈E

In other words, H1 is the first homology space of the graph G. Note that dim H1 = |E| − |V | + 1 by Euler’s rule. Interpreting weights x = (xe )e∈E ∈ (0, ∞)E as electric conductivities of the edges, the power consumption of the network required to support a current distribution y ∈ H1 equals  y2 e . Ax (y) = xe e∈E

(Of course, in order to drive this current in a real network, appropriate batteries must be wired into the edges. We ignore this fact.) Note that Ax is a quadratic form on H1 . We will also need its determinant det Ax , which is defined as the determinant of the matrix representing Ax with respect to any Z-basis of the lattice H1 ∩ ZE . Note that the determinant det Ax does not depend on the choice of the basis, since any base change has determinant ±1. We endow the homology space H1 with the Lebesgue measure, normalized such that the unit cell spanned by any Z-basis of H1 ∩ ZE gets volume 1. It turns out that the sequence (αt )t∈N converges almost surely to a random limit. Surprisingly, the limiting distribution can be determined explicitly: Theorem 3.1. The sequence (αt )t∈N converges Pv0 ,a -almost surely. The distribution of the limit is absolutely continuous with respect to the surface measure on ∆

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with density given by

φv0 ,a (x) = Zv−1 0 ,a



a − 21

xe e

e∈E av0 2 v0

x



av +1 2

xv



det Ax ,

x = (xe )e∈E ∈ ∆,

(3.6)

v∈V \{v0 }

where Zv0 ,a denotes a normalizing constant. Theorem 3.1 was discovered by Coppersmith and Diaconis [3]. A proof of the result was independently found by Keane and Rolles ([16], Theorem 1). In the special case of a triangle, a derivation of Theorem 3.1 was published by Keane [15]. The normalizing constant Zv0 ,a is explicitly known; see [16]. Let S denote the set of all subtrees T in G, viewed as set of edges T ⊆ E, which need not necessarily visit all vertices. The set of all spanning trees in G is denoted by T ; it is a subset of S. Using a matrix tree theorem (see e.g. [19], page 145, theorem 3’), one can rewrite the determinant of Ax as a sum over all spanning trees. This yields the following representation of the density: ae −1 xe  e∈E −1 φv0 ,a (x) = Zv0 ,a av0 xe . (3.7) av +1 xv 2 xv02 T ∈T e∈T v∈V \{v0 }

Given a path π = (v0 , v1 , . . . , vt ) of vertices in G, we define its corresponding “chain” [π] = ([π]e )e∈E ∈ RE as follows: Imagine an electric current of size 1 entering the network G at the starting vertex v0 , flowing along π, and then leaving the network at the last vertex vt . Let [π] ∈ RE denote the corresponding current distribution. More formally, for e ∈ E, [π]e equals the number of times the path π traverses the edge e in counting direction minus the number of times the path π traverses the edge e in opposite direction to the counting direction. Note that [π] ∈ / H1 unless the path π is closed, since Kirchhoff’s vertex rule is violated at the starting point v0 and at the last vertex vt . Now, for any time t, let πt = (X0 , X1 , . . . , Xt ) denote the random path the reinforced random walker follows up to time t. We define 1 (3.8) βt = √ [πt ] ∈ RE t to be the √ corresponding rescaled (random) current distribution. Indeed, the diffusive scale t turns out to be appropriate for studying the random currents [πt ] in the limit as t → ∞. Note that in general βt ∈ / H1 , due to the violation of Kirchhoff’s rule at the starting vertex X0 and the last vertex Xt . However, any weak limit of βt as t → ∞, if it exists, must be supported on the homology space H1 ⊆ RE . Indeed, the violation of Kirchhoff’s rule at any vertex is not larger than t−1/2 , which is negligible as t → ∞. For a (random) path πt = (X0 , . . . , Xt ) and any vertex v among {X0 , . . . , Xt−1 }, let elast exit (v, πt ) denote the edge by which v is left when it is visited by πt for the last time. Let Ttlast exit ∈ S denote the random tree graph consisting of all the edges elast exit (v, πt ), v ∈ {X0 , . . . , Xt−1 }. Since the reinforced random walk on the finite graph G visits every vertex almost surely, Ttlast exit ∈ T holds for all large t almost surely.

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Theorem 3.2. The sequence (αt , βt , Ttlast exit )t∈N converges weakly in RE ×RE ×S. The limiting distribution Q is supported on the subset ∆ × H1 × T of RE × RE × S. It is absolutely continuous with respect to the product of the surface measure on ∆, the Lebesgue measure on H1 , and the counting measure on T . Its density is given by

Φv0 ,a (x, y, T ) = Z˜v−1 0 ,a



a − 32

xe e

e∈E av0 2 v0

x



av +1 2

xv



e∈T

xe



  1 exp − Ax (y) , 2

(3.9)

v∈V \{v0 }

where Z˜v0 ,a denotes a normalizing constant. Theorem 1 of [16] states weak convergence of (αt , βt )t∈N to the limiting distribution with density  Φv0 ,a (x, y, T ). (3.10) T ∈T

However, the proof implicitly contains the proof of Theorem 3.2. Recall that the transition probabilities of any irreducible reversible Markov chain on G can be described by weights x = (xe )e∈E , xe ≥ 0, on the edges of the graph; the probability to traverse an edge is proportional to its weight. More precisely, denoting the distribution of the Markov chain induced by the edge weights x with starting vertex v0 by Qv0 ,x , one has Qv0 ,x (Xt+1 = v  |Xt = v) =

x{v,v
} , xv

(3.11)

whenever {v, v  } is an edge; the weight xv is defined in (3.3). The following representation of the edge-reinforced random walk on a finite graph as a mixture of reversible Markov chains is shown in Theorem 3.1 of [28]. Theorem 3.3. For any event B ⊆ V N0 , one has  Pv0 ,a ((Xt )t∈N0 ∈ B) = Qv0 ,x (B) Q(dx dy dT ),

(3.12)

∆×H1 ×T

where Q denotes the limiting measure from Theorem 3.2. In other words, the edgereinforced random walk on any finite graph G has the same distribution as a random walk in a random environment. The latter is given by random weights on the edges, distributed according to the limiting distribution of (αt )t∈N , namely the distribution with density φv0 ,a with respect to the Lebesgue measure on the simplex. This representation as a random walk in a random environment has been extremly useful in studying the edge-reinforced random walk on infinite ladders. The proof of Theorem 3.3 relies on the fact that the edge-reinforced random walk is partially exchangeable: the probability that the process traverses a particular finite path π = (v0 , . . . , vt ) depends only on the starting point of π and the number of transition counts of the undirected edges. If one knows that the process returns to the starting point infinitely often with probability one (which is the case for the edge-reinforced random walk on a finite graph), one can apply a de Finetti Theorem for Markov chains of Diaconis and Freedman [10]. A refinement for reversible Markov chains ([28], Theorem 1.1) yields a mixture of reversible chains.

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4. An application to Bayesian statistics Consider the following statistical situation: We observe X0 = v0 , X1 = v1 , . . ., Xt = vt generated by a reversible Markov chain. The transition kernel k(·, ·) and the stationary measure ν are unknown. Let V be the set of possible observations. We assume V to be a known finite set. Furthermore, we assume that k(v, v  ) > 0 if and only if k(v  , v) > 0. Hence, V together with the set E := {{v, v  } : k(v, v  ) > 0} defines a finite undirected graph. This graph is assumed to be known. It is a natural question how to model this in the framework of Bayesian statistics. One is interested in “natural” prior distributions on the set of reversible Markov chains. Because of reversibility, ν(v)k(v, v  ) = ν(v  )k(v  , v) for all v, v  ∈ V . The distribution of a reversible Markov chain with transition kernel k, stationary distribution ν, and starting point v0 is given by Qv0 ,x as defined in (3.11) with edge weights x{v,v
} = ν(v)k(v, v  ). Thus, one can describe the prior distributions as measures on the set of possible edge weights, namely as measures on ∆. The following theorem was proved by Diaconis and Rolles (Proposition 4.1 of [12]): Theorem 4.1. Let σ denote the Lebesgue measure on ∆ and set Pv0 ,a := φv0 ,a dσ. The family   Pv0 ,a : v0 ∈ V, a = (ae )e∈E ∈ (0, ∞)E (4.1)

of prior distributions is closed under sampling. More precisely, under the prior distribution Pv0 ,a with observations X0 = v0 , X1 = v1 , . . ., Xt = vt , the posterior is given by Pvt ,(ae +kt (e))e∈E , where kt (e) denotes the number of i with {vi , vi+1 } = e.

These prior distributions were further analyzed in [12]: They can be generalized to finite graphs with direct loops; thus one can include the case k(v, v) > 0 for some v. The set of linear combinations of the priors Pv0 ,a is weak-star dense in the set of all prior distributions on reversible Markov chains on G. Furthermore, it is shown that these prior distributions allow to perform tests: several hypotheses are tested for a data set of length 3370 arising from the DNA sequence of the humane HLA-B gene, for instance H0 : i.i.d.(unknown) versus H1 : reversible Markov chain and H0 : reversible Markov chain versus H1 : full Markov. The tests are based on the Bayes factor P (data|H0 )/P (data|H1 ) which can be easily computed. The priors Pv0 ,a generalize the well-known Dirichlet priors. The latter are obtained as a special case for star-shaped graphs. The Dirichlet prior was characterized by W.E. Johnson; see [39]. In [12], a similar characterization is given for the priors Pv0 ,a ; in this sense they are “natural”. 5. A characterization of reinforced random walk In this section, we review a characterization of the edge-reinforced random walk from [28]. We need some assumptions on the underlying graph G: Assumption 5.1. For all v ∈ V , degree(v) = 2. Furthermore, the graph G is 2-edge-connected, i.e. removing an edge does not make G disconnected. Let P be the distribution of a nearest-neighbor random walk on G such that the following hold: Assumption 5.2. There exists v0 ∈ V with P (X0 = v0 ) = 1.

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Assumption 5.3. For any admissible path π of length t ≥ 1 starting at v0 , we have P ((X0 , . . . , Xt ) = π) > 0. Assumption 5.4. The process (Xt )t∈N0 with distribution P is partially exchangeable. For t ∈ N0 , v ∈ V , and e ∈ E, we define kt (v) := |{i ∈ {0, . . . , t} : Xi = v}|,

kt (e) := |{i ∈ {1, . . . , t} : {Xi−1 , Xi } = e}|.

Assumption 5.5. For all v ∈ V and e ∈ E, there exists a function fv,e taking values in [0, 1] such that for all t ∈ N0 P (Xt+1 = v|X0 , . . . , Xt ) = fXt ,{Xt ,v} (kt (Xt ), kt ({Xt , v})). In other words, the conditional distribution for the next step, given the past up to time t, depends only on the position Xt at time t, the edge {Xt , v} to be traversed, the local time accumulated at Xt , and the local time on the edge {Xt , v}. It is not hard to see that an edge-reinforced random walk and a non-reinforced random walk starting at v0 satisfy Assumptions 5.2–5.5. The following theorem is the content of Theorem 1.2 of [28]: Theorem 5.1. Suppose the graph G satisfies Assumption 5.1. If P is the distribution of a nearest-neighbor random walk on G satisfying Assumptions 5.2–5.5, then, for all t, P (Xt+1 = v|X0 , . . . , Xt ) agrees on the set {kt (Xt ) ≥ 3} with the corresponding conditional probability for an edge-reinforced random walk or a non-reinforced random walk starting at v0 . In this sense, the above assumptions characterize the edge-reinforced random walk. Theorem 5.1 is used to give a charactarization of the priors Pv0 ,a . 6. Reinforced random walk on acyclic graphs The edge-reinforced random walk on acyclic graphs is much easier to analyze than on graphs with cycles. Let us briefly explain why: Consider edge-reinforced random walk on a tree. If the random walker leaves vertex v via the neighboring vertex v  , then in case the random walker ever returns to v, the next time she does so, she has to enter via the same edge {v, v  }. Hence, if the random walker leaves v via the edge {v, v  }, the weight of this edge will have increased by precisely 2, the next time the random walker arrives at v. Obviously, this is only true on an acyclic graph. Due to this observation, instead of recording the edge weights, one can place Polya urns at the vertices of the tree. Each time the random walker is at location v, a ball is drawn from the urn U (v) attached to v. Then, the ball is put back together with two balls of the same color. The different colors represent the edges incident to v; the number of balls in the urn equal the weights of the edges incident to v, observed at times when the random walker is at v. The initial composition of the urns is determined by the starting point and the initial edge weights of the reinforced random walk. The sequence of drawings from the urn U (v) at v is independent from the sequences of drawings from the urns U (v  ), v = v  , at all other locations. Using de Finetti’s theorem, one finds that the edge-reinforced random walk has the same

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distribution as a random walk in a random environment, where the latter is given by independent Dirichlet-distributed transition probabilities. This representation as a random walk in a random environment was observed by Pemantle [24]. It seems to be the most powerful tool to analyze the process on acyclic graphs. Pemantle used it to prove a phase transition in the recurrence/transience behavior on a binary tree. Later, Takeshima [33] characterized recurrence in one dimension for spaceinhomogeneous initial weights. (For all initial weights being equal, recurrence in one dimension follows for instance from a well-known recurrence criterion for random walk in random environment; see e.g. [32].) Collevecchio [4] proved a law of large numbers and a central limit theorem for the edge-reinforced random walk on b-ary trees, under the assumption that b is large. For linear edge-reinforcement on arbitrary directed graphs, so called directededge-reinforced random walk, a similar correspondance with a random walk in an independent random environment can be established. Using this correspondance, recurrence for Z × G for any finite graph G was proved by Keane and Rolles in [17]. A different criterion for recurrence and transience of a random walk in an independent random environment on a strip was established by Bolthausen and Goldsheid [2]. 7. Reinforced random walk on ladders Studying reinforced random walks on graphs of the form Z × {1, . . . , d} seems to be a challenging task. Only recently, recurrence results were obtained. The following result was proved in [22]. Theorem 7.1. For all a > 3/4, the edge-reinforced random walk on Z × {1, 2} with all initial weights equal to a is recurrent. The result was extended in [29], under the assumption the initial weight a is large: Theorem 7.2. Let G be a finite tree. For all large enough a, the edge-reinforced random walk on Z × G with all initial weights equal to a is recurrent. In particular, this applies to ladders Z × {1, 2, . . . , d} of any finite width d. In the following, we consider the process on N0 × G or Z × G with a finite tree G. We always assume that all initial edge weights are equal to the same large enough constant a and the random walk starts in a vertex 0 at level 0. Just as edge-reinforced random walk on finite graphs, the edge-reinforced random walk on infinite ladders turns out to be equivalent to a random walk in a random environment, as studied in Section 3: In particular, there is an infinitevolume analogue to Theorem 3.3 for the infinite graphs N0 × G and Z × G. Here, the law Q of the random environment in Theorem 3.3 gets replaced by an infinitevolume Gibbs measure, which we also denote by Q. Also in the infinite-volume setup, the fractions αt ∈ RE of times spent on the edges converge almost surely to random weights x ∈ RE as t → ∞, just as in Theorem 3.1. These random weights x are governed by the infinite-volume Gibbs measure Q. It turns out that Q-almost surely, the random weights decrease exponentially in space, i.e. Q-a.s.

lim sup |e|→∞

1 log xe ≤ −c(a, G) |e|

(7.1)

with a deterministic constant c(G, a) > 0. Here, |e| denotes the distance of an edge e from the starting point. For more details, see [21].

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Fig 1. Fraction of time spent on the edges on a logarithmic scale. Time horizon = 109 , initial weight a = 1.

This exponential decay of the weights x can be also observed in simulations: For the ladder Z × {1, 2, . . . , 30}, the simulation in Figure 1 shows one typical sample for the fraction of time αt spent on any edge up to time t = 109 . The fractions αt are displayed logarithmically as gray scales. The starting point is located in the center of the picture. The time t = 109 in the simulation is already so large that this αt is a good approximation for a typical sample of the random weights x. Note that the fractions αt in the simulation vary over many orders of magnitude, and that they decrease roughly exponentially as one gets farther away from the starting point. Entropy estimates and deformation arguments from statistical mechanics are used in [21] to derive the exponential decay of the weights. As a consequence, one obtains the following estimates for the position of the random walker, also proved in [21]: Theorem 7.3. There exist constants c1 , c2 > 0, depending only on G and a, such that for all t, n ∈ N0 , the following bound holds: P0,a (|Xt | ≥ n) ≤ c1 e−c2 n .

(7.2)

Note that the bounds are uniform in the time t. This is √ different from the behavior of simple random walk, which has fluctuations of order t. Corollary 7.1. There exists a constant c3 = c3 (G, a) > 0 such that P0,a -a.s., max |Xs | ≤ c3 ln t

s=0,...,t

for all t large enough.

(7.3)

A simulation shown in Figure 2 illustrates this corollary: For one sample path of an edge-reinforced random walk on the ladder Z × {1, 2, . . . , 30} with initial weight a = 1, the farthest level reached so far, maxs=0,...,t |Xs |, is displayed as a function of t. Note that the time t is plotted on a logarithmic scale. Figure 3 shows a simulation of reinforced random walk on Z2 with initial weights a = 1. It is still unknown whether this reinforced random walk is recurrent. Simulations show that there are random regions in Z2 , maybe in far distance from the starting point, which are visited much more frequently than other regions closer to the starting point. It remains unclear whether more and more extreme “favorable regions” arise farther away from the origin. Thus, the recurrence problem in Z2 remains open, even on a heuristic level. Acknowledgment We would like to thank Frank den Hollander and Guido Elsner for carefully reading our manuscript.

Linearly edge-reinforced random walks

Fig 2. Maximal distance from the starting point as a function of time. Initial weight a = 1.

Fig 3. Reinforced random walk on Z2 .

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References [1] Bena¨ım, M. (1997). Vertex-reinforced random walks and a conjecture of Pemantle. Ann. Probab. 25, 1, 361–392. MR1428513 [2] Bolthausen, E. and Goldsheid, I. (2000). Recurrence and transience of random walks in random environments on a strip. Comm. Math. Phys. 214, no. 2, 429–447. MR1796029 [3] Coppersmith, D. and Diaconis, P. (1986) Random walk with reinforcement. Unpublished manuscript. [4] Collevecchio, A. (2005). Limit theorems for Diaconis walk on certain trees. Preprint. [5] Dai, J. J. (2003). A note on vertex-reinforced random walks. Statist. Probab. Lett. 62, 3, 275–280. MR1966379 [6] Dai, J. J. (2004). Some results regarding vertex-reinforced random walks. Statist. Probab. Lett. 66, 3, 259–266. MR2045471 [7] Davis, B. (1989). Loss of recurrence in reinforced random walk. In Almost everywhere convergence (Columbus, OH, 1988). Academic Press, Boston, MA, 179–188. MR1035245 [8] Davis, B. (1990). Reinforced random walk. Probab. Theory Related Fields 84, 2, 203–229. MR1030727 [9] Davis, B. (1999). Reinforced and perturbed random walks. In Random walks (Budapest, 1998). Bolyai Soc. Math. Stud., Vol. 9. J´ anos Bolyai Math. Soc., Budapest, 113–126. MR1752892 [10] Diaconis, P. and Freedman, D. (1980). de Finetti’s theorem for Markov chains. Ann. Probab. 8, 1, 115–130. MR556418 [11] Diaconis, P. (1988). Recent progress on de Finetti’s notions of exchangeability. In Bayesian Statistics, 3 (Valencia, 1987). Oxford Sci. Publ. Oxford Univ. Press, New York, 111–125. MR1008047 [12] Diaconis, P. and Rolles, S.W.W. (2006). Bayesian analysis for reversible Markov chains. The Annals of Statistics 34, no. 3. [13] Durrett, R., Kesten, H., and Limic, V. (2002). Once edge-reinforced random walk on a tree. Probab. Theory Related Fields 122, 4, 567–592. MR1902191 ´th, L. and Shao, Q.-M. (1998). Limit distributions of directionally [14] Horva reinforced random walks. Adv. Math. 134, 2, 367–383. MR1617789 [15] Keane, M. S. (1990). Solution to problem 288. Statistica Neerlandica 44, 2, 95–100. [16] Keane, M. S. and Rolles, S. W. W. (2000). Edge-reinforced random walk on finite graphs. In Infinite Dimensional Stochastic Analysis (Amsterdam, 1999). Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet., Vol. 52. R. Neth. Acad. Arts Sci., Amsterdam, 217–234. MR1832379 [17] Keane, M. S. and Rolles, S. W. W. (2002). Tubular recurrence. Acta Math. Hungar. 97, 3, 207–221. MR1933730 [18] Limic, V. (2003). Attracting edge property for a class of reinforced random walks. Ann. Probab. 31, 3, 1615–1654. MR1989445 [19] Maurer, S. B. (1976). Matrix generalizations of some theorems on trees, cycles and cocycles in graphs. SIAM J. Appl. Math. 30, 1, 143–148. MR392635 ¨cker, H. (1996). Di[20] Mauldin, R. D., Monticino, M., and von Weizsa rectionally reinforced random walks. Adv. Math. 117, 2, 239–252. MR1371652 [21] Merkl, F. and Rolles, S.W.W. (2004). Asymptotic behavior of edgereinforced random walks. To appear in The Annals of Probability.

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[22] Merkl, F. and Rolles, S.W.W. (2005). Edge-reinforced random walk on a ladder. The Annals of Probability 33, 6, 2051–2093. [23] Othmer, H. G. and Stevens, A. (1997). Aggregation, blowup, and collapse: the ABCs of taxis in reinforced random walks. SIAM J. Appl. Math. 57, 4, 1044–1081. MR1462051 [24] Pemantle, R. (1988). Phase transition in reinforced random walk and RWRE on trees. Ann. Probab. 16, 3, 1229–1241. MR942765 [25] Pemantle, R. (1992). Vertex-reinforced random walk. Probab. Theory Related Fields 92, 1, 117–136. MR1156453 [26] Pemantle, R. (2001). Random processes with reinforcement, preprint. [27] Pemantle, R. and Volkov, S. (1999). Vertex-reinforced random walk on Z has finite range. Ann. Probab. 27, 3, 1368–1388. MR1733153 [28] Rolles, S. W. W. (2003). How edge-reinforced random walk arises naturally. Probab. Theory Related Fields 126, 2, 243–260. MR1990056 [29] Rolles, S.W.W. (2004). On the recurrence of edge-reinforced random walk on Z × G. To appear in Probability Theory and Related Fields. [30] Sellke, T. (1993). Recurrence of reinforced random walk on a ladder. Technical report 93-10, Purdue University. [31] Sellke, T. (1994). Reinforced random walk on the d-dimensional integer lattice. Technical report 94-26, Purdue University. [32] Solomon, F. (1975). Random walks in a random environment. Ann. Probability 3, 1–31. MR362503 [33] Takeshima, M. (2000). Behavior of 1-dimensional reinforced random walk. Osaka J. Math. 37, 2, 355–372. MR1772837 [34] Takeshima, M. (2001). Estimates of hitting probabilities for a 1-dimensional reinforced random walk. Osaka J. Math. 38, 4, 693–709. MR1864459 `s, P. (2004). Vertex-reinforced random walk on Z eventually gets stuck [35] Tarre on five points. Ann. Probab. 32, 3B, 2650–2701. MR2078554 ´ th, B. (1997). Limit theorems for weakly reinforced random walks on Z. [36] To Studia Sci. Math. Hungar. 33, 1-3, 321–337. MR1454118 [37] Vervoort, M. (2000). Games, walks and grammars. Problems I’ve worked on. PhD thesis, Universiteit van Amsterdam. [38] Volkov, S. (2001). Vertex-reinforced random walk on arbitrary graphs. Ann. Probab. 29, 1, 66–91. MR1825142 [39] Zabell, S. L. (1982). W. E. Johnson’s “sufficientness” postulate. Ann. Statist. 10, 4, 1090–1099. MR673645

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 78–84 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000112

Recurrence of cocycles and stationary random walks Klaus Schmidt1,2 University of Vienna and Erwin Schr¨ odinger Institute Abstract: We survey distributional properties of Rd -valued cocycles of finite measure preserving ergodic transformations (or, equivalently, of stationary random walks in Rd ) which determine recurrence or transience.

Let (Xn , n ≥ 0) be an ergodic stationary Rd -valued stochastic process, and let (Yn = X0 + · · · + Xn−1 , n ≥ 1) be the associated random walk. What can one say about recurrence of this random walk if one only knows the distributions of the random variables Yn , n ≥ 1? It turns out that methods from ergodic theory yield some general sufficient conditions for recurrence of such random walks without any assumptions on independence properties or moments of the process (Xn ). Most of the results described in this note have been published elsewhere. Only the Theorems 12 and 14 on recurrence of symmetrized random walks are — to my knowledge — new. Let us start our discussion by formulating the recurrence problem in the language of ergodic theory. Let T be a measure preserving ergodic automorphism of a probability space (X, S, µ), d ≥ 1, and let f : X −→ Rd be a Borel map. The cocycle f : Z × X −→ Rd is given by  n−1  x) + · · · + f (x) if n > 0, f (T f (n, x) = 0 (1) if n = 0,   n −f (−n, T x) if n < 0, and satisfies that

f (m, T n x) + f (n, x) = f (m + n, x)

for all m, n ∈ Z and x ∈ X. Definition 1. The cocycle f (or the function f ) in (1) is recurrent if lim inf f (n, x) = 0 n→∞

for µ-a.e. x ∈ X, where  ·  is a norm on Rd . If we start with a stationary Rn -valued random walk (Xn ) on a probability space (Ω, T, P ) we may assume without loss in generality that the sequence of random variables Xn : Ω −→ Rd is two-sided and generates the sigma-algebra T. 1 Mathematics

Institute, University of Vienna, Nordbergstraße 15, A-1090 Vienna, Austria. Schr¨ odinger Institute, for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria, e-mail: [email protected] AMS 2000 subject classifications: primary 37A20, 60G50. Keywords and phrases: cocycles, random walks. 2 Erwin

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Then there exists an ergodic, probability-preserving automorphism T of (Ω, T, P ) with Xm+n = Xm ◦ T n for all m, n ∈ Z, and we set f = X0 and obtain that f (n, ·) = Yn = X0 + · · · + Xn−1 for every n ≥ 1. Our notion of recurrence for f coincides with the usual probabilistic notion of recurrence of the random walk (Yn , n ≥ 1). We return to the ergodic-theoretic setting. The cocycle f (or the function f ) is a coboundary if there exists a Borel map b : X −→ Rd such that f (x) = b(T x) − b(x) for µ-a.e. x ∈ X or, equivalently, if

(2)

f (n, x) = b(T n x) − b(x)

for every n ∈ Z and µ-a.e. x ∈ X. If f  : X −→ Rd is a second Borel map and f  : Z × X −→ Rd the resulting cocycle in (1), then f and f  (or f and f  ) are cohomologous if there exists a Borel map b : X −→ Rd with f  (x) = b(T x) + f (x) − b(x) for µ-a.e. x ∈ X.

(3)

Recurrence is easily seen to be a cohomology invariant. Proposition 2. Let T be an ergodic measure preserving automorphism of a standard probability space (X, S, µ), and let f, f  : X −→ Rd be Borel maps. Then f is recurrent if and only if f  is recurrent. The proof of Proposition 2 can be found, for example, in [8] (cf. also [9]). The key to this proposition is the observation that recurrence is equivalent to the following condition: for every B ∈ S with µ(B) > 0 and every ε > 0 there exists a nonzero n ∈ Z with µ(B ∩ T −n B ∩ {x ∈ X : T n x = x and f (n, x) < ε}) > 0.

(4)

The recurrence properties of Rn -valued random walks arising from i.i.d. processes are understood completely. For random walks arising from ergodic stationary processes the question of recurrence is much more complex. The first results in this direction appear to be due to [1] and [6]. Theorem 3. Let T be an ergodic measure preserving automorphism of a standard probability space (X, S, µ) and f : X −→ R a Borel map. If f is integrable, then it is recurrent if and only if f dµ = 0.

Integrable functions with zero integral satisfy the strong law of large numbers by the ergodic theorem. For real-valued functions satisfying the weak law of large numbers, recurrence was proved in [3], [9] and by B. Weiss (unpublished). Theorem 4. Let T be an ergodic measure preserving automorphism of a standard probability space (X, S, µ). If a Borel map f : X −→ R satisfies the weak law of large numbers (i.e. if limn→∞ f (n, ·)/n = 0 in measure), then f is recurrent. The failure of f to be recurrent may be due to very simple reasons. For example, if d = 1 and f is integrable with nonzero integral, then f is nonrecurrent, but by subtracting the integral we add a drift term to the cocycle f which makes it recurrent. This motivates the following definition.

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Definition 5. The recurrence set of a Borel map f : X −→ Rd is defined as R(f ) = {c ∈ Rd : f − c is recurrent}. It is not difficult to see that R(f ) is a Borel set (cf. [9]). In order to discuss R(f ) further we consider the skew-product transformation Tf : X × Rd −→ X × Rd defined by Tf (x, g) = (T x, f (x) + g) (5) for every (x, g) ∈ X × Rd . It is clear that Tf preserves the infinite measure µ × λ, where λ is the Lebesgue measure on Rd . For the following result we recall that a Borel set D ⊂ X ×Rd is wandering under Tf if Tf D ∩ D = ∅ for every nonzero n ∈ Z. The transformation Tf is conservative if every Tf -wandering set D ⊂ X × Rd satisfies that (µ × λ)(D) = 0. Proposition 6 ([9]). Let T be an ergodic measure preserving automorphism of a standard probability space (X, S, µ) and f : X −→ Rd a Borel map. Then f is recurrent if and only if Tf is conservative. If f is transient (i.e. not recurrent), then there exists a Tf -wandering set D ⊂ X × Rd with    n d (6) Tf D = 0 (µ × λ) (X × R )  n∈Z

(such a set is sometimes called a sweep-out set). Since recurrence is a cohomology invariant, we obtain the following result. Corollary 7. The recurrence set is a cohomology invariant. In particular, R(f ) = R(f ◦ T ). Here are some examples of recurrence sets. Examples 8. Let f : X −→ R be a Borel map.  (1) If f is integrable  then R(f ) = { f dµ}. (2) If f ≥ 0 and f dµ = ∞ then R(f ) = ∅. (3) If (Xn ) is the i.i.d. Cauchy random walk and f = X0 then R(f ) = R. (4) The recurrence set R(f ) can be equal to any given countable closed subset of R (unpublished result by B. Weiss). (5) The sets R(f ) and R  R(f ) can simultaneously be dense in R. An example appears in [9]: let T be the tri-adic adding machine on X = {0, 1, 2}N , and let φ : X −→ X be the map which interchanges the digits ‘1’ and ‘2’ in each coordinate. Then φ2 = Identity and the automorphisms T and T  = φ ◦ T ◦ φ have the same orbits in the complement of a µ-null set (µ is the Haar measure of X). Hence there f (x) x for µ-a.e. x ∈ X, and both R(f ) exists a Borel map f : X −→ Z with T x = T  and R  R(f ) are dense in R. Corollary 7 and Example 8 (1) shows that the recurrence set behaves somewhat like an invariant integral for possibly nonintegrable functions (except for linearity). Furthermore, if R(f ) = ∅ or |R(f )| > 1 (where | · | denotes cardinality), then f cannot be cohomologous to an L1 -function. In [9] the problem was raised how one could recognize whether the recurrence set of a function f is nonempty. For real-valued functions a partial answer to this question was given in [5].

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Theorem 9. Let T be an ergodic measure preserving automorphism of a standard probability space (X, S, µ) and f : X −→ R a Borel map. If there exist ε, K > 0 such that µ({x ∈ X : |f (n, x)| ≤ K}) > ε for every n ≥ 1, then R(f ) = ∅. In particular, if the distributions of the random variables f (n, ·), n ≥ 1, are uniformly tight, then R(f ) = ∅. The proof of Theorem 9, as well as of several related results, depends on a ‘local limit formula’ in [10]: (d) Let f take values in Rd , d ≥ 1, and let σk be the distribution of the map k (d) (d) f (k, ·)/k1/d and τk = k1 l=1 σl . If f is transient there exist an integer L ≥ 1 and an ε > 0 such that (d)

lim sup τk (B(η)) ≤ 2d Lε−d λ(B(η)) k→∞

and lim sup k→∞

N

(d)

2n τ2n k (B(2−n/d η)) ≤ 2d+1 dLd ε−d λ(B(η))

(7)

(8)

n=0

for every η > 0 and N ≥ 1, where  ·  denotes the maximum norm on Rd and B(η) = {v ∈ Rd : v < η.} The proof of these formulae uses abstract ergodic theory (existence of a sweep-out set for the skew-product, cohomology and orbit equivalence). (d) The inequality (7) shows that the possible limits of the sequences (τk , k ≥ 1) are absolutely continuous at 0 and gives a bound on their density functions there. (d) As a corollary we obtain that f must be recurrent if any limit point of (τk , k ≥ 1) has an atom at 0 (which proves Theorem 4, for example). In order to give a very scanty idea of the proof of Theorem 9 we choose an (d) increasing sequence (km ) such that the vague limits ρn = limm→∞ τ2n km , n ≥ 1, exist, and obtain from (8) that N

2n ρn (B(2−n/d η)) ≤ 2d+1 dLd ε−d λ(B(η))

n=0

for every η > 0. This shows that some of the ρn must have arbitrarily small density at 0. If, under the hypotheses of Theorem 4, R(f ) were empty, one could construct (1) limit points of certain averages of the τk with arbitrarily small total mass, which would violate the hypotheses of Theorem 9. The details can be found in [5]. For d = 2, these considerations lead to the following special case of a result in [10]. Theorem 10 ([2]). Let T be an ergodic measure preserving automorphism of a standard probability space (X,√S, µ) and f : X −→ R2 a Borel map. If the distributions of the functions f (n, ·)/ n, n ≥ 1, in (1) converge to a Gaussian distribution on R2 , then f is recurrent. 2 One might be tempted to conjecture that, for a Borel map f : X −→ √ R , uniform tightness of the distributions of the random variables (f (n, ·)/ n, n ≥ 1), would imply recurrence. However, an example by M. Dekking in [3] shows that the distributions of the functions f (n, ·), n ≥ 1, can be uniformly tight even if f is transient. Let me mention a version of Theorem 10 for Rd -valued maps.

K. Schmidt

82

Theorem 11. Let T be an ergodic measure preserving automorphism of a standard probability space (X, S, µ) and f : X −→ Rd a Borel map, where d ≥ 1. Suppose that there exists an ε > 0 such that every limit point ρ of the distributions of the random variables f (n, ·)/n1/d , n ≥ 1 satisfies that lim inf ρ(B(η))/η d ≥ ε. η→0

Then f is recurrent. This result may not look very interesting for d > 2, but it (or at least an analogue of it) can be used to prove recurrence and ergodicity of skew-product extensions for cocycles with values in certain (noncommutative) matrix groups, such as the group of unipotent upper triangular d × d matrices (cf. [4]). Theorem 10 can be used to prove recurrence of an R2 -valued function, but I don’t know of any useful information about nonemptiness of the recurrence set of a transient function f : X −→ R2 . A possible approach to this problem is to investigate recurrence of the ‘symmetrized’ version f˜: X × X −→ Rd of f , defined by f˜(x, y) = f (x) − f (y) (9) for every (x, y) ∈ X × X. We denote by ˜f : Z × (X × X) −→ Rd the cocycle for the transformation S = T × T on X × X defined as in (1) (with S and f˜ replacing T and f ). For d = 1 we have the following result. Theorem 12. Let T be an ergodic measure preserving automorphism of a standard probability space (X, S, µ) and f : X −→ R a Borel map. Suppose that the distributions of the random variables f (n, ·)/n, n ≥ 1, are uniformly tight. Then the map f˜: X × X −→ R in (9) is recurrent. Conversely, if the distributions of the random variables ˜f (n, ·)/n, n ≥ 1, are uniformly tight, then R(f ) = ∅. (1)

Proof. Let σn = σn be the distribution of the random variable f (n, ·)/n, n ≥ 1, and let hδ = 1[−δ/2,δ/2] : R −→ R be the indicator function of the interval [−δ/2, δ/2] ⊂ 1 R for every δ ∈ (0, 1). We set gδ = δ2 · hδ ∗ hδ , where ∗ denotes convolution. Then gδ dλ = 1, where λ is Lebesgue measure on R, 0 ≤ gδ ≤ and

· 1[−δ,δ]

(10)



gδ (x + y + z) dσn (x) dσn (−y) dz  1 = 2· (hδ ∗ σn )(u + z)(hδ ∗ σn )(u) du dz. δ  φδ (z) = gδ (x + y + z) dσn (x) dσn (−y)

1=

We put

1 δ

for every z ∈ R. Assume that the probability measures (σn , n ≥ 1) are uniformly tight and choose K > 0 so that σn ([−K/2, K/2]) > 1/2 for all n ≥ 1. Then 

K+1

−K−1

φδ (u) du ≥ (σn ∗ σn )([−K, K]) > 1/4

Recurrence of cocycles and stationary random walks

83

for every n ≥ 1. Hence λ({u ∈ R : φδ (u) > 1/(8K + 8)}) > 0. Since φδ assumes its maximum at 0 we conclude that φδ (0) > 1/(8K + 8)

(11)

for every δ > 0. We set σ ¯n (B) = σn (−B) for every Borel set B ⊂ R, denote by σ ˜n = σn ∗ σ ¯n the distribution of the random variable ˜f (n, ·)/n, and obtain that  k −k −k 2 ·σ ˜n ([−2 η, 2 η]) ≥ η · g2−k η d˜ σn ≥ η/(8K + 8) for every η > 0 and k, n ≥ 1, by (10)–(11). By comparing this with (8) we see that f˜ is recurrent. Now suppose that the distributions σ ˜n of the ˜f (n, ·)/n, n ≥ 1, are uniformly tight. Theorem 2.2 in [7] implies the existence of a sequence (an , n ≥ 1) in R such that the probability measures pan ∗ σn , n ≥ 1, are uniformly tight, where pt is the unit point-mass at t for every t ∈ R. We set σ0 = p0 , a0 = 0, and σ−n = σ ¯n and a−n = −an for every n ≥ 1, and obtain that the family of probability measures {pan ∗ σn : n ∈ Z} is uniformly tight. It follows that the map (m, n) → c(m, n) = am+n − am − an from Z × Z to R is bounded, and we choose a translation-invariant mean M (·) on ∞ (Z, R) and set bn = M (c(n, ·)) for every n ∈ Z. Then the set {bn : n ∈ Z} is bounded, c(m, n) = bm+n − bm − bn for every m, n ∈ Z, and the maps n → an and n → bn differ by a homomorphism from Z to R of the form n → tn for some t ∈ R. Our choice of t implies that the distributions of the random variables f (n, ·)/n + an −bn = f (n, ·)/n+tn, n ≥ 1, are uniformly tight. For every ε > 0 we can therefore choose a K > 0 such that µ({x ∈ X : −nK − tn2 ≤ f (n, x) ≤ nK − tn2 }) > 1 − ε for every n ∈ Z. If t = 0 we obtain a contradiction for sufficiently small ε. This proves that the sequence (ρn , n ≥ 1) is uniformly tight and that R(f ) = ∅ by Theorem 9. Remarks 13. (1) Under the hypotheses of Theorem 12, R(f˜) = ∅ by Theorem 9. It should really be obvious that in this case 0 ∈ R(f˜), but I can’t think of any direct reason why this should be true. (2) Let T be an ergodic measure preserving automorphism of a standard probability space (X, S, µ) and f : X −→ R a Borel map. If the distributions of the random variables (f (n, ·), n ≥ 1) are uniformly tight, then Theorem 12 implies that there exists, for (µ × µ)-a.e. (x, y) ∈ X × X, an increasing sequence (nk , k ≥ 1) of natural numbers with limk→∞ f (nk , x) − f (nk , y) = 0. (3) From the proof of Theorem 12 it is clear that we can replace the uniform tightness of the distributions of the f (n, ·), n ≥ 1, by the condition that there exist ε, K > 0 such that µ({x ∈ X : |f (n, x)| ≤ K} > ε) for every n ≥ 1. We turn to the much more interesting case where d > 1. For a function f : X −→

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Rd , Dekking’s example in [3] shows that uniform tightness of the distributions of of the random variables f (n, ·)/n1/d , n ≥ 1, need not imply recurrence. However, the function f˜ in (9) is recurrent under this condition. At present there appears to be no analogue of the reverse implication of Theorem 12. Theorem 14. Let T be an ergodic measure preserving automorphism of a standard probability space (X, S, µ) and f : X −→ Rd a Borel map. Suppose that the distributions of the random variables f (n, ·)/n1/d , n ≥ 1, are uniformly tight (or that there exist ε, K > 0 such that µ({x ∈ X : f (n, x)/n1/d  ≤ K}) > ε for every n ≥ 1). Then the map f˜: X × X −→ Rd in (9) is recurrent. Proof. The proof is essentially the same as that of the first part of Theorem 12. (d) Let σn = σn be the distribution of the random variable f (n, ·)/n1/d , and let hδ = 1[−δ/2,δ/2]d : Rd −→ R be the indicator function of [−δ/2, δ/2]d ⊂ Rd for every δ ∈ (0, 1). We set gδ = δ12d · hδ ∗ hδ and put  φδ (z) = gδ (x + y + z) dσn (x) dσn (−y) for every z ∈ Rd . As in the proof of Theorem 12 we see that 2dk · σ ˜n ([−2−k η, 2−k η]d ) ≥ η d · φδ (0) > η d /4(2K + 2)d for every δ > 0 and conclude from (8) that f˜ is recurrent. References [1] Atkinson, G. (1976). Recurrence of co-cycles and random walks, J. London Math. Soc. (2) 13, 486–488. MR0419727 [2] Conze, J.-P. (1999). Sur un crit`ere de r´ecurrence en dimension 2 pour les marches stationnaires, Ergod. Th. & Dynam. Sys. 19, 1233–1245. MR1721618 [3] Dekking, F.M. (1982). On transience and recurrence of generalized random walks, Z. Wahrsch. verw. Gebiete 61, 459–465. MR682573 [4] Greschonig, G. (2005). Recurrence in unipotent groups and ergodic nonabelian group extensions. Israel J. Math. 147, 245–267. [5] Greschonig, G. and Schmidt, K. (2003). Growth and recurrence of stationary random walks, Probab. Theor. Relat. Fields 125, 266–270. MR1961345 [6] Kesten, H. (1975). Sums of stationary sequences cannot grow slower than linearly, Proc. Amer. Math. Soc. 49, 205–211. MR0370713 [7] Parthasarathy, K.R. (1967). Probability Measures on Metric Spaces, Academic Press, New York–London. MR0226684 [8] Schmidt, K. (1977). Cocycles on ergodic transformation groups, Macmillan Company of India, Delhi. MR0578731 [9] Schmidt, K. (1984). On recurrence, Z. Wahrsch. verw. Gebiete 68, 75–95. MR767446 [10] Schmidt, K. (1998). On joint recurrence, C. R. Acad. Sci. Paris S´er. I Math. 327, 837–842. MR1663750

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 85–99 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000121

Heavy tail properties of stationary solutions of multidimensional stochastic recursions Yves Guivarc’h1 Universit´ e de Rennes 1 (France) Abstract: We consider the following recurrence relation with random i.i.d. coefficients (an , bn ): xn+1 = an+1 xn + bn+1 where an ∈ GL(d, R), bn ∈ Rd . Under natural conditions on (an , bn ) this equation has a unique stationary solution, and its support is non-compact. We show that, in general, its law has a heavy tail behavior and we study the corresponding directions. This provides a natural construction of laws with heavy tails in great generality. Our main result extends to the general case the results previously obtained by H. Kesten in [16] under positivity or density assumptions, and the results recently developed in [17] in a special framework.

1. Notation and problem 1.1. General notation We consider the d-dimensional vector space V = Rd , endowed with the scalar product d
xi yi
x, y = 1

and the corresponding norm x =



d


|xi |

2

1/2

.

1

Let G = GL(V ) be the general linear group, and H = Af f (V ) the affine group. An element h ∈ H can be written in the form h(x) = ax + b

(x ∈ V )

where a ∈ G, b ∈ V . In reduced form we write h = (b, a) and we observe that the projection map (b, a) → a is a homomorphism from H to G. We consider also the projection (b, a) → b of H on V and we observe that H can be written as a semi-direct product H = G
V where V denotes also the translation group of V . For a locally compact second countable (l.c.s.c.) space X, we denote by M 1 (X) the convex set of probability measures on X and we endow M 1 (X) with the weak 1 IRMAR, CNRS Rennes I, Universit´ e de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France, e-mail: [email protected] AMS 2000 subject classifications: primary 60G50, 60H25; secondary 37B05. Keywords and phrases: random walk, stationary measure, random matrix, heavy tail, Mellin transform.

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topology. For a l.c.s.c. semi-group L and a probability measure ρ ∈ M 1 (L), we denote by Tρ the closed semi-group generated by supp ρ, the support of ρ. In particular let η ∈ M 1 (H) and denote by µ (resp. η) its projection on G (resp. V ). Then Tη (resp. Tµ ) will denote the corresponding semi-group in H (resp. G). Let L (resp. X) be a l.c.s.c. semi-group (resp. l.c.s.c. space) and assume X is a L-space. Then for any ρ ∈ M 1 (L) and σ ∈ M 1 (X), we define the convolution ρ ∗ σ ∈ M 1 (X) by:   ρ ∗ σ = δhx dρ(h) dσ(x) = h∗ (σ) dρ(h) where h∗ (σ) ∈ M 1 (X) is the push-forward of σ under the map h ∈ L. We are interested in the special cases L = H or G, X = V or V \ {0} and the actions of H, G are the natural ones. We need to consider a class of Radon measures λ on V \ {0} which are homogeneous under dilations. Such a measure can be written in dt is the natural the form λ = ρ ⊗ s where ρ is a measure on Sd−1 and s (dt) = ts+1 s-homogeneous measure on R+ = {t ∈ R ; t > 0}. 1.2.

The recursion

We consider the product space Ω = H N endowed with the product measure P = η ⊗N and the shift map θ. The corresponding expectation is denoted by E. We are interested in the stationary solutions of the equation: xn+1 = an+1 xn + bn+1

(1.0)

where an ∈ GL(d, R), bn ∈ Rd , i.e., sequences (xn )n∈N such that the law of (xn )n∈N is shift-invariant. As is well known from the theory of Markov chains, this reduces to finding probability measures ν ∈ M 1 (V ) (the law of x0 ) which are η-stationary, i.e.: ν=η ∗ ν

(1.1)

Here, we are interested in ”the shape at infinity” of ν, if such a ν exists. A sufficient condition for the existence of ν is the convergence of the random series ∞
a1 · · · ak bk+1 . Then z satisfies the functional equation: z= 0

(1.2)

z = a1 (z ◦ θ) + b1

and ν is the law of z. In particular, the above series converges if E(log+ a + log+ b) < +∞ and 1 α = lim E(log a0 · · · ak ) < 0. We denote by k k Sd−1 = {u ∈ V : u = 1} the unit sphere in Rd , and for u ∈ Sd−1 , t ≥ 0, we write Htu = {v ∈ V :
u, v ≥ t}. Denote the Radon transform of ν by ϕu (t), and the asymptotic behavior of ϕu (t) = ν(Htu ) (as t → +∞) is the so-called tail of ν in the direction u. In order to describe

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ν, we compactify the Euclidean space V by adding the sphere at infinity Sd−1 ∞ : d−1 d−1 V = V ∪ S∞ . A sequence vn ∈ V converges to u ∈ S∞ if lim vn  → ∞ n

and

lim n

vn = u, vn 

d−1 where u ∈ Sd−1 is naturally identified with the point at infinity in S∞ with the same direction. The space V is endowed with the metric

d(v, v  ) = v − v  , and for Y ⊂ V, x ∈ V we denote d(x, Y ) = inf x − y. y∈Y

In a sense, the problem reduces to finding the asymptotics of ϕu (t) for large t. For the case of positive matrices an and positive vectors bn , it was shown by H. Kesten in [16] that, under some non-degeneracy conditions, there exist χ > 0 and c > 0 such that lim tχ ν(Bt ) = c > 0, t→+∞

where Bt = {x ∈ V : x ≥ t}. Furthermore, there exists a function c(u) on Sd−1 which is not identically zero, such that ∀ u ∈ Sd−1 ,

lim tχ ν(Htu ) = c(u).

t→+∞

Under some assumption on η to be detailed below, our main result implies in particular the following: Theorem 1.1. There exist χ > 0 and a non-zero positive measure ν∞ on Sd−1 , such that lim tχ (δt−1 ∗ ν)(ϕ) = (ν∞ ⊗ χ )(ϕ) t→+∞

for every Borel function ϕ, where the set of discontinuities of ϕ has ν∞ ⊗χ measure 0, and for some ε > 0, sup v−χ | log v|1+ε |ϕ(v)| < +∞. v=0

Under irreducibility conditions, we will describe in Section 6 the properties of ν∞ , depending on the geometry of Tη and Tν . A subtle point is the calculation of the support of ν∞ . For d = 1, this reduces to the study of the positivity of c+ and c− (see [9] and Theorem 3.1). Under special hypothesis, the calculation of the support of ν∞ is done in [17], which gives supp ν∞ = Sd−1 (see Corollary 6.4). In particular, Theorem 6.2 and Theorem 3.1 below can be considered as extensions of the results in [9], [16], [17]. The method of proof that we will use is strongly inspired by [16]. It can be roughly described as ”linearization at infinity” of the stationarity equation ν = η ∗ ν, (see Section 4). More precisely, this equation will be replaced by (1.3)

λ=µ∗λ .

Thus the problem reduces to comparing ν and λ at infinity. In a sense to be explained below, (1.3) is the homogeneous equation associated with the inhomogeneous linear equation (1.1). We will be able to express ν in terms of the Green

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kernel of equation (1.3), i.e., the potential kernel of µ (see Section 7). It is known that equation (1.3) has only one or two extremal solutions which are relevant to the problem. In the proofs, we will rely strongly on the analytic tools of [18], which have also been essential in [17]. We will also use the dynamical aspects of linear group actions on real vector spaces (see the recent surveys [13], [14]). For information on products of random matrices, we refer to [2], [11], [12] and [13]. For an account of our main result in a special case, see [6]. The present exposition is on results which extend the results in the joint work [6]. It is also an improved version of part of the unpublished work [18]. 2. The stationary measure and its support 2.1.

Notation

For any g ∈ G, we denote

r(g) = lim g n 1/n . n

We say that g ∈ G, or h = (b, a) ∈ H, is quasi-expanding (resp. contracting) if r(g) > 1 (resp. r(g) < 1). For a semi-group T ⊂ G or T ⊂ H, we will denote by T e ⊂ T (resp. T c ⊂ T ) the subset of its quasi-expanding (resp. contracting) elements. We observe that, if h = (b, a) is contracting, then h has a unique (attractive) fixed point ha ∈ V : ∞
a −1 ak b . h = (I − a) b = 0

For a semi-group T ⊂ H we denote: Λa (T ) = Closure{ha : h ∈ T c } ⊂ V . In what follows, a measure η ∈ M 1 (H) will be given and the following moment conditions will be assumed:   + (M ) log adµ(a) < +∞ and log+ bdη(b) < +∞ . We will consider the largest Lyapunov exponent α of the random product a0 · · · an :  1 α = lim log adµn (a) , n n where µn is the nth convolution power of µ and the limit exists by subadditivity. We observe that, if α < 0, then there exists a ∈ Tµ with a < 1, and hence r(a) < 1. It follows that Tµc = 0 and Λa (Tη ) = φ. 2.2.

Existence and uniqueness of ν

As a preliminary result we state the following, the first part of which is well known [3]. Proposition 2.1. Assume that η ∈ M 1 (H) satisfies condition (M )

E(log+ (a) < +∞

and

E(log+ b) < +∞ ,

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1 and α = lim E(log a0 · · · an ) < 0. Assume also that Tη has no fixed point. Then n n there exists a unique stationary solution of ν = η ∗ ν.  ∞ The probability measure ν is non-atomic, the series 0 a1 · · · ak bk+1 converges P − a.s. with ν as its limiting law. The support of ν is equal to Λa (Tη ) and Λa (Tη ) is the unique Tη -minimal set in V . For every initial point x0 = x, the random vector xn defined by xn+1 = an+1 xn + bn+1 converges in law to ν and we have the P − a.s. convergence lim d[xn , Λa (Tη )] = 0 . n

Furthermore, if Tµe = φ, then Λa (Tη ) is non-compact. 2.3.

Some examples

As an illustration, we consider some special examples. We first consider the special case d = 1, bk = 1 and ak > 0, i.e. the recursion xn+1 = an+1 xn + 1 . If E(log a) < 0, then its stationary solution has the same law as z =1+

∞


a1 · · · ak .

1

We take µ of the form µ = 21 (δu + δu
) with α = E(log a) = 21 (log u + log u ) < 0. (a) Choose u = 12 , u = 13 , then α = − 12 log 6 < 0, Tµe = φ and supp ν is a Cantor subset of the interval [ 32 , 2]. Hence, the tail of ν vanishes. (b) Choose u = 1/3, u = 2, then α = − 12 log(3/2) < 0 and 2 ∈ Tµe = φ. It follows from [16] (see also [5], [7], [9], [10]) that 3 supp ν = [ , ∞[ 2

and

lim tχ ν(t, ∞) = c > 0 ,

t→+∞

where χ > 0 is defined by E(aχ ) = 1. Such a χ exists since the log-convex function k(s) = E(|a|s ) satisfies α = k (0) < 0

and

lim k(s) = +∞.

s→+∞

An example in two dimension was proposed by H. Kesten in [16]. A simplified version is the following p, p > 0 , η = pδh + p δh
,      cos θ − sin θ λ 0  h=ρ , h = b, , sin θ cos θ 0 λ θ/π ∈ / Q, b = 0, 0 < ρ < 1, 0 < λ < 1 < λ. Since λ > 1, we have h ∈ Tµe = φ. Also, if ρ is sufficiently small, then we have α < 0. It will follow from our main result (Section 6) that for any u with u = 1, there exist χ > 0 and c(u) > 0, such that lim tχ ν(Htu ) = c(u) > 0 .

t→+∞

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3. The case of the line (d = 1) For the multidimensional case (d > 1), we will impose geometric conditions on Tµ , which imply non-arithmeticity properties. The case d = 1 will not be covered by these general assumptions. Furthermore, for d = 1, we will remove the condition det(a) = 0, which means that we replace the group H by the affine semi-group H1 of V : H1 = {g = (b, a) ; b ∈ R , a ∈ R}. However the final results will be similar. As a comparison and introduction to the general case, we first give the result for d = 1. We need to consider the Mellin transform of µ:  k(s) =

|a|s dµ(a) .

We suppose that k(s) is defined for some s > 0, i.e., s∞ = sup{s ≥ 0; k(s) < ∞} > 0. Our main condition on µ, which is responsible for the tail behavior of ν, will be the following contraction-expansion condition: (C–E)

α = k (0) < 0,

s∞ > 0,

lim k(s) ≥ 1.

s→s∞

It is satisfied if s∞ = +∞, α < 0 and Tµ ⊂ [−1, 1]. Using condition (C − E), we can define χ > 0 by k(χ) = 1. We also need to consider the corresponding probability measure µχ (da) = |a|χ µ(da). We denote by (C) the following set of conditions:  (M ) (log+ |a| + log+ |b|)dη(h) < +∞ , (C–E)

(Mχ )

α < 0, 

s∞ > 0,

lim k(s) ≥ 1,

s→s∞

[ |a|χ log+ |a| + |b|χ ]dη(h) < +∞ ,

Tη has no fixed point.

(N –F )

Theorem 3.1 (H1 = ”ax + b”). Assume that η ∈ M 1 (H1 ) satisfies hypothesis (C), and for any ρ > 0, (N –A)

supp µ ⊂ {±ρn : n ∈ Z} ∪ {0},

then ν is diffuse and there are only 3 cases. Case I: supp µ ⊂ R+ ∪ {0}, then supp ν = R and there exists c > 0 such that lim tχ ν(t, ∞) = lim |t|χ ν(−∞, t) = c.

t→+∞

Case II: supp µ ⊂ R+ ∪ {0}.

t→−∞

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Case II 1: +∞ ∈ Λa (Tη ) and −∞ ∈ Λa (Tη ), then supp ν = R, and there exist c+ > 0, c− > 0 such that lim tχ ν(t, ∞) = c+

and

t→+∞

lim |t|χ ν(−∞, t) = c− .

t→−∞

Case II 2: +∞ ∈ Λa (Tη ) but −∞ ∈ / Λa (Tη ), then there exist c > 0 and m ∈ R such that supp ν = [m, +∞)

lim tχ ν(t, ∞) = c.

and

t→+∞

Remark. For previous work on the case d = 1, see [9], where it is proved that c+ + c− > 0 and expressions for c+ and c− are given in terms of ν. Our results on supp ν and the tail of ν in case II 1 are new. For an application to tail estimates in a different context, see [10]. 4. The linearization procedure (d > 1) Here we develop a heuristic approach which suggests that, at infinity, the solution ν of ν = η ∗ ν should be compared to a ”stable solution at infinity” of the linear homogeneous equation λ = µ ∗ λ, where λ is a Radon measure on V \ {0}. We think of ν as a perturbation of the trivial solution δ0 of the unperturbed equation δ0 = µ ∗ δ0 . 4.1.

Homogeneity at infinity

For s ≥ 0, we denote 

1/n , g dµ (g) s

n

k(s)

=

lim

s∞

=

sup{s ≥ 0 : k(s) < +∞} ,

n

and we assume s∞ > 0,

lim k(s) ≥ 1,

s→s∞

1 α = lim n n



log gdµn (g) < 0.

We then define χ > 0 by k(χ) = 1. It follows that if s ∈ (0, χ), then k(s) < 1. We ∞
a1 · · · ak bk+1 : estimate the s-th moment of z = 0

E(zs ) ≤ sup(1, s)

∞


E(a1 · · · ak bk+1 s ).

0

Since lim E(a1 · · · ak bk+1 s )1/n = k(s) < 1 for s ∈ (0, χ), we conclude that n

E(zs ) < +∞. This calculation is not valid for s = χ. This suggests that E(zχ ) = +∞, and that if η is sufficiently non-degenerate, then ν will have the following homogeneity property at infinity: lim tχ (δt−1 ∗ ν) = λ ,

t→+∞

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where λ is χ-homogeneous under dilations, i.e., λ = ν∞ ⊗ χ for some non-zero measure ν∞ on Sd−1 . Furthermore, if the convergence holds for bounded continuous functions, then ν∞ will satisfy: lim tχ νBt
= ν∞

t→+∞

and

u

lim tχ ν(Htu ) = ν∞ (H 0 ∩ Sd−1 ∞ ),

t→+∞

where νBt
is the restriction of ν to Bt = {v ∈ V : v ≥ t}, ν∞ is a positive measure u d−1 on S∞ identified with Sd−1 , and H 0 is the closure of H0u in V . 4.2.

Derivation of the linearized equation

We restrict to the case where η is a product measure on H. In this case, η =η∗µ ,

ν = η ∗ (µ ∗ ν).

Writing νt = tχ (δt−1 ∗ ν) and η t = δt−1 ∗ η ∗ δt , we get νt = η t ∗ (µ ∗ νt ). Since lim η t = δ0 and lim νt = λ, we obtain t→+∞

t→+∞

λ = µ ∗ λ,

λ = ν∞ ⊗ χ .

We introduce the natural actions of g ∈ G on Sd−1 and Pd−1 = Sd−1 /{±Id}. On gu , (u ∈ Sd−1 ). The projective action of g on x ∈ Pd−1 will Sd−1 we have g.u = gu also be denoted g.x. Then the above equation reduces to  ν∞ = gxχ δg.x dν∞ (x)dµ(g), and a similar equation on Pd−1 holds for the projection of ν∞ . Equations of this type were considered by H. Furstenberg (see [8], [11], [13]) in the context of harmonic measures for random walks. We introduce the representations rs on Sd−1 , Pd−1 : rs (g)(δx ) = gxs δg.x . In particular, the above integral equation can be written as rχ (µ)(ν∞ ) = ν∞ . 5. Limit sets on Pd−1 , Sd−1 5.1.

Notation

We recall briefly some definitions and results of [11], [13]. Definition 5.1. A semi-group T ⊂ GL(V ) is said to be strongly irreducible if there does not exist a finite union of proper subspaces of V which is T -invariant. Definition 5.2. An element g ∈ GL(V ) is said to be proximal if there exists a unique eigenvalue λg of g, such that |λg | = r(g) = lim g n 1/n . n

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93

This means that we can write V = Rvg ⊕ Vg< , where gvg = λg vg , Vg< is g-invariant, and the spectral radius of g in Vg< is strictly less than |λg |. We denote by g a ∈ Pd−1 the point corresponding to vg ∈ V . We observe that g a is attractive: ∀x∈ / Vg< ,

lim g n .x = g a . n

Definition 5.3. We say that T ⊂ G satisfies condition i.p. if T is strongly irreducible and contains a proximal element. We denote by T prox the subset of proximal elements. If for example, the Zariski closure of T contains SL(V ), then it is known that T satisfies i.p. (see [12], [13]). Furthermore, condition i.p. for T is equivalent to condition i-p for the Zariski closure of T [13]. This Zariski closure is a Lie group with a finite number of components with a special structure described in [14], Lemma 2.7. 5.2.

The dynamics of T on Pd−1 , Sd−1

Condition i.p. ensures that the dynamics of T on Pd−1 and Sd−1 can be described in a simple way. Definition 5.4. Assume T ⊂ GL(V ) satisfies condition i.p. We denote Λ(T ) = Closure{g a ∈ Pd−1 : g ∈ T prox } , Λ1 (T ) = Closure{vg ∈ Sd−1 : g ∈ T prox } . Proposition 5.5. Assume that T ⊂ GL(V ) satisfies condition i.p. of Definition 5.3. Then Λ(T ) is the unique T -minimal set of Pd−1 . The action of T on Sd−1 has either one or two minimal sets, whose union is Λ1 (T ). Case I: T does not preserve a convex cone in V , then Λ1 (T ) is the unique T minimal set. Case II: T preserves a convex cone C ⊂ V , − then the action of T on Sd−1 has two minimal sets Λ+ 1 (T ), Λ1 (T ) with d−1 ∩C Λ+ 1 (T ) ⊂ S

,

+ Λ− 1 (T ) = −Λ1 (T ).

The existence of a convex cone preserved by T is not related to the fact that T is a semigroup and not a group. Exemples of Zariski dense groups preserving a convex cone exist in abondance (See [1]). For the action of µ on measures we have the following Proposition 5.6. Assume T = Tµ satisfies condition i.p., s∞ > 0, lim k(s) ≥ 1. s→s∞

Let χ > 0 be defined by k(χ) = 1. Then the equation rχ (µ)(ρ) = ρ,

ρ ∈ M 1 (Sd−1 )

has one or two extremal solutions. In case I as above, ρ = ν1 . In case II, there are two extremal solutions ν1+ , ν1− with supp (ν1+ ) = Λ+ 1 (Tµ ),

supp (ν1− ) = Λ− 1 (Tµ ) ,

and ν1− , ν1+ are symmetric with respect to each other.

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An important consequence of i.p. which guarantees the χ-homogeneity of λ at infinity is given by the following (see [11], [14]). Together with Proposition 5.6, it is one of the main algebraic facts which play a role in Theorem 6.2 below. Proposition 5.7 (d > 1). Assume that T ⊂ GL(V ) satisfies i.p. Then the subgroup of R generated by the spectrum of T , Σ(T ) = {log |r(g)| ; g ∈ T prox }, is dense in R. 6. The main theorem (d > 1) 6.1.

Main theorem

As in Section 3, we will use the following set of hypothesis (C):  (M ) [log+ a + log+ b]dη(h) < +∞ , s∞ > 0,

(C–E) 

(Mχ )

χ

+

α < 0,

lim k(s) ≥ 1 ,

s→s∞

a log adµ(a) < +∞,



bχ dη(b) < +∞ ,

Tη has no fixed point .

(N –F )

We observe that k(s) is a log-convex function and is finite at s if gs dµ(g) < +∞. Also, if α < 0 and lim k(s) ≥ 1, then there exists χ > 0 with k(χ) = 1. If α < 0, s→s∞

Tµ contains a quasi-expanding element and s∞ = +∞, then lim k(s) = +∞. A s→+∞

detailed study of k(s) under condition i.p. can be found in [13], where k(s) is shown to be analytic. Condition i-p will also be used in Theorem 6.2 below. The above conditions are inspired by [16], where the case of positive matrices is considered. There, it is not assumed that det(a) = 0, and reducibility is allowed. However, hypothesis (C) and existence of a proximal element are implicitly assumed. On the other hand, for d > 1, the non-arithmeticity condition of [16] does not appear explicitly here, although it is valid (compare with Theorem 3.1). d−1 . In parWe will identify the sphere at infinity, Sd−1 ∞ , with the unit sphere S − + d−1 ticular we consider Λ1 (Tµ ), Λ1 (Tµ ) and Λ1 (Tµ ) as subsets of S∞ . The convex envelope of a closed subset Y ⊂ Rd will be denoted by Co(Y ). If Y ⊂ Sd−1 , we also denote by Co(Y ) the intersection of Sd−1 with the convex envelope of the cone generated by Y . We will need a concept of direct Riemann integrability as in [16], [17]. Definition 6.1. Let X be a compact metric space, ρ a probability measure on X, and ϕ a Borel function on X × R+ . We say that ϕ is ρ-directly Riemann integrable if +∞
(a) sup{|ϕ(x, t)| : (x, t) ∈ X × [2k , 2k+1 ]} < +∞ , k=−∞

(b) The set of discontinuities of ϕ has ρ ⊗  measure 0.

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Using polar coordinates, we can write V \ {0} = Sd−1 × R+ , and the above definition will be used for the case X = Sd−1 , ρ = ν1 (see Section 5). Theorem 6.2 (d > 1). Assume that η ∈ M 1 (H) satisfies hypothesis (C) and condition i.p. above. Then for any Borel function ϕ on V \ {0} with v−χ ϕ(v) ν1 -directly Riemann integrable, we have lim tχ (δt−1 ∗ ν)(ϕ) = (ν∞ ⊗ χ )(ϕ),

t→+∞

where ν∞ is a non-zero measure on Sd−1 . There are only 3 cases Case I: Tµ has no convex invariant cone, then Co (supp ν) = Rd and there exists C > 0 such that ν∞ = Cν1 . Case II : Tµ has a convex invariant cone C ⊂ Rd . Case II 1: Λa (Tη ) ∩ Sd−1 ∞ ⊃ Λ1 (T ), then Co(supp ν) = Rd and there exist C+ > 0, C− > 0 with ν∞ = C+ ν1+ +C− ν1− . Case II 2: Λa (Tη ) ∩ Sd−1  Λ− ∞ ⊃ 1 (T ), d then Co(supp ν) = R and there exists C+ > 0 with ν∞ = C+ ν1+ . For a subset Y ⊂ Sd−1 , we define the polar subset Y ⊥ by Y ⊥ = {u ∈ Sd−1 : ∀x ∈ Y, < u, x >≤ 0}. Then Co(Y ⊥ ) = Y ⊥ = (CoY )⊥ and Y ⊥ = φ if Y is symmetric. With these notation we have Corollary 6.3. With the above hypothesis, there exists a continuous function c(u) on Sd−1 , such that ∀u∈ / (supp ν∞ )⊥ ,

lim tχ ν(Htu ) = c(u) > 0 .

t→+∞

Furthermore the set of zeros of c is equal to (supp ν∞ )⊥ . d−1 In particular, the conditions Co(supp ν∞ ) = S∞ and Co(supp ν) = Rd are equivalent. This corollary extends the results of [9], [16], [17] to the general case. A simplified situation is described in the following (see [6]).



Corollary 6.4. Assume that for any s > 0, as dµ(a) < +∞ and bs dη(b) <

+∞. Also assume that α = lim n1 log adµn (a) < 0, Tη has no fixed point in n V , Tµ contains a quasi-expanding element, Tµ is Zariski dense in G and do not preserve a convex cone in Rd . Then, there exist χ > 0 and c(u) > 0, such that

∀ u ∈ Sd−1 , lim tχ ν(Htu ) = c(u) . t→+∞

Furthermore, Co(supp ν) = Rd . In particular, if η has a density on H which is non-zero at e = (0, Id) ∈ H, α < 0, and for any s > 0, [as + bs ]dη(h) < +∞, then the conditions of corollary 6.4 are satisfied, hence c(u) > 0 on Sd−1 and supp ν∞ = Sd−1 . Situations of this type were considered in [16], [17].

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6.2. Remarks (a) It can be seen that hypothesis (C) in the theorem is necessary for the validity of the first conclusion. Condition i.p. of Definition 5.3 is not necessary, but our  set of conditions is r generically satisfied, as we explain now. If η is of the form η = 1 pi δhi with r ≥ 2, pi > 0, hi ∈ H, and h(r) = (h1 , h2 , · · · hr ) varies in a certain Zariski open subset U of H (r) , then all the conditions of the theorem are satisfied. Hence our conditions are generically satisfied in the weak topology of measures on H, and stability of the conclusions under perturbation is valid. In particular, and in contrast to the case d = 1, Diophantine conditions do not appear explicitly in Theorem 6.2 and corollaries 6.2 and 6.4. This is a consequence of Proposition 5.7. (b) However, interesting special cases are not covered by the theorem. For example, if Tµ consists of diagonal matrices with real or complex entries, then condition i.p. is violated. Estimation at infinity of Poisson kernels on homogeneous nilpotent groups also leads to such problems and to analogous results (See [4]). (c) In the simple example mentioned in section 2 (d = 2), i.e., η = pδh + q  δh
with p, p > 0, and      cos θ − sin θ λ 0  h=ρ , h = b, , sin θ cos θ 0 λ where b = 0, ρ < 1, θ/π ∈ / Q and λ > 1 > λ > 0, the condition α < 0 is satisfied for ρ sufficiently small. Then lim k(s) = +∞, the conditions of the theorem are s→+∞

satisfied and we are in case I. In particular, the support of ν∞ is the whole unit circle, and c(u) is positive for every u. Also Co(supp ν) = R2 . d−1 is strictly larger than supp (ν∞ ⊗ χ ) ∩ Sd−1 = (d) In general supp ν ∩ S∞ ∞ χ supp ν∞ . The set supp (ν∞ ⊗  ) can be thought of as a kind of ”spine at infinity” of supp ν. In general, and in contrast to supp ν, it has a transversal Cantor structure given by supp ν∞ (see [13,14]). (e) We have assumed det(a) = 0 in order to rely on the group framework of [11], [13]. However, the above statements remain valid if H is replaced by the affine semi-group of V , i.e., if det(a) = 0 is allowed, as in Theorem 3.1. 7. Some tools of the proof 7.1.

The scheme

We consider the functional equation (1)

z − b = a (z ◦ θ) .

Let ν  he the law of z − b. Then we obtain ν  = µ ∗ ν,

ν − ν = ν − µ ∗ ν .

It can be shown that ν is given by the potential ν=

∞
0

µk ∗ (ν − ν  ) .

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The advantage of this implicit formula is that it involves only convolution of measures on GL(V ) and V \ {0}. It can be shown that ν − ν  is ”small at infinity”, then the general renewal theorem of [15] can be applied to the operator Pµ on V \ {0} = Sd−1 × R+ defined by Pµ (v, .) = µ ∗ δv , and to its potential kernel ∞


Pµk (v, .)

0

=

∞


µk ∗ δv .

0

Thus the invariant measures λ of Pµ , defined by λ = µ ∗ λ, play an essential role in the problem (see Section 4). A technically important step is to consider the space V˜ = V /{±Id}, since the geometric properties of convolution operators on V˜ are simpler than on V . In order to verify the conditions of validity for the renewal theorem of [15], we need to use the following properties, which are developed in [11], [13]. (a) The equation rχ (µ)(ρ) = ρ , ρ ∈ M 1 (Pd−1 ), has a unique solution on Pd−1 . (b) The spectrum Σ(T ) = {log[r(a)] : a ∈ Tµprox } generates a dense subgroup of R (see also [14]). (c) The Markov chain on Pd−1 associated with the above equation has strong mixing properties and the first Lyapunov exponent of µ is simple. These properties depend on the condition i.p., of Definition 5.3 which justifies its introduction. (d) Finally we need to study the positivity of C, C+ and C− . Here we use the geometry of Tη and Tµ , and we study an auxiliary Markov chain on the space of affine hyperplanes of Rd . This allows us to control the function ν(Htu ) using the ergodic properties of this chain (see [18]). 7.2. Analytic argument for d = 1 For d = 1, a special proof of the positivity of c+ and c− can be given under some stronger hypothesis than in Theorem 3.1. We sketch it here for case II under the simplifying condition: |b| ≤ B, s∞ = ∞, a > 0 . We restrict to the study of c+ . Then, the above functional equation gives (z − b)+ = a(z + ◦ θ) , where z + = sup{z, 0}. We denote by ν+ the law of z + , and write h(s) = E(|z + |s ), u(s) = E(|z + |s − |(z − b)+ |s ). Observe that h is defined for s < χ while |u(s)| ≤ B s u(s) is meromorphic in the half plane Re(s) > 0. for any s, hence the function 1−k(s) The above equation gives for 0 ≤ s < χ h(s)[1 − k(s)] = u(s) ,

Y. Guivarc’h

98

and from the renewal theorem (see [9], [10]), lim tχ (δt−1 ∗ ν+ ) = C+ χ ,

t→+∞

where C+ is the residue at χ of

u(s) 1−k(s)

(see [5]).

u(s) If C+ = 0, then it follows that the function 1−k(s) is holomorphic at χ, and hence on the whole line R+ .

∞ We recall that the Mellin transform γ (s) = 0 xs dγ(x) of a positive measure on R+ cannot be extended holomorphically to a neighbourhood of its abcisse of convergence τ ([19] p. 58). In particular, this is valid for u(s) = E(|z + |s ), hence the condition C+ = 0 implies that E(|z + |s ) < +∞ in a neighborhood of χ. The same argument gives

τ = +∞,

[k(s) − 1]E(|z + |s ) ≤ B s

(s > χ).

It follows that |a|∞ |z + |∞ = lim [k(s)]1/s E(|z + |s )1/s ≤ B . s→+∞

Since |a|∞ > 0, this implies that +∞ ∈ / Λa (T ), which contradicts assumptions II 1, II 2. References [1] Benoist, Y. (2004). Convexes divisibles I. In Algebraic Groups and Arithmetics, 339-374. Tata Inst. Fund. Res., Mumbai. MR2094116 [2] Bougerol, P., and Lacroix, J. (1985). Products of random matrices with applications to Schro¨edinger operators. Progress in Probability and Statistics, 8, Birkha¨ user. MR0886674 [3] Brandt, A., Franken, P. , and Lisek, B. (1990). Stationary Stochastic Models. Wiley, Chichester. MR1137270 [4] Buraczewski, D., Damek, E., Guivarc’h, Y., Hulanicki, A., and Urban, R. (2003). Stochastic difference equations on N A groups. Preprint. Wroclaw. [5] De Calan, C., Luck, J.-M., Nieuwenhuizen, T.-M., and Petritis, D. (1985). On the distribution of a random variable occuring in i.i.d disordered systems. J. Phys. A 18, no. 3, 501–523 MR0783195 [6] De Saporta, B., Guivarc’h, Y., and Le Page, E. (2004). On the multidimensional stochastic equation Yn+1 = An Yn + Bn . CRAS Paris I, 339, 499–502. MR2099549 [7] Derrida, B., and Hilhorst, H.-J. (1983). Singular behaviour of certain infinite products of random 2 × 2 matrices. J. Phys. A, A16, 2641–2654. MR0715727 [8] Furstenberg, H. (1973). Boundary theory and stochastic processes on homogeneous spaces. In: Harmonic Analysis on Homogeneous Spaces. Proc. Sympos. Pure Math, vol. XXVI. American Mathematical Society, 193–229. MR0352328 [9] Goldie, C.-M. (1991). Implicit renewal theory and tails of solutions of random recurrence equations. Ann. Appl. Probab., 1, 126–166. MR1097468 [10] Guivarc’h, Y. (1990). Sur une extension de la notion de loi semi-stable. Ann I.H.P, 26, no. 2, 261–285. MR1063751

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[11] Guivarc’h, Y., and Raugi, A. (1986). Products of random matrices: convergence theorems. In: Random Matrices and Their Applications, Workshop Brunswick Maine 1984, Eds. J.-E. Cohen, H. Kesten, and C.-M. Newman. Contemp. Math, 50, 31–54. MR0841080 [12] Goldsheid, I.-Ya., and Guivarc’h, Y. (1996). Zariski closure and the dimension of the Gaussian law of the product of random matrices. Probab. Theory Related Fields, 105, 109–142. MR1389734 [13] Guivarc’h, Y., and Le Page, E. (2004). Simplicit´e de spectres de Lyapunov et propri´et´e d’isolation spectrale pour une famille d’op´erateurs de transfert sur l’espace projectif. In: Random Walks and Geometry, Workshop Vienna 2001. Ed. V. Kaimanovich. De Gruyter, 181–259. MR2087783 [14] Guivarc’h, Y., and Urban, R. (2005). Semi-group actions on tori and stationary measures on projective spaces. Studia Math. 171, no. 1, 33–66. [15] Kesten, H. (1974). Renewal theory for functionals of a Markov chain with general state space. Ann. Probab., 2, 355–386. MR0365740 [16] Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248. MR0440724 ¨ppelberg, C., and Pergamentchikov, S. (2004). On the tail of [17] Klu the stationary distribution of a random coefficient AR(q) model. Ann. App. Probab., 14-2 (1985), 971–1005. MR2052910 [18] Le Page, E. (1983). Th´eor`emes de renouvellement pour les produits de matrices al´eatoires. Equations aux diff´erences al´eatoires. Publ. S´em. Math. Univ. Rennes. MR0863321 [19] Widder, D.V. (1941). The Laplace Transform. Princeton Mathematical Series 6. Princeton University Press, Princeton, NJ. MR0005923

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 100–108 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000130

Attractiveness of the Haar measure for linear cellular automata on Markov subgroups Alejandro Maass1,∗ , Servet Mart´ınez1,∗ , Marcus Pivato2,† and Reem Yassawi2,† Universidad de Chile and Trent University Abstract: For the action of an algebraic cellular automaton on a Markov subgroup, we show that the Ces` aro mean of the iterates of a Markov measure converges to the Haar measure. This is proven by using the combinatorics of the binomial coefficients on the regenerative construction of the Markov measure.

1. Introduction and main concepts Let (A, +) be a finite Abelian group with unit 0 and (AZ , +) be the product group with the componentwise addition. The shift map σ : AZ → AZ is defined on x ∈ AZ and n ∈ Z by (σx)n = xn+1 .
r i Let µ be a probability measure on AZ and Φ = i=
ci σ a linear cellular Z automaton on A , where , r, c
, . . . , cr are integers such that  ≤ r. The study of the convergence of the sequence (Φn (µ) : n ∈ N) started with the pioneering work of Lind in [8]. This work, devoted to the linear cellular automaton Φ = σ −1 + id + σ on {0, 1}Z and µ a Bernoulli measure, stated that the Ces` aro mean of the sequence converges to the uniform Bernoulli measure. So Φ randomizes any Bernoulli measure. After such work two different strategies were developed to prove the same kind of results but for more general classes of linear cellular automata and other kind of starting measures. The combinatorial properties of the Pascal triangle and the technique of regeneration of measures proposed in [1] were exploited in [2, 3, 5, 9, 10] to prove Lind’s result in the case Φ = id + σ and µ is a measure with complete connections and summable memory decay for any finite group A. In [12, 13] an harmonic analysis formalism was developed to prove that diffusive linear cellular automata randomize harmonically mixing measures (both concepts were defined in [12]). Other related results were proved in [5] and [14] In this paper we consider the same problem for linear cellular automata defined in a Markov subgroup. Before state our main result we need some precise notations and definitions. More details and references on the subject can be found in [6, 7, 15]. * Supported

by Nucleus Millennium Information and Randomness P04-069-F. supported by the NSERC Canada. 1 Departamento de Ingenier´ ıa Matem´ atica and Centro de Modelamiento Matem´ atico, Universidad de Chile, Av. Blanco Encalada 2120 5to piso, Santiago, Chile, e-mail: [email protected] e-mail: [email protected] 2 Department of Mathematics, Trent University, Canada, e-mail: [email protected] e-mail: [email protected] AMS 2000 subject classifications: primary 54H20; secondary 37B20. Keywords and phrases: cellular automata, maximal entropy measure, Markov measures, algebraic topological Markov chain. † Partially

100

Attractiveness of Haar measure by cellular automata

101

Let G ⊆ AZ be a Markov subgroup, that is, it is a topological Markov shift defined by a 0 − 1 incidence matrix M and it is a subgroup of AZ . For every g ∈ A, let F(g) = {h ∈ A : Mgh = 1} be the set of followers of g and P(g) = {h ∈ A : Mhg = 1} be the set of predecessors of g. Then F := F(0) and P := P(0) are subgroups of A and F(g) and P(g) are cosets of F and P in A respectively. It holds that |F| = |P|, where | · | means the cardinality of a set. We fix a function f : A → A such that f (g) ∈ F(g) for every g ∈ A, and then F(g) = f (g) + F. For every g ∈ A and n ≥ 1 define F n (g) = {h ∈ A : Mngh > 0}. One has F n+1 (g) = ∪h∈F (g) F n (h) and |F n (g)| = |F n (0)| for every g ∈ A. The Markov shift G is transitive if and only if M is irreducible, that is, there exists n ≥ 1 such that F n (0) = A. Denote by r the smallest n verifying this condition. In this case we have the mixing property F n (g) = A for every n ≥ r and g ∈ A. For n ≥ 0 and g0 , gn ∈ A such that Mng0 gn > 0 define Cn (g0 , gn ) = {(g1 , . . . , gn−1 ) ∈ An−1 : Mgi gi+1 = 1, i ∈ {0, . . . , n − 1}}. Then |Cn (g0 , gn )| = |Cn (0, 0)|. Therefore, in the transitive case, |Cn (g, h)| = |Cn (0, 0)| for every g, h ∈ A and n ≥ r. The Haar measure of (G, +) is denoted by ν. It is the Markov measure defined by the stochastic matrix L = (Lgh : g, h ∈ A) with Lgh = |F|−1 if h ∈ F(g) and Lgh = 0 otherwise, and the L−stationary vector ρ = (ρ(g) = |A|−1 : g ∈ A). The Haar measure is the maximal entropy measure for the Markov shift (G, σ). It is useful to introduce a notation for the finite-dimensional distributions associated to ν. Thus, for  ≥ 1, ν (
) is the finite-dimensional distribution of ν in A
, and it is concentrated in the subset G
= {(g0 , . . . , g
−1 ) ∈ A
: gi+1 ∈ F(gi ), ∀i ∈ {0, . . . ,  − 2}}. Hence,  γ
if g ∈ G
, 1 , then ν (
) (g) = if γ
=
−1 |A| · |F| 0 if g ∈ G
. 1.1. Main Result Let µ be a Markov probability measure in AZ that is defined by a stochastic matrix P and by some probability vector π invariant for P (π · P = π). The measure µ is said to be compatible with M if Pgh > 0 if and only if Mgh > 0. This property is equivalent to the fact that the support of µ is G. If x ∈ AZ and m, n ∈ Z are two integers with m ≤ n, define xnm = (xm , xm+1 , . . . , xn ) . Consider the endomorphism Φ = id + σ on AZ , that is (Φx)n = xn + xn+1 for x ∈ AZ and n ∈ Z. Our main result is the following one. Theorem 1.1. Assume µ is a Markov probability measure in AZ compatible with M. Furthermore, assume the Abelian group A is ps -torsion for some prime number p and s ≥ 1 and that M is irreducible. Then the Ces` aro mean of µ, under the action of Φ, converges to the Haar measure ν. That is, for any m ∈ N and g ∈ Gm : N −1  1   n m−1 µ (Φ x)0 = g = ν(g). N →∞ N n=0

lim

(1.1)

A. Maass et al.

102

The elements of our proof includes a regenerative construction of the Markov measure and the combinatorics of the binomial coefficients associated with the iterates of the cellular automaton. In [11] a more general result is shown: for Markov fields verifying a ‘filling property’ it is shown that the attractive property of the Theorem holds. In the one-dimensional case this ‘filling property’ is always satisfied, therefore the result follows. 2. Construction of a Markov measure and renewal properties We use the Athreya–Ney [1] representation of Markov chains. It says that it is possible to enlarge the probability space we are considering in order to include a family of integer random times. 2.1. Construction of a Markov measure Let G ⊆ AZ be a Markov subgroup with incidence matrix M. Let µ be a Markov probability measure in AZ compatible with M defined by a pair (π, P) as in subsection 1.1. We describe a procedure to construct the restriction of µ to AN . In this purpose it is useful to introduce the following notation: for g ∈ A we put µg the measure induced on AN by µ conditioned to the event {x−1 = g}; then
N g∈A π(g)µg coincides with the restriction of µ to A . Let α > 0 be a strictly positive number such that α < min{Pgh : g, h ∈ A, Mgh > 0}. We consider a probability space (Ω, B, P) and three independent processes of i.i.d. random variables defined on this space U = (Un : n ∈ N), W = (Wn : n ∈ N) and V = (Vn : n ∈ N) whose marginal distributions are as follows: Un is Bernoulli(α), that is P(Un = 1) = α = 1 − P(Un = 0); Wn is uniformly distributed in F, so P(Wn = g) = |F|−1 for g ∈ F; and Vn is uniformly distributed in the unit interval [0, 1]. Let us construct a sequence (xn : n ∈ N) ∈ AN as a deterministic function of (U, W, V). For any g, h ∈ A define Qgh =

Pgh − α|F|−1 Mgh . 1−α

Thus, Q = (Qgh : g, h ∈ A) is a stochastic matrix compatible with M. For each ˜ gh ⊆ [0, 1] : h ∈ F(g)} a measurable partition of the interval [0, 1] such g ∈ A fix {Q ˜ gh is Qgh . For all g ∈ A, u ∈ {0, 1}, w ∈ F and that the Lebesgue measure of Q v ∈ [0, 1] define    H(g, u, w, v) = u f (g)+w + (1−u) h 11Q gh (v). h∈F (g)

Now, for any x−1 ∈ A and for n ≥ 0 we put

xn = H(xn−1 , Un , Wn , Vn ). It is clear that the distribution of the sequence x = (xn : n ∈ N) is µx−1 . If x−1 ∈ A is a random variable with distribution π, then the distribution of x is µ.

Attractiveness of Haar measure by cellular automata

103

2.2. The associated renewal process For any s, t ∈ N with s ≤ t define     Uts = 1 ⇐⇒ Uk = 1, for all k ∈ {s, ..., t} . (m)

: n ∈ N) given by:

(m)

: Uii+m = 1},

For every m ≥ 1 define a renewal process (Tn (m)

T0 and

(m)

= 0, T1

= min{i > T0 (m)

Tn(m) = min{i > Tn−1 + m : Uii+m = 1} for n ≥ 2. (m)

(m)

It is clear that (Tn+1 −Tn : n ≥ 1) is a family of i.i.d. random variables. Also, from (m) our computations below, it follows that the distribution of T1 has a geometrical (m) tail; that is, there exists β := β(m) ∈ [0, 1) such that P(T1 > t) ≤ β t for any t ≥ 0. (m) Let N(m) be the renewal process induced by (Tn : n ∈ N). That is, for any A ⊆ N, (m) N(m) (A) = {n ∈ A : for some  ∈ N, T
= n}. Let n ∈ A with n > 0. One has that Un−1 = 0 and Unn+m = 1 implies n ∈ N(m) (A). Then P(n ∈ N(m) (A)) ≥ αm+1 (1 − α) := δ > 0. Clearly if 0 ∈ A then P(0 ∈ N(m) (A)) = 1 ≥ δ. For A a finite subset of N and m ∈ N we say A is m-separated if, for any a, b ∈ A with a = b, |a − b| ≥ m + 1. One gets,       A is m-separated =⇒ P N(m) (A) = ∅ ≥ 1−(1−δ)|A| . (2.1) Let A(m) be the largest m-separated subset of A; then |A(m) | ≥ |A|/(m + 1). Thus,      |A| P N(m) (A) = ∅ ≤ P N(m) (A(m) ) = ∅ ≤ (1 − δ)1/(m+1) . (m)

Hence, the distribution of T1

has a geometric tail.

3. Convergence of the Ces` aro mean 3.1.

Independence lemmas

Assume G is transitive and recall r is the smallest integer verifying F r (0) = A. Also recall that γ
= |A|−1 |F|−(
−1) . Let m ≤ n in Z. If x is a random variable in n AZ with distribution µ define Fm to be the sigma-algebra generated by xnm . Lemma 3.1. Let k ≥ r and m ≥ 0. Then the random variable xkk+m conditioned  k+m  to Uk−r = 1, F0k−r−1 is ν (m+1) −distributed. That is, for any g ∈ Gm+1 ,   k+m P xkk+m = g Uk−r = 1, F0k−r−1 = γm+1 . k+m+r n Also, the variable xkk+m conditioned to (Uk−r = 1, F0k−r−1 ∨ Fk+m+r+1 ) is (m+1) ν −distributed for any n ≥ k + m + r + 1. That is, for any g ∈ Gm+1 ,   k+m+r k+m n = γm+1 . = 1, F0k−r−1 ∨ Fk+m+r+1 P xk = g Uk−r

A. Maass et al.

104

Proof. Let g = (g0 , . . . , gm ) ∈ Gm+1 and put n = k − r. For any fixed h = (h0 , . . . , hn−1 ) ∈ Gn ,   k+m P xk = g xn−1 = h and Unk+m = 1    = P xnn−1 = z and xk+m = g x0n−1 = h and Unk+m = 1 z∈Cr+1 (hn−1 ,g0 )

=



|F|−(r+m+1) = |Cr+1 (0, 0)| · |F|(r+m+1)

z∈Cr+1 (hn−1 ,g0 )

=

|F|r+1 1 · |F|−(r+m+1) = = γm+1 |A| |A| · |F m |

This proves the first part, the second one is entirely analogous. Now assume that A is ps -torsion for some prime number p and some s ≥ 1, with s being the smallest number verifying this property. That is: mg = 0 for every g ∈ A and m ∈ ps Z, and for every m < ps there exists some g ∈ A such that mg = g. Observe that for every c ∈ Zps relatively prime to p, there exists a multiplicative inverse c−1 ∈ Zps , such that cc−1 = 1 mod (ps ) and c−1 is also relatively prime to p. Thus, for any g ∈ A, cc−1 g = g. Moreover, (cg = h) ⇐⇒ (g = c−1 h). 3.2. The transformation Recall σ is the shift map in AZ and Φ = id + σ is an endomorphism of AZ . Fix x ∈ AZ . Then for all n ≥ 0 and i ∈ Z, one has n   n xi+k . (Φ x)i = k n

(3.1)

k=0

For every m ∈ N denote by m(s) its equivalent class all n ≥ 0 and i ∈ Z, equality (3.1) can be written n

(Φ x)i =

n (s)  n

k=0

k

xi+k .

mod (ps ) in Zps . Hence, for

(3.2)

 (s) is Let m,  ≥ 0. For n ≥ 0 and 0 ≤ k ≤ n we say k is (m, )−isolated in n if nk  n (s)   relatively prime to p, while, if k ∈ {k−m, . . . , k+} with k = k then = 0. k
 n (s) Here the convention is k
= 0 whenever k  < 0 or k > n. Lemma 3.2. Let m ∈ N and n ≥ 2r+2m+1. If 0 ≤ k ≤ n is (r+m, r+m)-isolated in n, then for every i ∈ Z and g ∈ Gm+1 , one has   i+k+r+m n i+m P (Φ x)i = g Ui+k−r = 1 = γm+1 .

Proof. Since µ is σ-invariant then Φn (µ) is σ-invariant too; hence it suffices to prove the result for i = 0. In other words, it suffices to show that   k+r+m n m P (Φ x)0 = g Uk−r = 1 = γm+1 .

Attractiveness of Haar measure by cellular automata

(s) n xj+k and Consider j ∈ {0, ..., m}. Define Yj = k n (s) k−r−m−1   n (s) n
xj+k (∗) Xj = xj+k
+   k k
k
=0 k =0

k
=k

n 

k
=k+r+m+1

105

(s) n xj+k
, k

(3.3) where (∗) is because k is (r + m, r + m)-isolated. Thus, (Φn x)j = Xj + Yj . If X = (X0 , . . . , Xm ) and Y = (Y0 , . . . , Ym ), then n m (Φ x)0 = X + Y. One gets,     k+r+m k+r+m U U P (Φn x)m = g = 1 = P X + Y = g = 1 0 k−r k−r    k+r+m P Y = g − h X = h, Uk−r =1 = h∈Am+1

  k+r+m =1 . × P X = h Uk−r

(3.4)

 (s) Let c = nk ; then c is relatively prime to p, and Y = c · xkk+m . Thus, if c−1 is the (mod ps ) inverse of c, then xkk+m = c−1 · Y. Thus, for any fixed h ∈ Am+1 ,   k+r+m P Y = g − h X = h and Uk−r =1 (3.5)   k+r+m = P xkk+m = c−1 (g − h) X = h and Uk−r =1 γ , (†) m+1 where (†) is because equation (3.3) implies that X is a function only of x0k−r−1 and xn+m Substituting (3.5) into (3.4) yields k+r+m+1 ; and it allows to apply Lemma 3.1.   k+r+m U = g P (Φn x)m = 1 = 0 k−r



h∈Am+1

= γm+1 .

  k+r+m γm+1 · P X = h Uk−r =1

Therefore the result follows. 3.3.

Elementary facts on the Pascal triangle


Now we use the following result on the Pascal triangle. Let n = j∈N nj pj be the decomposition of n in base p, so nj
∈ {0, . . . , p − 1} for every
j ∈ N. For 0 ≤ k ≤ n j j consider the decompositions k = j∈N kj p and n − k = j∈N (n − k)j p . The Kummer’s Theorem on binomial coefficients, whose proof can be found in [4], states the following result.   Lemma 3.3. The biggest integer  such that p
divides nk is the number of carries needed to sum k and n − k in base p. We introduce the following notation. For n ∈ N and i ≥ 0 we put Ji (n) = {j ≥ i : nj = 0}, ξi (n) = |Ji (n)|. For a real number c denote by c the integer part of c and for a ≥ 0 define pa N = {pa · n : n ∈ N} = {n ∈ N : nj = 0 for all 0 ≤ j < a}.

A. Maass et al.

106

Lemma 3.4. Let m ≥ 1 and a ≥ 2s+1 be such that pa/2 > m. For every n ∈ pa N and i ≥ a, {0 ≤ k ≤ n : k is (m, m)-isolated in n} ≥ 2ξa+i (n) − 1.

Proof. Fix a nonempty subset J ⊆ Ji+a (n), and define 0 ≤ k ≤ n by kj = nj if j ∈ J and kj = nj − 1 for j ∈ J. Therefore n − k verifies (n − k)j = 1 for j ∈ J and (n − k)j = 0 for j ∈ J. Then there is no carry in the sum of k and n − k, so Lemma  (s)   3.3 says that nk is relatively prime to p. It remains to show that kn
= 0 for all k ∈ {k − m, . . . , k + m} \ {k}. Case 1: Let 1 ≤ v ≤ m and k = k − v. Then the p-ary decomposition of k has some nonzero elements in coordinates between 0 and a − 1 (because pa divides k, but does not divide v); moreover, it has at least b = a/2 zeros in {0, . . . , a − 1}. However, the p-ary decomposition of n has no nonzero elements in {0, . . . , a − 1}, so there must be at least b carries in the addition: (n − k ) + k  = n. Thus, Lemma 3.3 says that pb divides kn
. But b = a/2 ≥ (2s + 1)/2 ≥ s, so we conclude that  n (s) = 0. k
Case 2: Let k  = k + v with 1 ≤ v ≤ m. Then kj ≥ nj for every j < a + i and for some j  < a we have kj
> nj
. Hence, the sum in base p of k and n − k  will have  (s)   = 0. at least a carries, so Lemma 3.3 says that pa divides kn
and finally kn


Lemma 3.5. Let a ≥ 0. Then, the set M0 = n ∈ pa N : ξa+ 1 log 2

1 logp (n) (n) ≥ p (n) 5

is of density 1 in pa N. That is, |M0 ∩ {0, . . . , N − 1}| = 1. N →∞ |pa N ∩ {0, . . . , N − 1}| lim

 Proof. Let M = n ∈ N : ξ 1 log (n) (n) ≥ p 2

1 5 (logp (n)

 + a) . Then

1 M ∩ {0, . . . , N − 1} = 1. N →∞ N lim

To see this, let n ∈ N be a ‘generic’ large integer; then the Law of Large Numbers says that only about p1 of the p-ary digits of n are zero; hence p−1 p are nonzero. 1 1 Since there are 2 logp (n) digits in the range [ 2 logp (n), logp (n)], we conclude, with 1 asymptotic probability 1, that at least p−1 2p logp (n) digits in [ 2 logp (n), logp (n)] are 1 nonzero; hence ξ 1 log (n) (n) ≥ p−1 2p logp (n) ≥ 5 (logp (n) + a) (assuming p ≥ 2 2

p

and logp (n) > 4a). Now define bijection ψ : N → pa N by ψ(n) = pa n; then ψ(M) ⊇ M0 . The lemma follows. 3.4. Proof of Theorem 1.1 Proof. Fix m ≥ 1. If N ⊂ N, we say that the m-dimensional marginal of the Ces` aro mean converges along N if for any g ∈ Gm 1 N →∞ |N ∩ {0, . . . , N − 1}| lim



n∈N ∩{0,...,N −1}

µ



(Φn x)0m−1 = g



= ν(g).

(3.6)

Attractiveness of Haar measure by cellular automata

107

Let m = m + r and consider a ≥ 2s + 1 with pa/2 > m (as in Lemma 3.4). Let M0 be as in Lemma 3.5. We claim that the m-dimensional marginal of the Ces` aro mean converges along M0 .   Let n ∈ M0 be enough large such that i = 21 logp (n) ≥ a. Define A = {0 ≤ k ≤ n : k is (m , m ) -isolated in n }. Therefore, by Lemma 3.4 and the definition of M0 , 1

1

|A| ≥ 2ξa+i (n) − 1 ≥ 2 5 logp (n) − 1 = 2 C log2 (n) − 1 = n1/C − 1, where C = 5 log2 (p).Thus,     k+m
(2m
)  µ ∃k ∈ A with Uk−r = 1 ≥ µ N (A − m ) = ∅ ≥

1(1 − δ)|A|

(2.1)

≥

1 − (1 − δ)n

1/c

(3.7)

−1

,

(3.7)

where “(3.7)” is by equation (3.7) and “(2.1)” is by equation (2.1) (since A is (2m )-separated). 1/C Finally, Lemma 3.2 implies |µ((Φn x)ii+m−1 = g)−γm | ≤ (1−δ)n −1 , and thus, 1/C limn→∞,n∈M0 |µ((Φn x)i+m−1 = g) − γm | = limn→∞ (1 − δ)n −1 = 0, as desired. i Lemma 3.5 then implies that the m-dimensional marginal of the Ces` aro mean converges along pa N. Now, since ν is invariant for powers of Φ, we find that for any 0 ≤ j < pa , the m-dimensional marginal of the Ces` aro mean also converges along Mj = {n + j; n ∈ M0 }. Therefore (1.1) follows from the fact that N −1  1   n m−1 µ (φ x)0 =g p lim N →∞ N n=0  1 = |M ∩ {0, . . . , N − 1}| a a

0≤j≤p



n∈Mj ∩{0,...,N −1}

  µ (φn x)0m−1 = g

= ν(g) References [1] Athreya, K. B., and Ney, P. (1978). A new approach to the limit theory of recurrent Markov chains. Transactions of the AMS 248, 493–501. MR511425 [2] Ferrari, P. A., Maass, A. and Mart´ınez, S. (1999). Ces`aro mean distribution of group automata starting from Markov measures. Preprint. [3] Ferrari, P. A., Maass, A., Mart´ınez, S. and Ney, P. (2000). Ces`aro mean distribution of group automata starting from measures with summable decay. Ergodic Theory and Dynamical Systems 20, 6, 1657–1670. MR1804951 [4] Heye, T. Kummer’s Theorem. http://planetmath.org/encyclopedia/KummersTheorem.html [5] Host, B., Maass, A. and Mart´ınez, S. (2003). Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete Contin. Dyn. Syst. 9, 6, 1423–1446. MR2017675 [6] Kitchens, B. P. (1987). Expansive dynamics on zero-dimensional groups. Ergodic Theory and Dynamical Systems 7, 2, 249–261. MR896796 [7] Kitchens, B. and Schmidt, K. (1989). Automorphisms of compact groups. Ergodic Theory and Dynamical Systems 9, 4, 691–735. MR1036904 [8] Lind, D. A. (1984). Applications of ergodic theory and sofic systems to cellular automata. Phys. D 10, 1-2, 36–44. MR762651

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[9] Maass, A. and Mart´ınez, S. (1998). On Ces` aro limit distribution of a class of permutative cellular automata. J. Statist. Phys. 90, 1–2, 435–452. MR1611088 [10] Maass, A. and Mart´ınez, S. (1999). Time averages for some classes of expansive one-dimensional cellular automata. In Cellular Automata and Complex Systems (Santiago, 1996). Nonlinear Phenom. Complex Systems, Vol. 3. Kluwer Acad. Publ., Dordrecht, 37–54. MR1672858 [11] Maass, A., Mart´ınez, S., Pivato, M. and Yassawi, R. (2004). Asymptotic randomization of subgroup shifts by linear cellular automata. Ergodic Theory and Dynamical Systems. To appear. [12] Pivato, M. and Yassawi, R. (2002). Limit measures for affine cellular automata. Ergodic Theory and Dynamical Systems 22, 4, 1269–1287. MR1926287 [13] Pivato, M. and Yassawi, R. (2004). Limit measures for affine cellular automata II. Ergodic Theory and Dynamical Systems 24, 6, 1961–1980. MR2106773 [14] Pivato, M. and Yassawi, R. (2003). Asymptotic randomization of sofic shifts by linear cellular automata. Preprint, available at: http://arXiv.org/abs/math.DS/0306136. [15] Schmidt, K. (1995). Dynamical systems of algebraic origin. Progress in Mathematics, Vol. 128. Birkh¨ auser Verlag, Basel. MR1345152

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 109–118 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000149

Weak stability and generalized weak convolution for random vectors and stochastic processes Jolanta K. Misiewicz1 University of Zielona G´ ora Abstract: A random vector X is weakly stable iff for all a, b ∈ R there exists d

a random variable Θ such that aX + bX
= XΘ. This is equivalent (see [11]) with the condition that for all random variables Q1 , Q2 there exists a random variable Θ such that d XQ1 + X
Q2 = XΘ, where X, X
, Q1 , Q2 , Θ are independent. In this paper we define generalized convolution of measures defined by the formula L(Q1 ) ⊕µ L(Q2 ) = L(Θ), if the equation (∗) holds for X, Q1 , Q2 , Θ and µ = L(Θ). We study here basic properties of this convolution, basic properties of ⊕µ -infinitely divisible distributions, ⊕µ -stable distributions and give a series of examples.

1. Introduction The investigations of weakly stable random variables started in seventies by the papers of Kucharczak and Urbanik (see [8, 15]). Later there appeared a series of papers on weakly stable distributions written by Urbanik, Kucharczak and Vol’kovich (see e.g. [9, 16–18]). Recently there appeared a paper written by Misiewicz, Oleszkiewicz and Urbanik (see [11]), where one can find a full characterization of weakly stable distributions with non-trivial discrete part, and a substantial attempt to characterize weakly stable distributions in general case. In financial mathematics, insurance mathematics and many different areas of science people are trying to predict future behaviour of certain processes by stochastic modelling. Using independent random variables in a variety of constructions turned out to be not sufficient for modelling real events. Multidimensional stable distributions have nice linear properties and enable more complicated structures of dependencies. On the other hand stable distributions are very difficult in calculations because of the complicated density functions and because of the possibility of unbounded jumps stable stochastic processes. Also the distributions called copulas, extensively investigated recently, are giving the possibility of modelling complicated structure of dependencies. Namely, for every choice of parameters such as covariance matrix or a wines structure of conditional dependency coefficients one can find an arbitrarily nice copula with this parameters. 1 Department

of Mathematics Informatics and Econometry, University of Zielona G´ ora, ul. Podg´ orna 50, 65-246 Zielona G´ ora, Poland, e-mail: [email protected] AMS 2000 subject classifications: 60A10, 60B05, 60E05, 60E07, 60E10. Keywords and phrases: weakly stable distribution, symmetric stable distribution,
α -symmetric distribution, scale mixture. 109

110

J. K. Misiewicz

In this situation weakly stable distributions and processes seem to be good candidates for using in stochastic modelling. They have nice linear properties, i.e. if (Xi ) is a sequence of independent identically distributed random
vectors with the weakly stable distribution µ then every linear combination ai Xi has the same distribution as X1 · Θ for some random variable Θ independent of X1 . This condition holds not only when (ai ) is a sequence of real numbers, but also when (ai ) is a sequence of random variables for ai , Xi , i = 1, 2, . . . , mutually
independent. This means that dependence structure of the linear combination ai Xi and dependence structure of the random vector X1 are the same, and the sequence (ai ) is responsible only for the radial behaviour. Moreover weak stability is preserved under taking linear operators A(X1 ), under taking projections or functionals
ξ, X1 . On the other hand radial properties of a distribution can be arbitrarily defined by choosing a proper random variable Θ independent of X1 and considering the distribution of Θ · X1 . Similar properties of tempered stable distributions (see e.g. [12]) are the reason why they are so important now in statistical physics to model turbulence, or in mathematical finance to model stochastic volatility. Throughout this paper we denote by L(X) the distribution of the random vector d

X. If random vectors X and Y have the same distribution we will write X = Y. By P(E) we denote the set of all probability measures on a Banach space (or on a set) E. We will use the simplified notation P(R) = P, P([0, +∞)) = P + . For every a ∈ R and every probability measure µ we define the rescaling operator Ta : P(E) → P(E) by the formula:  µ(A/a) for a = 0; Ta µ(A) = δ0 (A) for a = 0, for every Borel set A ∈ E. Equivalently Ta µ is the distribution of the random vector aX if µ is the distribution of the vector X. The scale mixture µ ◦ λ of a measure µ ∈ P(E) with respect to the measure λ ∈ P is defined by the formula:  def µ ◦ λ(A) = Ts µ (A) λ(ds). R

It is easy to see that µ ◦ λ is the distribution of random vector XΘ if µ = L(X), λ = L(Θ), X and Θ are independent. In the language of characteristic functions we obtain  µ (ts)λ(ds). µ
◦ λ(t) = R

It is known that for a symmetric random vector X independent of random variable d Θ we have XΘ = X|Θ|. From this property we obtain that if µ is a symmetric probability distribution then µ ◦ λ = µ ◦ |λ|, where |λ| = L(|Θ|). Definition 1. A probability measure µ ∈ P(E) is weakly stable (or weakly stable on [0, ∞)) if for every choice of λ1 , λ2 ∈ P (λ1 , λ2 ∈ P+ ) there exists λ ∈ P (λ ∈ P+ ) such that (λ1 ◦ µ) ∗ (λ2 ◦ µ) = λ ◦ µ. If µ is not symmetric then the measure λ is uniquely determined. This fact was proven in [11] in the case of a weakly stable measure µ, and in [15] in the case of µ

Weak stability and generalized weak convolution

111

weakly stable on [0, ∞). If the measure µ is symmetric then only the symmetrization of λ is uniquely determined (see [11], Remark 1). In this case we can always replace the measure λ by its symmetrization ( 12 δ1 + 12 δ−1 ) ◦ λ. For the convenience in this paper we will assume that for symmetric µ the measure λ is concentrated on [0, ∞) taking if necessary |λ| instead of λ. The most important Theorem 1 in the paper [11] states that the distribution µ is a weakly stable if and only if for every a, b ∈ R there exists a probability distribution λ ∈ P such that Ta µ ∗ Tb µ = µ ◦ λ. Moreover we know that (Th. 6 in [11]) if µ is weakly stable probability measure on a separable Banach space E then either there exists a ∈ E such that µ = δa , or there exists a ∈ E \ {0} such that µ = 12 (δa + δ−a ), or µ({a}) = 0 for every a ∈ E. Many interesting classes of weakly stable distributions are already known in the literature. Symmetric stable random vectors are weakly stable, strictly stable vectors are weakly stable on [0, ∞). Uniform distributions on the unit spheres S n−1 ⊂ Rn , their k-dimensional projections and their linear deformations by linear operators are weakly stable. The beautiful class of extreme points in the set of 1 -symmetric distributions in Rn given by Cambanis, Keener and Simons (see [4]) is weakly stable. 2. Generalized weak convolution Definition 2. Let µ ∈ P(E) be a nontrivial weakly stable measure, and let λ1 , λ2 be probability measures on R. If (λ1 ◦ µ) ∗ (λ2 ◦ µ) = λ ◦ µ, then the generalized convolution of the measures λ1 , λ2 with respect to the measure µ (notation λ1 ⊕µ λ2 ) is defined as follows  λ if µ is not symmetric; λ1 ⊕µ λ2 = |λ| if µ is symmetric. If Θ1 , Θ2 are random variables with distributions λ1 , λ2 respectively then the random variable with distribution λ1 ⊕µ λ2 we will denote by Θ1 ⊕µ Θ2 . Thus we have  d  Θ1 X
+ Θ2 X

= Θ1 ⊕µ Θ2 X,

where X, X
, X

have distribution µ, Θ1 , Θ2 , X
, X

and Θ1 ⊕µ Θ2 , X are independent. One can always choose such versions of Θ1 ⊕µ Θ2 and X that the above equality holds almost everywhere. Now it is easy to see that the following lemma holds. Lemma 1. If the weakly stable measure µ ∈ P(E) is not trivial then (1) (2) (3) (4) (5)

λ1 ⊕µ λ2 is uniquely determined; λ1 ⊕µ λ2 = λ2 ⊕µ λ1 ;  λ ⊕µ δ0 =λ; λ1 ⊕µ λ2 ⊕µ λ3 = λ1⊕µ λ2 ⊕µλ3 ); Ta λ1 ⊕µ λ2 = Ta λ1 ⊕µ Ta λ2 .

Example 1. It is known that the random vector Un = (U1 , . . . , Un ) with the uniform distribution ωn on the unit sphere Sn−1 ⊂ Rn is weakly stable. The easiest way to see this is using the characterizations of a rotationally invariant vectors.

J. K. Misiewicz

112

Let us recall that the random vector X ∈ Rn is rotationally invariant (spherically d

symmetric) if L(X) = X for every unitary linear operator L : Rn → Rn . It is known (see [5, 14] for the details) that the following conditions are equivalent (a) X ∈ Rn is rotationally invariant d

(b) X = ΘUn , where Θ = X2 is independent of Un , (c) the characteristic function of X has the form Eeiξ,X = ϕX (ξ) = ϕ(ξ2 ) for some symmetric function ϕ : R → R. Now let L(Θ1 ) = λ1 , L(Θ2 ) = λ2 be such that Θ1 , Θ2 , Un1 , Un2 are independent, d

d

Un1 = Un2 = Un . In order to prove weak stability of Un we consider the characteristic function ψ of the vector Θ1 Un1 + Θ2 Un2   ψ(ξ) = E exp i
ξ, Θ1 Un1 + Θ2 Un2          = E exp i
ξ, Θ1 Un1  E exp i
ξ, Θ2 Un2  = ϕ1 ξ2 ϕ2 ξ2 .

It follows from the condition (c) that Θ1 Un1 + Θ2 Un2 is also rotationally invariant. d

Using condition (b) we obtain that Θ1 Un1 + Θ2 Un2 = ΘUn for some random variable Θ, which we denote by Θ1 ⊕ωn Θ2 . This means that Un is weakly stable and Θ1 ⊕ωn Θ2 = Θ1 Un1 + Θ2 Un2 2

n 1/2   2 = , Θ1 Ukn1 + Θ2 Ukn2 k=1

where Uni = (U1ni , . . . , Unni ), i = 1, 2. Since U2 = (cos ϕ, sin ϕ) for the random variable ϕ with uniform distribution on [0, 2π], then in the case n = 2 we get  1/2 Θ1 ⊕ωn Θ2 = Θ21 + Θ22 + 2Θ1 Θ2 cos(α − β) , where Θ1 , Θ2 , α, β are independent, α and β have uniform distribution on the interval [0, 2π]. It is easy to check that cos(α − β) has the same distribution as cos(α), thus we have 1/2 d  . Θ1 ⊕ωn Θ2 = Θ21 + Θ22 + 2Θ1 Θ2 cos(α) Definition 3. Let L(Θ) = λ, and let µ = L(X) be a weakly stable measure on E. We say that the measure λ (random variable Θ) is µ-weakly infinitely divisible if for every n ∈ N there exists a probability measure λn such that λ = λn ⊕µ · · · ⊕µ λn ,

(n-times),

where (for the uniqueness) λn ∈ P+ if µ is weakly stable on [0, ∞) or if µ is symmetric, and λn ∈ P if µ is weakly stable nonsymmetric. Notice that if λ is µ-weakly infinitely divisible then the measure λ ◦ µ is infinitely divisible in the usual sense. However if λ ◦ µ is infinitely divisible then it does not have to imply µ-infinite divisibility of λ.

Weak stability and generalized weak convolution

113

Example 2. If γα is a strictly α-stable (symmetric α-stable) distribution on a separable Banach space E then it is weakly stable on [0, ∞) (weakly stable). Simple application of the definition of stable distribution shows that 

d d α α 1/α α 1/α = (|Θ | + |Θ | ) Θ + Θ ) , Θ ⊕ . Θ1 ⊕γα Θ2 = (Θα 1 2 2 1 γα 1 2 Now we see that Θ is γα - weakly infinitely divisible if and only if Θα (respectively |Θ|α ) is infinitely divisible in the usual sense. Lemma 2. Let µ be a weakly stable distribution, µ = δ0 . If λ is µ-weakly infinitely divisible then there exists a family {λr : r ≥ 0} such that (1) λ0 = δ0 , λ1 = λ; (2) λr ⊕µ λs = λr+s , r, s ≥ 0; (3) λr ⇒ δ0 if r → 0. Proof. If λ is µ-weakly infinitely divisible then for every n ∈ N there exists a measure λn such that ∗n (λn ◦ µ) = λ ◦ µ, where ν ∗n denotes the n’th convolution power of the measure ν. We define λ1/n := λn . Weak stability of the measure µ implies that for every k, n ∈ N there exists a probability measure which we denote by λk/n such that

∗k ∗k/n λk/n ◦ µ = λ1/n ◦ µ = (λ ◦ µ) . The last expression follows from the infinite divisibility of the measure λ ◦ µ. We see here that for every n, k, m ∈ N we have λkm/nm = λk/n , since

∗km/nm

(λ ◦ µ)

= (λ ◦ µ)

∗k/n

.

Now let x > 0 and let (rn )n be a sequence of rational numbers such that rn → x ∗r ∗x when n → ∞. Since (λ ◦ µ) n → (λ ◦ µ) and   ∗r {λrn ◦ µ : n ∈ N} = (λ ◦ µ) n : n ∈ N

then this family of measures is tight. Lemma 2 in [11] implies that also the family {λrn : n ∈ N} is tight, so there exists a subsequence λrnk weakly convergent to a probability measure which we call λx . Since λx ◦ µ = (λ ◦ µ)∗x then uniqueness of the measure λx follows from the uniqueness of (λ ◦ µ)∗x , Remark 1 in [11] and our assumptions. To see (3) let rn → 0, rn > 0. Since λrn ◦ µ = (λ ◦ µ)

rn

⇒ δ0 = δ0 ◦ µ,

then {(λ ◦ µ)rn : n ∈ N} is tight, and by Lemma 2 in [11] the set {λrn : n ∈ N} is
also tight. Let {rn
} be the subsequence of {rn } such that λrn converges weakly to some probability measure λ0 . Then we have


λrn ◦ µ ⇒ λ0 ◦ µ, and therefore λ0 ◦ µ = δ0 ◦ µ. If µ is not symmetric then Remark 1 in [11] implies that λ0 = δ0 . If µ is symmetric then by our assumptions λ and λrn are concentrated on [0, ∞), thus also λ0 is concentrated on [0, ∞). Since by Remark 1 in [11] the symmetrization of the mixing measure is uniquely determined in this case we also conclude that λ0 = δ0 .

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3. µ-weakly stable random variables and vectors Definition 4. Let µ be a weakly stable distribution, µ = δ0 . We say that the probability measure λ is µ-weakly stable if ∀ a, b > 0 ∃ c > 0 such that Ta λ ⊕µ Tb λ = Tc λ. We say that the random variable Θ is µ-weakly stable if d

∀ a, b > 0 ∃ c > 0 such that (aΘ)X + (bΘ
)X
= (cΘ)X, where the random variable Θ
is an independent copy of Θ, the vectors X and X
have distribution µ and Θ, Θ
, X, X
are independent. Directly from the definition we see the following Lemma 3. Let µ be a weakly stable distribution, µ = δ0 . A probability measure λ is µ-weakly stable iff the measure λ ◦ µ is strictly stable in the usual sense, thus there exists α ∈ (0, 2] such that λ ◦ µ is strictly α-stable. In such a case we will say that λ is µ-weakly α-stable. Proof. Let us define itXΘ

ψ(t) = Ee

=



µ (ts)λ(ds),

where X has distribution µ, Θ has distribution λ, X and Θ are independent. The condition of µ-weak stability of λ can be written in the following way ∀ a, b > 0 ∃ c > 0 such that ψ(at)ψ(bt) = ψ(ct), which is the functional equation defining strictly stable characteristic functions. Example 3. Let Un be a random vector with the uniform distribution µ = ωn on the unit sphere Sn−1 ⊂ Rn (or Un,k -any its projection into Rk , k < n). The ωn -weakly Gaussian random variable Γn is defined by the following equation: d

Un · Γn = (X1 , . . . , Xn ) = X, where Un and Γn are independent, X is an n-dimensional Gaussian random vector with independent identically distributed coordinates. For convenience we can assume that each Xi has distribution N (0, 1). It follows from the condition (b) in the characterization of rotationally invariant random vectors given in Example 1 that Γn has the same distribution as X2 . Simple calculations show that Γn has density 2 2 f2,n (r) = n/2 n rn−1 e−r /2 . 2 Γ( 2 ) For n = 2 this is a Rayleigh distribution with parameter λ = 2, thus the Rayleigh distribution is ω2 -weakly Gaussian. For n = 3 this is the Maxwell distribution with parameter λ = 2, thus Maxwell distribution is ω3 -weakly Gaussian. Let us remind that the generalized Gamma distribution with parameters λ, p, a > 0 (notation Γ(λ, p, a)) is defined by its density function  a x a p−1 x exp − f (x) = , for x > 0. p/a λ Γ(p/a)λ

Weak stability and generalized weak convolution

115

Thus we have that the generalized Gamma distribution Γ(λ, n, 2) is ωn -weakly Gaussian. Now let Θnα be an ωn -weakly α-stable random variable. Then Un Θnα is a rotationally invariant α-stable random vector for Un independent of Θnα . On the other hand,every rotationally invariant α-stable random vector has the same distribution as Y Θα/2 , where Y is rotationally invariant Gaussian random vector indepenα/2

dent of the nonnegative variable Θα/2 with the Laplace transform e−t we have  d Un · Θnα = Un Γn Θα/2 ,

. Finally

for Un , Γn and Θα/2 independent. This implies that the density of a ωn -weakly α-stable random variable Θnα is given by    ∞ r 1 √ fα/2 (s)ds. fα,n (r) = f2,n √ s s 0 In particular if we take α = 1 then 1 x > 0. f1/2 (s) = √ x−3/2 e−1/(2x) , 2π √ Simple calculations and the equality πΓ(2s) = 22s−1 Γ(s)Γ(s + 12 ), s > 0, show that 22−n Γ(n) rn−1 f1,n (r) = , r > 0, Γ(n/2)Γ(n/2) (r2 + 1)(n+1)/2 is the density function of a ωn -weakly Cauchy distribution. Example 4. We know that for every symmetric α-stable random vector X with distribution γα on any separable Banach space E and every p ∈ (0, 1) the random 1/α vector XΘp is symmetric αp-stable for X independent of Θp with the distribution p λp and the Laplace transform e−t . Since symmetric stable vectors are weakly stable we obtain that ∀ γα ∀ p ∈ (0, 1) λp is γα − weakly αp − stable. 4. (λ, µ)-weakly stable L´ evy processes In this section we construct a L´evy process based on a nontrivial weakly stable probability measure µ. The measure µ can be defined on the real line, on Rn or on a Banach space E. By λ we will denote in this section a µ-weakly infinitely divisible distribution on R. Let T = [0, ∞) and let m be a Borel measure on T . We say that {Xt : t ∈ T } is a (λ, µ)-weakly stable L´evy process if the following conditions hold: (a) X0 ≡ 0; (b) Xt has independent increments; (c) Xt has distribution λm[0,t) ◦ µ. If m is equal to the Lebesgue measure on T then this process has stationary increments. Notice that for λ = δ1 and µ = N (0, 1) we obtain with this construction the Brownian motion.

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J. K. Misiewicz

Example 5. Let γα be a strictly stable distribution on a separable Banach space E with the characteristic function exp{−R(ξ)} and let Θr , r > 0, denote the random variable with distribution   r+k−1 (1 − p)k pr , k = 0, 1, 2 . . . , P{Θr = k} = k for some p ∈ (0, 1). Since R(tξ) = |t|α R(ξ) it is easy to see that the measure 1/α 1/α λ = L(Θ1 ) is γα -weakly infinitely divisible, λr = L(Θr ), and λr ⊕γα λs = λr+s . Let {Xt : t ∈ T } be the (λ, γα )-weakly stable L´evy process. Then Xt has the following characteristic function: E exp{i
ξ, Xt } = E exp{−R(ξ)Θm[0,t) }   ∞  m[0, t) + k − 1 (1 − p)k pm[0,t) exp{−R(ξ)k} = k k=0  m[0,t) p . = 1 − (1 − p) exp{−R(ξ)} Example 6. For γα being a strictly stable distribution on a separable Banach space E with the characteristic function exp{−R(ξ)} and λ = L(Q1/α ), where Q has Gamma distribution with parameters p = 1 and a > 0 we obtain (λ, γα )-weakly stable Levy process with the distribution defined by the following characteristic function E exp{i
ξ, Xt } = E exp{−R(ξ)Qm[0,t) }  ∞ am[0,t) m[0,t)−1 −as s e ds = exp{−R(ξ)s} Γ(m[0, t)) 0  m[0,t) a = . a + R(ξ) 1/α

To see this it is enough to notice that λr = L(Qr bution Γ(r, a).

), where Qr has gamma distri-

5. µ-weakly one-dependent processes Let us recall that the stochastic process {Yn : n ∈ N} taking values in a separable Banach space E is one-dependent if for each n ∈ N the sequences {Y1 , . . . , Yn−1 } and {Yn+1 , Yn+2 , . . . } are independent. It is evident that if f : R → R is a measurable function and {Y1 , Y2 , . . . } is a one-dependent process then also the process {f (Y1 ), f (Y2 ), . . . } is one-dependent. A simples possible one-dependent process can be obtained as {f (Xi , Xi+1 ) : i = 1, 2, . . . }, where {Xi } is a sequence of independent (often identically distributed) random variables. A nice counterexample that not all one-dependent processes have this construction is given in [1, 2]. There are several possibilities for constructing one-dependent processes with distributions which are mixtures of a fixed weakly stable measure µ on a separable Banach space E. In the first of the following examples we give this construction assuming that the mixing measure is µ-weakly infinitely divisible. In the second example this assumption is omitted, but the weakly stable measure µ must be stable.

Weak stability and generalized weak convolution

117

Example 7. Let µ be a weakly stable distribution on a separable Banach space E and let λ be a µ-weakly infinitely divisible measure on R. Assume that ∞m is a σ-finite measure on a rich enough measure space (S, B) such that S = n=1 An , m(An ) < ∞, n = 1, 2, . . . and Ai ∩Aj = ∅ for i = j. With each set An we connect the random variable Zn with distribution λm(An ) such that Z1 , Z2 , . . . are independent. Let also X1 , X2 , . . . be the sequence of independent identically distributed random vectors with distribution µ. Now we define Yn = Zn Xn + Zn+1 Xn+1 ,

n = 1, 2, . . . .

It is easy to see that {Yn : n ∈ N} is a one-dependent process with the distribution L(Yn ) = λm(An ∪An+1 ) ◦ µ. This process is stationary if m(Ai ) = m(Aj ) for all i, j ∈ N. If µ = ωk then {Yn : n ∈ N} is elliptically contoured. If µ = γα then {Yn : n ∈ N} is α-substable. If µ = γα and for some β ∈ (0, 1) λ = L(Θβ ), where Θβ is nonnegative β-stable random variable, then {Yn : n ∈ N} is αβ-stable α-substable. Example 8. Assume that γp is a symmetric p-stable distribution on a separable Banach space E, and let {Zn : n ∈ N} be any one-dependent stochastic process taking values in [0, ∞). For the sequence X1 , X2 , . . . of i.i.d. random vectors with distribution µ we define Yn = Xn Zn1/p , n ∈ N. Directly from the construction, it follows that the process {Yn : n ∈ N} is onedependent. This process is also p-substable and the characteristic function of the
linear combination of its components an Yn can be easily calculated using the Laplace transform for {Zn : n ∈ N}. Namely for every ξ ∈ E∗ we have       E exp i
ξ, an Yn  = E exp i an
ξ, Xn Zn1/p    p p = E exp − |an | (ξ)p Zn > , where  is the linear operator fom East into some Lp -space such that   E exp {i
ξ, X1 } = E exp −(ξ)pp . References [1] Aaronson, J., Gilat, D., Keane, M., , and de Valk, V. (1989). An algebraic construction of a class of one-dependent processes. Ann. Probab. 17, 1, 128–143. MR972778 [2] Aaronson, J., Gilat, D., and Keane, M. a. (1992). On the structure of 1-dependent Markov chains. J. Theoret. Probab. 5, 3, 545–561. MR1176437 [3] Bretagnolle, J., Dacunha Castelle, D. and Krivine, J.-L. (1966) Lois stables et espaces Lp , Ann. Inst. H. Poincar´e Sect. B (N.S.) 2, 231–259. MR0203757 [4] Cambanis, S., Keener, R., and Simons, G. (1983). On α-symmetric multivariate distributions. J. Multivariate Anal. 13, no. 2, 213–233. MR0705548 [5] Crawford, J. J. (1977). Elliptically contoured measures on infinitedimensional Banach spaces. Studia Math. 60, 1, 15–32. MR436243

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[6] Dudley, R.M. (1989). Real Analysis and Probability. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA. MR0982264 [7] Feller, W. (1966). An Introduction to Probability Theory and its Applications, vol. II, John Wiley, New York. MR0210154 [8] Kucharczak, J. and Urbanik,K. (1974) Quasi-Stable functions. Bulletin of Polish Academy of Sciences, Mathematics 22(3), 263–268. MR0343338 [9] Kucharczak, J. and Urbanik, K. (1986). Transformations preserving weak stability. Bull. Polish Acad. Sci. Math. 34, no. 7–8, 475–486. MR0874894 ´cs, E.. (1960). Characteristic Functions. Griffin’s Statistical Monographs [10] Luka & Courses 5. Hafner Publishing Co., New York MR0124075 [11] Misiewicz, J. K., Oleszkiewicz, K., and Urbanik, K. (2005). Classes of measures closed under mixing and convolution. Weak stability. Studia Math. 167, 3, 195–213. MR2131418 ´ski, J., Tempering stable processes. Preprint. [12] Rosin [13] Samorodnitsky, G. and Taqqu, M. (1994), Stable Non-Gaussian Random Processes. Stochastic models with infinite variance. Stochastic Modeling. Chapman & Hall, New York. MR1280932 [14] Schoenberg, I. J. (1938). Metric spaces and completely monotonic functions, Ann. Math. (2) 39, pp. 811–841. MR1503439 [15] Urbanik, K. (1976). Remarks on B-stable Probability Distributions. Bulletin of Polish Academy of Sciences, Mathematics, 24(9), pp. 783–787. MR0423472 [16] Vol’kovich, V. (1992). On symmetric stochastic convolutions. J. Theoret. Probab. 5, no. 3, 417–430. MR1176429 [17] Vol’kovich, V. (1985). On infinitely decomposable measures in algebras with stochastic convolution, Stability Problems of Stochastic models. Proceedings of VNIICI Seminar, Moscow, 15–24, in Russian. MR0859210 [18] Vol’kovich, V. (1984) Multidimensional B-stable distributions and some generalized convolutions. Stability Problems of Stochastic models. Proceedings of VNIICI Seminar, Moscow, 40–53, in Russian. [19] Zolotarev, V.M. (1986) One-Dimensional Stable Distributions, Transl. Math. Monographs 65, Amer. Math. Soc., Providence. MR0854867

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 119–127 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000158

Coverage of space in Boolean models Rahul Roy1,∗ Indian Statistical Institute Abstract: For a marked point process {(xi , Si )i≥1 } with {xi ∈ Λ : i ≥ 1} being a point process on Λ ⊆ Rd and {Si ⊆ Rd : i ≥ 1} being random sets consider the region C = ∪i≥1 (xi + Si ). This is the covered region obtained from the Boolean model {(xi + Si ) : i ≥ 1}. The Boolean model is said to be completely covered if Λ ⊆ C almost surely. If Λ is an infinite set such that s + Λ ⊆ Λ for all s ∈ Λ (e.g. the orthant), then the Boolean model is said to be eventually covered if t + Λ ⊆ C for some t almost surely. We discuss the issues of coverage when Λ is Rd and when Λ is [0, ∞)d .

1. Introduction A question of interest in geometric probability and stochastic geometry is that of the complete coverage of a given region by smaller random sets. This study was initiated in the late 1950’s. An account of the work done during that period may be found in Kendall and Moran (1963). A similar question is that of the connectedness of a random graph when two vertices u and v are connected with a probability pu−v independent of other pairs of vertices. Grimmett, Keane and Marstrand (1984) and Kalikow and Weiss (1988) have shown that
barring the ‘periodic’ cases, the graph is almost surely connected if and only if i pi = ∞. Mandelbrot (1972) introduced the terminology interval processes to study questions of coverage of the real line R by random intervals, and Shepp (1972) showed that if S is an inhomogeneous Poisson point process on R × [0, ∞) with density measure λ × µ where λ is the Lebesgue measure on the x-axis and µ is a given measure on the y-axis, then ∪(x,y)∈S (x, x + y) = R almost surely if and only if 1 ∞ dx exp( (y − x)µ(dy)) = ∞. Shepp also considered random Cantor sets de0 x fined as follows: let 1 ≥ t1 ≥ t2 ≥ . . . be a sequence of positive numbers decreasing to 0 and let P1 , P2 , . . . be Poisson point processes on R, each with density λ. The set V := R \ (∪i ∪x∈Pi (x, x + ti )) is the
random Cantor set. He showed that V has Lebesgue
measure 0 if and only if i ti = ∞. Moreover, P (V = ∅) = 0 or 1 ∞ according as n=1 n−2 exp{λ(t1 + · · · + tn )} converges or diverges. In recent years the study has been re-initiated in light of its connection to percolation theory. Here we have a marked point process {(xi , Si )i≥1 } with {xi : i ≥ 1} being a point process on Λ ⊆ Rd and Si ⊆ Rd being random sets. Let C = ∪i≥1 (xi +Si ) be the covered region of the Boolean model {(xi + Si ) : i ≥ 1}. The simplest model to consider is the Poisson Boolean model, i.e., the process {xi : i ≥ 1} is a stationary Poisson point process of intensity λ on Rd and Si = [0, ρi ]d , i ≥ 1 ∗ This

(1)

research is supported in part by a grant from DST. and Mathematics Unit Indian Statistical Institute, 7 SJS Sansanwal Marg, New Delhi 110016, India, e-mail: [email protected] AMS 2000 subject classifications: primary 05C80, 05C40; secondary 60K35. Keywords and phrases: Poisson process, Boolean model, coverage. 1 Statistics

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are d-dimensional cubes the lengths of whose sides form an independent i.i.d collection {ρi : i ≥ 1} of positive random variables. (Alternately, Si ’s are d-dimensional spheres with random radius ρi .) In this case, Hall (1988) showed that Theorem 1.1. C = Rd almost surely if and only if Eρd1 = ∞. More generally, Meester and Roy (1996) obtained Theorem 1.2. For {xi : i ≥ 1} a stationary point process, if Eρd1 = ∞ then C = Rd almost surely. The above results relate to the question of complete coverage of the space Rd . Another question which arises naturally in the Poisson Boolean model is that of eventual coverage (see Athreya, Roy and Sarkar [2004]). Let {xi : i ≥ 1} be a stationary Poisson process of intensity λ on the orthant Rd+ and the Boolean model is constructed with random squares Si as above yielding the covered region C = ∪i≥1 [xi (1), xi (1) + ρi ] × · · · × [xi (d), xi (d) + ρi ]. In that case, P (Rd+ ⊆ C) = 0, however we may say that Rd+ is eventually covered if there exists 0 < t < ∞ such that (t, ∞)d ⊆ C. In this case, there is a dichotomy vis-a-vis dimensions in the coverage properties. In particular, while eventual coverage depends on the intensity λ for d = 1, for d ≥ 2 there is no such dependence. Theorem 1.3. For d = 1, (a) if 0 < l := lim inf x→∞ xP (ρ1 > x) < ∞ then there exists 0 < λ0 ≤ such that  0 if λ < λ0 Pλ (R+ is eventually covered by C) = 1 if λ > λ0 ; (b) if 0 < L := lim supx→∞ xP (ρ1 > x) < ∞ then there exists 0 < such that  0 if λ < λ1 Pλ (R+ is eventually covered by C) = 1 if λ > λ1 ;

1 L

1 l

<∞

≤ λ1 < ∞

(c) if limx→∞ xP (ρ1 > x) = ∞ then for all λ > 0, R+ is eventually covered by C almost surely (Pλ ); (d) if limx→∞ xP (ρ1 > x) = 0 then for any λ > 0, R+ is not eventually covered by C almost surely (Pλ ). Theorem 1.4. Let d ≥ 2, for all λ > 0. (a) Pλ (Rd+ is eventually covered by C) = 1 whenever lim inf x→∞ xP (ρ1 > x) > 0. (b) Pλ (Rd+ is eventually covered by C) = 0 whenever limx→∞ xP (ρ1 > x) = 0. In 1-dimension for the discrete case we may consider a Markov model as follows: X1 , X2 , . . . is a {0, 1} valued Markov chain and Si := [0, ρi ], i = 1, 2, . . . are i.i.d. intervals where ρi is as employed in (1). The region ∪(i + Si )1Xi =1 is the covered region. This model has an interesting application in genomic sequencing (Ewens and Grant [2001]). If pij = P (Xn+1 = j | Xn = i), i, j = 0 or 1, denote the transition probabilities of the Markov chain then we have Theorem 1.5. Suppose 0 < p00 , p10 < 1.

Coverage of space in Boolean models

121

(a) If l := lim inf j→∞ jP (ρ1 > j) > 1, then P {C eventually covers N} = 1 when01 ever p10p+p > 1/l. 01 (b) If L := lim supj→∞ jP (ρ1 > j) < ∞, then P {C eventually covers N} = 0 01 whenever p10p+p < 1/L. 01 Molchanov and Scherbakov (2003) considered the case when the Boolean model is non-stationary. For a Poisson point process {xi : i ≥ 1} of intensity λ, we place a d-dimensional ball B(xi , ρi h(||xi ||)) centred at xi and of radius ρi h(||xi ||) where ρi is as before and h : [0, ∞) → (0, ∞) is a nondecreasing function. Let C = ∪∞ i=1 B(xi , ρi h(||xi ||)) denote the covered region. Let πd denote the volume of a ball of unit radius in d 1/d  dimensions and take h0 (r) = λπd d log r . Theorem 1.6. Suppose Eρd+η < ∞ for some η > 0 and h is as above. (a) If d  h(r) 1 > then P (C = Rd ) > 0, lh := lim inf d r→∞ h0 (r) E(ρ1 ) and (b) if 0 ≤ Lh := lim sup r→∞



h(r) h0 (r)

d

<

1 E(ρd1 )

then P (C = Rd ) = 0.

The result in (a) above cannot be translated into an almost sure result because of the lack of ergodicity in the model. 2. Complete coverage We now sketch the proofs of Theorems 1.1, 1.2 and 1.6. Let V = [0, 1]d \ C denote the ‘vacant’ region in the unit cube [0, 1]d ;  E((V )) = E 1{x is not covered} dx [0,1]d

= exp(−λEρd1 )

(2)

where  stands for the d-dimensional Lebesgue measure. Hence, if Eρd1 < ∞ then E((V )) > 0 and so P ([0, 1]d ⊆ C) < 1. Conversely, Eρd1 = ∞ implies E((V )) = 0 and thus by stationarity, E((Rd \ C)) = 0. Using the convexity of the shapes Si we may conclude that P (C = Rd ) = 1. Here the Poisson structure was used to obtain the expression (2); for a general process we need to extract, if possible, an ergodic component of the process and show that the Boolean model obtained from this ergodic component covers the entire space when Eρd1 = ∞. To this end let {xi : i ≥ 1} be an ergodic point process with density 1. Let Dn = [0, 2n/d ]d and En = {there exists xi in the annulus Dn+1 \ Dn such that D0 ⊆ (xi + Si )}. Also let Am be the event that m is the first index such that #{i : xi ∈ Dn+1 \ Dn } ≥ a2n for all n ≥ m and for some

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122

fixed constant a. By ergodicity, {Am : m ≥ 1} forms a partition of the probability space and we obtain c ∞ k/d + 1} | Am ) P (∩∞ k=m Ek | Am ) ≤ P (∩k=m ∩i:xi ∈Dk+1 \Dk {ρi ≤ 2 ∞ k+1 ≤ P (ρ1 ≤ 2k/d + 1)a2

≤

k=m ∞

k=m+1 ∞

≤

k=m+1

=



P (ρd1 ≤ 2k )a2 

k 2 −1



j=1

k

a

P (ρd1 ≤ q2k + j)

∞   1 − P (ρd1 > k)

a

k=2m

= 0 if and only if

∞ 

P (ρd1 > k) = ∞.

k=2m

This completes the proof of Theorems 1.1 and 1.2. 1/d To prove Theorem 1.6 (a) we study the case when h(r) = lh h0 (r) and lh Eρd1 > 1 + δ for some δ > 0. It may be easily seen that for z ∈ Zd , P {z + (−1/2, 1/2]d ⊆ C} ≤ exp{−λµ(Rz )}, where Rz = {(r, x) ∈ [0, ∞) × Rd : z + (−1/2, 1/2]d ⊆ B(x, rh(x)) and µ is the product measure of the measure governing ρ1 and Lebesgue measure. From the properties of h0 it may be seen, after some calculations, that given  > 0, there exists r such that for ||z|| > r, µ(Rz ) ≥ (1 − )πd (h(z))d Eρd1 . Thus we obtain, for some constant K,   exp{−d(1 + δ)(1 − ) log(||z||)} P {z + (−1/2, 1/2]d ⊆ C} ≤ K z∈Zd

z∈Zd

= K



||z||−d(1+δ)(1−) < ∞

z∈Zd

whenever (1 + δ)(1 − ) > 1. Invoking the Borel-Cantelli lemma we have that z + (−1/2, 1/2]d ⊆ C occurs for only finitely many z ∈ Zd . Using this we now complete the proof of Theorem 1.6 (a). The proof of Theorem 1.6(b) is more delicate and we just present the idea here. For d ≥ 2, if we place points in a spherical shell of radius nγ , such that the interpoint distances are maximum and are of the order of nβ where 0 < β < γ and γ > 1 then the number of points one can place on this shell is of the order of nγ−β)(d−1) . Let Vn be the event that one such point is not covered by C. It may be shown that there is a choice of γ and β such that infinitely many events Vn occur with probability 1. For d = 1, the same idea may be used and, in fact, the proof is much simpler. 3. Eventual coverage We discretise the space Rd+ by partitioning it into unit cells {(i1 , . . . , id ) + (0, 1]d : i1 , . . . , id = 0, 1, . . .} and call a vertex i := (i1 , . . . , id ) green if x ∈ i + (0, 1]d for some point x of the point process. Consider two independent i.i.d. collections of random variables {ρui } and {ρli } where the distribution of ρui and ρli are identical

Coverage of space in Boolean models

123

to that of 2 + max{ρ1 , . . . , ρN } and max{0, max{ρ1 , . . . , ρN } − 1} respectively; here N is an independent Poisson random variable with mean λ conditioned to be 1 or more. We now define two discrete models, an upper model and a lower model: in both these models a vertex i is open or closed independently of other vertices, and the covered region for the upper model is ∪{i open} (i + [0, ρui ]), and that for the lower model is ∪{i open} (i + [0, ρli ]). Observe that the eventual coverage of the Poisson model ensures the same for the upper model and eventual coverage of the lower model ensures the same for the Poisson model. Thus it suffices to consider the eventual coverage question for a discrete model, as in Proposition 3.1 below, and check that the random variables ρu and ρl satisfy the conditions of the proposition. We take {Xi : i ∈ Nd } to be an i.i.d. collection of {0, 1} valued random variables with p := P (Xi = 1) and {ρi : i ∈ Nd } to be another i.i.d. collection of positive integer valued random variables with distribution function F (= 1 − G) and independent of {Xi : i ∈ Nd }. Let C := ∪{i∈Nd | Xi =1} (i + [0, ρi ]d ). We first consider eventual coverage of Nd by Xi . Proposition 3.1. Let d ≥ 2 and 0 < p < 1. (a) if limj→∞ jG(j) = 0 then Pp (C eventually covers Nd ) = 0, (b) if lim inf j→∞ jG(j) > 0 then Pp (C eventually covers Nd ) = 1. We sketch the proof for d = 2. For i, j ∈ N let A(i, j) := {(i, j) ∈ C}. Clearly, P (A(k, j) ∩ A(i, j)) = P (A(k − i, j))P (A(i, j)) for k ≥ i, i.e.,
∞ for each fixed j the event A(i, j) is a renewal event. Thus, if, for every j ≥ 1, i=1 P (A(i, j)) = ∞ then, on every line {y = j}, j ≥ 1, we have infinitely many i’s for which (i, j) is uncovered with probability one and hence Nd can never be eventually covered. To calculate Pp (A(i, j)) we divide the rectangle [1, i] × [1, j] as in Figure 1. For any point (k, l), 1 ≤ k ≤ i − j and 1 ≤ l ≤ j, in the shaded region of Figure 1, we ensure that either X(k,l) = 0 or ρ(k,l) ≤ k + j − 1. The remaining square region in Figure 1 is decomposed into j sub squares of length t, 1 ≤ t ≤ j − 1 and we ensure that for each point (k, l) on the section of the boundary of the sub square t given by the dotted lines either X(k,l) = 0 or ρ(k,l) ≤ t. So, Pp (A(i, j)) = (1 − p)

j−1

(1 − p + pF (t − 1))

= (1 − p)

j−1

(1 − pG(t))2t+1

2t+1

t=1

i−j

(1 − p + pF (k + j − 1))j

k=1

t=1

i

(1 − pG(k))j .

(3)

k=j+1

Now choose  > 0 such that pj < 1 and get N such that, for all i ≥ N, iG(i) < . j−1 2t+1 from (3) we have that Taking cj := t=1 (1 − pG(t)) ∞ 

Pp (A(i, j)) = (1 − p)cj

i=N

= (1 − p)cj

∞ i−j 

i=N k=1 ∞ 

ei (say).

i=N

j

(1 − pG(k + j))

(4)

R. Roy

124

(1, j)

(i − j, j)

(i, j)

(1, 1)

(i − j, 1)

(i, 1)

Fig 1. Division of the rectangle formed by [1, i] × [1, j].

For m ≥ N we have em+1 j = (1 − pG(m + 1)) em j  p ≥ 1− m+1   j  p k k j = 1−j + (−p) m+1 k (m + 1)k k=2

= 1−

g(m, p, j, ) pj + , m+1 (m + 1)2

for some in m. Thus by Gauss’ test, as pj < 1 we
∞function g(m, p, j, ) bounded
∞ have i=N ei = ∞ and hence i=1 Pp (A(i, j)) = ∞. This completes the proof of the first part of the proposition. For the next part we fix η > 0 such that η < lim inf j→∞ jG(j) and get N1 such that for all i ≥ N1 we have iG(i) > η. Also, fix 0 < p < 1 and choose a such that 0 < exp(−pη) < a < 1. Let N2 be such that for all j ≥ N2 we have (1 − pηj −1 )j < a. For N := max{N1 , N2 }, let i, j ∈ N be such that j ≥ N and i > j. Define A(i, j) := {(i, j) ∈ C}. As in (3) we have Pp (A(i, j)) = (1 − p)

i−j

k=1

(1 − pG(j + k))

j

j−1 t=1

(1 − pG(t))

2t+1

.

(5)

Coverage of space in Boolean models

Taking cj :=

j−1 t=1

∞ 

(1 − pG(t))

2t+1

125

, we have from (5) and our choice of j,

Pp (A(i, j)) = (1 − p)cj

i=N

= (1 − p)cj

∞ i−j 

j

(1 − pG(k + j))

i=N k=1 ∞ 

bi (say).

i=N

For m ≥ N bm+1 j = (1 − pG(m + 1)) bm j  η ≤ 1−p m+1 h(m, p, j, η) pjη + = 1− m+1 (m + 1)2

(6)

for j, η) bounded in m; thus by Gauss’ test, if pjη > 1 then
∞some function h(m, p,
∞ i=N bi < ∞ and hence i=1 Pp (A(i, j)) < ∞. Now, for a given p, let j  := sup{j : pjη < 1} and j0 := max{j  + 1, N }. We next show that the region Qj0 := {(i1 , i2 ) ∈ Nd : i1 , i2 ≥ j0 } has at most finitely many points that are not covered by C almost surely; there by proving that C eventually covers Nd . For this we apply Borel-Cantelli lemma after showing that
(i1 ,i2 )∈Qj0 Pp (A(i1 , i2 )) < ∞. Towards this end we have  Pp (A(i1 , i2 )) i1 ,i2 ≥j0

= 2(1 − p)

∞ 

j

0 +m−1

m=1



(1 − pG(t))2t+1

t=1

×

∞ 

k−m

(1 − pG(j0 + m + i))j0 +m



k=m+1 i=1

+

∞ j0 +k−1 

k=1

(1 − pG(t))2t+1 .

t=1

Observe that σm := ≤

∞ 

k−m

k=m+1 i=1 ∞ s   s=1 i=1

(1 − pG(j0 + m + i))j0 +m

pη 1− j0 + m + s

j0 +m

,

hence as in (6) and the subsequent application of Gauss’ test, we have that, for every m ≥ 1, σm < ∞. j0 +m−1 (1 − pG(t))2t+1 σm . Note that an application of the ratio Now let γm := t=1

R. Roy

126

test yields γm+1 γm


∞

m=1

γm < ∞; indeed from (7),


∞ s (1 − pG(j0 + m + 1 + i))j0 +m+1 s=1 s
∞ i=1 = (1 − pG(j0 + m)) j0 +m i=1 (1 − pG(j0 + m + i)) s=1 2j +2m+1
∞ s (1 − pG(j0 + m + 1 + i))j0 +m+1 (1 − pG(j0 + m)) 0 s=1
i=1 = s ∞ j +m 1 + j0 +m (1 − pG(j0 + m + 1)) 0 i=2 (1 − pG(j0 + m + i)) s=2
∞ s (1 − pG(j0 + m + 1 + i))j0 +m+1 j0 +m+1 s=1
∞ i=1 s ≤ (1 − pG(j0 + m)) . 1 + s=1 i=1 (1 − pG(j0 + m + 1 + i))j0 +m 2j0 +2m+1

Since σm < ∞ for all m ≥ 1, both the numerator and the denominator in the fraction above are finite. Moreover, each term in the sum of the numerator is less than the corresponding term in the sum of the denominator; yielding that the fraction is at most 1. Hence, for 0 < a < 1 as chosen earlier γm+1 j +m+1 ≤ (1 − pG(j0 + m)) 0 γm ≤ a.
∞ This shows that m=1 γm < ∞ and completes the proof of part (b) of the proposition. It may now be seen easily that ρu and ρl satisfy the conditions of Proposition 3.1 and thus Theorem 1.4 holds. 4. Markov Model The relation between the Poisson model and the discrete model explained in Section 3 shows that Theorem 1.3 would follow once we establish Theorem 1.5. In the setup of the Theorem
1.5, for each k ∈ N let Ak := {k ∈ C}. To prove Theorem 1.5(a) we show that lemma k P (Ak ) < ∞ and an application of the Borel-Cantelli
yields the result, while to prove Theorem 1.5 (b) we show that k P (Ak ) = ∞. However, the Ak ’s are not independent and hence Borel-Cantelli lemma cannot be applied. Nonetheless using the Markov property one can show that P (Ak ∩ Ai ) = P (A )P (Ai ) and therefore, Ai ’s are renewal events; so by the renewal theorem,
k−i ∞ if i=1 P (Ai ) = ∞ then Ai occurs for infinitely many i’s with probability one. For k ≥ 1, let P0 (Ak ) = P (Ak | X1 = 0) and P1 (Ak ) = P (Ak | X1 = 1). The following recurrence relations may be easily verified P0 (Ak+1 ) = p00 P0 (Ak ) + p01 P1 (Ak ) (7) P1 (Ak+1 ) = F (k − 1) [p10 P0 (Ak ) + p11 P1 (Ak )] . (8)
∞ k We use this to prove Theorem 1.5(b) first. Let Ψ0 (s) = k=k0 P0 (Ak )s and
∞ k Ψ1 (s) = denote the generating functions of the sequences k=k0 P1 (Ak )s {P0 (Ak ) : k ≥ k0 } and {P1 (Ak ) : k ≥ k0 } respectively, where k0 is such that for a given  > 0 and C = L +  > 0 (where L is as in the statement of the theorem), C for k ≥ k0 . k0 +(1−C) > 0, P0 (Ak0 ) > 0, P1 (Ak0 ) > 0, and F (k−1) ≥ 1− k+1 Such a k0 exists by the condition of the theorem. Using the recurrence relations (7) and (8) we obtain Ψ1 (s)P (s) ≥ Q(s)B(s) + R(s),

(9)

Coverage of space in Boolean models

127

where P (s) = (1 − p00 s)2 (1 − p11 s) + p10 s(1 − p00 s)p01 s = (1 − p00 s)(1 − s)(1 − s(1 − p01 − p10 )) Q(s) = (1 − p00 s)2 (1 − C)p11 + (1 − C)p10 p01 s(1 − p00 s) +p10 sp01 (1 − p00 s) + p10 p00 p01 s2 R(s) = (1 − p00 s)2 k0 sk0 −1 P1 (Ak0 ) + (k0 + 1 − C)p10 sk0 (1 − p00 s)P0 (Ak0 ) +p10 sk0 +1 p00 P0 (Ak0 ). From (9) we have for any 0 < t < 1 Ψ1 (t) ≥ e

t

Q(s) ds 0 P (s)



t

e 0

s 0

−Q(r) dr P (r)

R(s) ds. P (s)

D E F Now for s < 1, Q(s) P (s) = 1−p00 s + 1−s + 1−s(1−p01 −p10 ) , for some real numbers D, E, F. It may now be seen that Ψ1 (1) = ∞ whenever E > 0. Also the recurrence relations show that Ψ0 (1) = ∞ whenever Ψ1 (1) = ∞. A simple calculation now 1 01 yields that E > 0 if and only if p10p+p < C1 = L+ . Since  is arbitrary, we obtain 01 Theorem 1.5(b). The proof of Theorem 1.5(a) is similar.

References [1] Athreya, S., Roy, R., and Sarkar, A. (2004). On the coverage of space by random sets. Adv. in Appl. Probab. 36, 1, 1–18. MR2035771 [2] Grimmett, G.R., Keane, M., and Marstrand, J.M. (1984). On the connectedness of a random graph. Math. Proc. Cambridge Philos. Soc. 94, 1, 151–166. MR0743711 [3] Hall, P. A. (1988). Introduction To the Theory of Coverage Processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc., New York. MR0973404 [4] Kalikow, S. and Weiss, B. (1988). When are random graphs connected. Israel J. Math.. 62, 3, 257–268. MR0955131 [5] Kendall, M.G, and Moran, P.A.P (1963). Geometrical Probability, Griffin’s Statistical Monographs & Courses, No. 10. Hafner Publishing Co., New York. MR0174068 [6] Mandelbrot, B. B. (1972). On Dvoretzky coverings for the circle. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22, 158–160. MR309163 [7] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge Tracts in Mathematics, Vol. 119. Cambridge University Press, Cambridge. MR1409145 [8] Molchanov, I. and Scherbakov, V. a. (2003). Coverage of the whole space. Adv. in Appl. Probab. 35, 4, 898–912. MR2014261 [9] Shepp, L. A. (1972). Covering the circle with random arcs. Israel J. Math. 11, 328–345. MR295402

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 128–143 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000167

Strong invariance principle for dependent random fields∗ Alexander Bulinski1 and Alexey Shashkin1 Moscow State University Abstract: A strong invariance principle is established for random fields which satisfy dependence conditions more general than positive or negative association. We use the approach of Cs¨ org˝ o and R´ ev´ esz applied recently by Balan to associated random fields. The key step in our proof combines new moment and maximal inequalities, established by the authors for partial sums of multiindexed random variables, with the estimate of the convergence rate in the CLT for random fields under consideration.

1. Introduction and main results Strong invariance principles are limit theorems concerning strong approximation for partial sums process of some random sequence or field by a (multiparameter) Wiener process. The first result of such type was obtained by Strassen [21] with the help of Skorokhod’s embedding technique. Another powerful method, introduced by Cs¨ org˝ o and R´ev´esz [10], is based on quantile transforms. It was used by K´ omlos, Major and Tusnady [15, 16] to achieve an unimprovable rate of convergence in the strong invariance principle for independent identically distributed random sequences. Berkes and Morrow [2] extended that method to mixing random fields. In this paper, we study random fields with dependence condition proposed by Bulinski and Suquet [7] (in the case of a random sequence it was given by Doukhan and Louhichi [11]). Namely, let X = {Xj , j ∈ Zd } be a real-valued random field on a probability space (Ω, F, P) with EXj2 < ∞ for any j ∈ Zd . We say that X is weakly dependent, or (BL, θ)−dependent, if there exists a sequence θ = (θr )r∈N of positive numbers, θr → 0 as r → ∞, such that for any pair of disjoint finite sets I, J ⊂ Zd and any pair of bounded Lipschitz functions f : R|I| → R and g : R|J| → R one has |cov(f (Xi , i ∈ I), g(Xj , j ∈ J))| ≤ Lip(f )Lip(g)(|I| ∧ |J|)θr .

(1.1)

Here and below |V | stands for the cardinality of a finite set V, r = dist(I, J) = min{i − j : i ∈ I, j ∈ J} with the norm z = maxi=1,...,d |zi |, z = (z1 , . . . , zd ) ∈ Zd , and, for F : Rn → R, Lip(F ) = sup x=y

|F (x) − F (y)| . |x1 − y1 | + · · · + |xn − yn |

Note that one can apply (1.1) to unbounded Lipschitz functions f and g whenever Ef 2 (Xi , i ∈ I) < ∞ and Eg 2 (Xj , j ∈ J) < ∞. * This

work is partially supported by the RFBR grant 03-01-00724, by the grant 1758.2003.1 of Scientific Schools and by the INTAS grant 03-51-5018. 1 Dept. of Mathematics and Mechanics, Moscow State University, Moscow, 119992, Russia, e-mail: [email protected] e-mail: [email protected] AMS 2000 subject classifications: primary 60F15, 60F17. Keywords and phrases: dependent random fields, weak dependence, association, covariance inequalities, strong invariance principle, law of the iterated logarithm. 128

Strong invariance principle

129

The interest in studying model (1.1) is motivated by the following fact. There are a number of important stochastic models in mathematical statistics, reliability theory and statistical physics involving families of positively and negatively associated random variables (see [13, 14, 18] for the exact definitions and examples, for further references see, e.g., [7]). As shown by Bulinski and Shabanovich [8], a positively or negatively associated random field with finite second moments satisfies (1.1), provided that the Cox-Grimmett coefficient  |cov(Xu , Xj )|, r ∈ N, θ
r = sup j∈Zd

u∈Zd :u−j≥r

is finite and θ
r → 0 when r → ∞. In this case one can take θ = (θ
r )r∈N . There are also examples of (1.1) which are not induced by association, see [11, 20]. A strong invariance principle for associated random sequences whose Cox-Grimmett coefficient decreases exponentially was proved by Yu [22]. Recently Balan [1] extended this result to associated random fields. The principal goal of this paper is to extend the strong invariance principle to (BL, θ)−dependent random fields. The new maximal inequality needed is given  in Theorem 1.1. For any finite V ⊂ Zd , we let S(V ) = j∈V Xj . The sum over empty set is zero, as usual. We call a block a set V = (a, b] := ((a1 , b1 ] × · · · × (ad , bd ]) ∩ Zd when a, b ∈ Zd , a1 < b1 , . . . , ad < bd . Given a block V, set M (V ) = maxW ⊂V |S(W )| where the supremum is over all the blocks W contained in V. Assume that (1.2) Dp := sup E|Xj |p < ∞ for some p > 2. j∈Zd

We will use condition (1.1) specialized to a sequence θ with a power or exponential rate of decreasing. Namely, either

or

θr ≤ c0 r−λ , r ∈ N, for some c0 > 1 and λ > 0,

(1.3)

θr ≤ c0 e−λr , r ∈ N, for some c0 > 1 and λ > 0.

(1.4)

Introduce a function   1)(x − 2)−1 , 2 < x ≤ 4, (x − √ √ ψ(x) = (3 − x)( x + 1)/2, 4 < x ≤ t20 , (1.5)    ((x − 1) (x − 2)2 − 3 − x2 + 6x − 11)(3x − 12)−1 , x > t20

where t0 ≈ 2.1413 is the maximal root of the equation t3 + 2t2 − 7t − 4 = 0.

Note that ψ(x) → 1 as x → ∞. Now let us formulate the first of the main results of this paper. Theorem 1.1. Let X be a centered (BL, θ)−dependent random field satisfying (1.2) and (1.3) with λ > dψ(p) for ψ(p) defined in (1.5). Then there exist δ > 0 and C > 1 depending only on d, p, Dp , c0 and λ such that for any block U ⊂ Zd one has E|S(U )|2+δ ≤ C|U |1+δ/2 , EM (U )2+δ ≤ AC|U |1+δ/2 where A = 5d (1 − 2δ/(4+2δ) )−d(2+δ) .

(1.6)

A. Bulinski and A. Shashkin

130

Remark 1. Moment and maximal inequalities for associated random fields were obtained in [4] and [6]. In the paper [19] similar inequalities were proved for weakly dependent random fields X = {Xj , j ∈ Zd } under a stronger moment condition supj∈Zd E|Xj |4+δ < ∞, δ > 0. An inequality for a (BL, θ)−dependent field X having only finite second moments when λ > 3d in (1.3) was also established there, permitting to prove a weak invariance principle in the strictly stationary case. That result does not comprise ours. Remark 2. The condition on the rate of decrease of θr determined by the function ψ in (1.5) is implied by a simple condition λ > d(p − 1)/(p − 2) because ψ(p) ≤ (p − 1)/(p − 2) for all p > 2. Now suppose that  cov(X0 , Xj ) = 0. (1.7) σ 2 := j∈Zd

Note that (1.1) entails the convergence of series in (1.7) for a field X with EXj2 < ∞, j ∈ Zd . As is generally known, for a wide-sense stationary field X one has var(SN ) ∼ σ 2 [N ] as N → ∞

(1.8)

where N ∈ Nd , [N ] = N1 . . . Nd , SN = S((0, N ]) and N → ∞ means that N1 → ∞, . . . , Nd → ∞. Following [2], for any τ > 0, we introduce the set Gτ =

d

 τ  . j ∈ Nd : js ≥ js


s=1

(1.9)

s
=s

Theorem 1.2. Suppose that X is a wide-sense stationary (BL, θ)−dependent centered random field satisfying (1.2), (1.4) and (1.7). Then one can redefine X, without changing its distribution, on a new probability space together with a d−parameter Wiener process W = {Wt , t ∈ [0, ∞)d }, so that for some ε > 0 the following relation holds SN − σWN = O([N ]1/2−ε ) a.s. (1.10) as N → ∞, N ∈ Gτ and τ > 0. Remark 3. The value ε in (1.10) depends on a field X. More precisely, ε is determined by τ, the covariance function of X and parameters d, p, Dp , c0 , λ. Note that Gτ = ∅ for τ > 1/(d − 1), d > 1. One can easily obtain an analogue of Theorem 1.2 for wide-sense stationary weakly dependent stochastic process (i.e. for d = 1). 2. Proof of Theorem 1.1 We fix some δ ∈ (0, 1], δ < p − 2. The exact value of δ will be specified later. Choose Aδ > 0 (e.g., Aδ = 5) to ensure that (x + y)2 (1 + x + y)δ ≤ x2+δ + y 2+δ + Aδ ((1 + x)δ y 2 + x2 (1 + y)δ ) for any x, y ≥ 0. Let h(n) = min{k ∈ Z+ : 2k ≥ n}, n ∈ N. For any block V ⊂ Zd having edges with lengths l1 , . . . , ld , we set h(V ) = h(l1 ) + · · · + h(ld ). We will show that for some C > 2(Dp ∨ 1) and all blocks U ⊂ Zd ES 2 (U )(1 + |S(U )|)δ ≤ C|U |1+δ/2 .

(2.1)

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131

This is proved by induction on h(U ). For h(U ) = 0 (i.e. when |U | = 1) inequality (2.1) is obviously true. Suppose now that (2.1) is verified for all U such that h(U ) ≤ h0 . Consider a block U having h(U ) = h0 + 1. Let L be any of the longest edges of U. Denote its length by l(U ). Draw a hyperplane orthogonal to L dividing it into two intervals of lengths [l(U )/2] and l(U )−[l(U )/2], here [·] stands for integer part of a number. This hyperplane divides U into two blocks U1 and U2 with h(U1 ), h(U2 ) ≤ h0 . Lemma 2.1. There exists a value τ0 = τ0 (δ) < 1 such that, for any block U ⊂ Zd with |U | > 1, one has |U1 |1+δ/2 + |U2 |1+δ/2 ≤ τ0 |U |1+δ/2 . Proof. Straightforward. Observe that for the considered field X condition (1.3) implies the bound ES 2 (U ) ≤ (D2 + c0 )|U |

(2.2)

for any block U ⊂ Zd . Set Qk = S(Uk ), k = 1, 2. By induction hypothesis and Lemma 2.1, ES 2 (U )(1 + |S(U )|)δ = E(Q1 + Q2 )2 (1 + |Q1 + Q2 |)δ ≤ C(|U1 |1+δ/2 + |U2 |1+δ/2 ) + Aδ E((1 + |Q1 |)δ Q22 + (1 + |Q2 |)δ Q21 )

 ≤ Cτ0 |U |1+δ/2 + Aδ E (1 + |Q1 |)δ Q22 + (1 + |Q2 |)δ Q21 .(2.3)

Our goal is to obtain upper bounds for E(1 + |Q1 |)δ Q22 and E(1 + |Q2 |)δ Q21 . We proceed with the first estimate only, the second one being similar. To this end, let us take positive ζ < (1 − τ0 )/(4Aδ ) and introduce a block V letting V = {j ∈ U2 : dist({j}, U1 ) ≤ ζ|U |1/d }. Note that the induction hypothesis applies to V, since V ⊆ U2 . Using the H¨ older inequality and (2.2), one shows that E(1 + |Q1 |)δ Q22 ≤ 2E(1 + |Q1 |δ )S 2 (V ) + 2E(1 + |Q1 |)δ S 2 (U2 \ V ) ≤ 2(D2 + c0 )|V | + 2(E|Q1 |2+δ )δ/(2+δ) (E|S(V )|2+δ )2/(2+δ) + 2E(1 + |Q1 |)δ S 2 (U2 \ V ) ≤ 2(D2 + c0 )|U | + 2Cζ|U |1+δ/2 + 2E(1 + |Q1 |)δ S 2 (U2 \ V ).

(2.4)

Fix any indices i, j ∈ U2 \V and assume first that i = j. Then dist({j}, {i}∪U1 ) = m > 0. For any y > 0, we define the function Gy by Gy (t) = (|t| ∧ y)sign(t), t ∈ R, and for some y, z ≥ 1 introduce the random variables

 δ I δ 1/δ QI1 = Gy (Q1 ), QII , 1 = (1 + |Q1 |) − (1 + |Q1 |) XiI = Gz (Xi ), XiII = Xi − XiI .

(2.5)

A. Bulinski and A. Shashkin

132

I II To simplify the notation we do not write QI1,y , QII 1,y , Xi,z and Xi,z . Obviously

|E(1 + |Q1 |)δ Xi Xj | ≤ |E(1 + |QI1 |)δ XiI Xj | δ I + E(1 + |Q1 |)δ |XiII Xj | + E|QII 1 | |Xi Xj |.

(2.6)

Note that Φ(v, w) = (1 + |Gy (v)|)δ Gz (w) is a bounded Lipschitz function with Lip(Φ) ≤ 2y δ + z. Since X is a weakly dependent centered field, we can write |E(1 + |QI1 |)δ XiI Xj | = |cov((1 + |QI1 |)δ XiI , Xj )| ≤ (2y δ + z)θm .

(2.7)

Let q be a positive number such that 1/q + δ/(2 + δ) + 1/p = 1, that is q = p(2 + δ)/(2p − 2 − δ) < p. By the H¨ older and Lyapunov inequalities,     E 1 + |Q1 |)δ |XiII Xj  ≤ (E|XiII |q )1/q Dp1/p 1 + (E|Q1 |2+δ )δ/(2+δ)  1/q Dp δ/(2+δ) δ/2 ≤ 2C |U | (2.8) Dp1/p , z p−q the last estimate being due to the induction hypothesis. For r ∈ (δ, 2 + δ) to be specified later,   δ I II δ II δ II δ E|QII | |X X | ≤ zE|Q | |X | ≤ z |cov(|Q |)| + E|X |E|Q | , |X | j j j j 1 i 1 1 1 ≤ zδy δ−1 θm + 2zy δ−r D1 C r/(2+δ) |U |r/2 .

(2.9)

The last inequality follows from the induction hypothesis and the fact that the function v → ((1 + |v|)δ − (1 + y)δ )I{|v| ≥ y} is a Lipschitz one. Now from (1.3) and (2.6)—(2.9), denoting T = 2c0 (1 ∨ Dp ), we conclude that   |E(1+|Q1 |)δ Xi Xj | ≤ T C r/(2+δ) (y δ +z)m−λ +|U |δ/2 z 1−p/q +zy δ−r |U |r/2 . (2.10) Let β, γ be positive parameters. Introduce y = |U |1/2 mβλ , z = mγλ . Then in view of (2.10) we obtain   3  |E(1 + |Q1 |)δ Xi Xj | ≤ T C r/(2+δ) |U |δ/2 m−λνk + m−λν4 k=1

where ν1 = 1 − δβ, ν2 = γ



 p − 1 , ν3 = (r − δ)β − γ, ν4 = 1 − γ. q

Our next claim is the following elementary statement. Lemma 2.2. For each d ∈ N, any block U ⊂ Zd , every ν > 0 and arbitrary i ∈ U one has  i − j−ν ≤ c(d, ν)f (|U |, d, ν) (2.11) j∈U,j=i

where c(d, ν) > 0 and  1−ν/d  , |U | f (|U |, d, ν) = (1 + ln |U |)|U |1−ν/d ,   1,

0 < ν < d, ν ∈ N, 0 < ν ≤ d, ν ∈ N, ν > d.

Strong invariance principle

133

Proof. The case d = 1 is trivial. For d ≥ 2, consider U = (a, b] ∩ Zd . Without loss of generality we can assume that l1 ≤ · · · ≤ ld where ls = bs − as , s = 1, . . . , d. It is easily seen that 

i − j

−ν

d

≤3 d

d−1 

ls+1

lm

s=0 1≤m≤s

j∈U,j=i



k d−s−1−ν

k=ls +1

where l0 = 0 and a product over an empty set is equal to 1. Using the well-known r2 estimates for sums k=r k γ by means of corresponding integrals and the estimates 1 

1≤m≤s



lm l

d−s−ν

≤



1≤m≤d

lm

1−ν/d

for l = ls and l = ls+1 we come to (2.11). The Lemma is complete. Now pick r close enough to 2 + δ and β, γ in such a way that λνk > d, k = 1, 2, 3. One can verify that this is possible if λ > λ1 (d) = d

(2 + δ)(2p − 4 − δ) . 4(p − 2 − δ)

(2.12)

Moreover, to have simultaneously λν4 > (1 − δ/2)d it suffices to require  δ 2+δ +1− . λ > λ2 (d) = d 2(p − 2 − δ) 2

(2.13)

The condition imposed on λ in Theorem 1.1 enables us to satisfy (2.12) and (2.13) √ taking δ small enough when p ≤ 4, respectively δ = p − p − 2 when 4 < p ≤ t20 and   2 δ= p − 2 − (p − 2)2 − 3 3 otherwise. For arbitrary i ∈ U2 \ V, set 2 = {j ∈ U2 \ V : j − i < ζ|U |1/d }. U2 = {j ∈ U2 \ V : j − i ≥ ζ|U |1/d }, U

By Lemma 2.2, for any i ∈ U2 \ V, we have          E(1 + |Q1 |)δ Xi Xj  E(1 + |Q1 |)δ Xi Xj  ≤   j=i,j∈U2 \V

j∈U2

     δ + E(1 + |Q1 |) Xi Xj 
2 ,j=i j∈U 4    −λν0 r/(2+δ) δ/2 4ζ ≤ TC c(d, λνk ) , (2.14) + |U | k=1

here ν0 = maxk=1,...,4 νk . Now we treat the case of i = j ∈ U2 \ V. Obviously, one has δp/(p − 2) < 2 + δ. Therefore, by H¨ older’s inequality and induction hypothesis we infer that  (p−2)/p   δp/(p−2) δ 2 p 2/p 1 + E|Q1 | E(1 + |Q1 |) Xi ≤ (E|Xi | ) ≤ T C δ/(2+δ) |U |δ/2 .

(2.15)

A. Bulinski and A. Shashkin

134

From (2.14) and (2.15) one deduces that       E(1 + |Q1 |)δ S 2 (U2 \ V ) ≤  E(1 + |Q1 |)δ Xi2 E(1 + |Q1 |)δ Xi Xj  + i,j∈U2 \V i=j

i∈U2 \V

  ≤ |U | max  i∈U2 \V



j=i,j∈U2 \V

δ/(2+δ)

  E(1 + |Q1 |)δ Xi Xj 

+ TC |U |1+δ/2 ≤ M C r/(2+δ) |U |1+δ/2 (2.16)  4 where M = T (1 + 4ζ −λν0 + k=1 c(d, λνk )). Employing (2.3), (2.4) and (2.16) we conclude that   ES 2 (U )(1 + |S(U )|)δ ≤ Cτ0 + 4Aδ (D2 + c0 )4CAδ ζ + 4C r/(2+δ) Aδ M |U |1+δ/2 . The first assertion of the Theorem is now easily verified on account of (2.1) if C is so large that (1 − τ0 − 4Aδ ζ)C > 4Aδ M C r/(2+δ) + 4Aδ (D2 + c0 ). The second assertion follows from the first one and the Moricz theorem [17].
3. Proof of Theorem 1.2 The proof adapts the approach of [1] and [2]. However, as the random field under consideration possesses a dependence property more general than association, we have to involve other results on normal approximation and partial sums behaviour. We also give simplified proofs of some steps. Let α > β > 1 be integers specified later. Introduce n0 = 0, nl :=

l  (iα + iβ ), l ∈ N.

(3.1)

i=1

For k ∈ Nd , put k − 1 = (k1 − 1, . . . , kd − 1) and Nk = (nk1 , . . . , nkd ). Set Bk = (Nk−1 , Nk ], Hk =

d

(nks −1 , nks −1 + ksα ], Ik = Bk \ Hk ,

s=1

(3.2)

uk = S(Hk ), σk2 = var(uk ), vk = S(Ik ), τk2 = var(vk ). We can redefine the random field {uk } on another probability space together with a random field {wk , k ∈ Nd } of independent random variables such that wk ∼ N (0, τk2 ) and the fields {uk } and {wk } are independent. Further on we will denote by C any positive factor which could depend only on d, c0 , λ, p, Dp , τ and the covariance function of the field X except when specially mentioned. Occasionally when C is a positive random variable, we write C(ω).1 Now we pass to a number of lemmas. In their formulations, the requirements “(1.3) holds” or “(1.4) holds” mean that the field X is (BL, θ)−dependent with the decrease of θ as mentioned. Due to weak stationarity and (1.7), we have σk2 > 0 for any k ∈ Nd . There exists k0 ∈ N such that τk2 > 0 for all k ∈ Nd with mins=1,...,d ks ≥ k0 . This is an immediate consequence of the following statement. 1A

point ω, in general, belongs to an extension of the initial probability space.

Strong invariance principle

135

Lemma 3.1. Let X be a random field such that D2 < ∞ and (1.4) holds. Then, for any finite union V of disjoint blocks Vq , one has σ2 −

1 var(S(V )) = O(l(V )−1/2 ), as l(V ) → ∞, |V |

(3.3)

where l(V ) is the minimal edge of all the blocks Vq . Proof. We have          2   cov(Xj , Xk ) = |Σ1 + Σ2 |, σ |V | − var(S(V )) =  j∈V,k∈V 

where Σ1 is the sum over k ∈ V and all j’s such that dist({j}, {Zd \ V }) ≥ l(V )1/2 , and Σ2 is over dependence  taken 1/2  k ∈ V and the rest j ∈d V. By weak −1/2 |Σ . Furthermore, |Σ2 | ≤ 2 C|V |l(V ) with C = 1 | ≤ C|V | exp −λl(V ) |cov(X , X )| < ∞ and the Lemma is true. d 0 k k∈Z In what follows, we will consider only k “large enough”, that is having mins ks ≥ k0 . For such k and x ∈ R let uk + wk , Fk (x) = P(ξk ≤ x). ξk =  2 σk + τk2

(3.4)

Then ξk has a density fk (x). Analogously to [1] we introduce the random variables  ηk = Φ−1 (Fk (ξk )), ek = σk2 + τk2 (ξk − ηk ), k ∈ Nd , x 2 where Φ(x) = (2π)−1/2 −∞ e−t /2 dt. Let ρ = τ /8, L be the set of all indices i corresponding to the (“good”) blocks Bi ⊂ Gρ , and H be the set of points in Nd which belong to some good block. For each point N = (N1 , . . . , Nd ) ∈ H, let N (1) , . . . , N (d) be the points defined as follows: (s) Ns
:= Ns
, s = s and Ns(s) := min ns . n∈H ns
=Ns
,s
=s

We consider also the sets (1)

(d)

Rk = (Mk , Nk ] where Mk = ((Nk )1 , . . . , (Nk )d ), Lk = {i : Bi ⊂ Rk }. Clearly, 

 σi2 + τi2 S(Rk ) = − σ ηi |Bi | ei + |Bi | i∈Lk i∈Lk     σ |Bi |ηi − + vi . wi + 

i∈Lk



i∈Lk

(3.5)

i∈Lk

If V = (a, b] ∩ Zd+ is a block in Zd+ for some a, b ∈ Zd+ , and V = (a, b] ⊂ Rd+ , then set W (V ) := W (V ). Here W (V ) is defined as usual (i.e. the signed sum of W (tl ) where tl , l = 1, . . . , 2d , are the vertices of V ).

A. Bulinski and A. Shashkin

136

Lemma 3.2. If (1.2), (1.3) and (1.7) are satisfied, then for any k ∈ L sup |Fk (x) − Φ(x)| ≤ C[k]−αµ x∈R

where µ > 0 does not depend on k. Proof. If X is a weakly dependent random field and we replace some of the variables Xj with independent from X and mutually independent random variables Yj , then the random field X  = {Xj , j ∈ Nd } obtained is again weakly dependent with the same sequence θ = (θr )r∈N (cf. [12]). Moreover, note that if Xj = Yj ∼ N (0, τk2 /|Ik |) for j ∈ Ik and Xj = Xj for other j ∈ Zd then for some C > 0,  σ2 + τ 2   1  Xj ≥ kd αk ≥ C var |Bk | 2 [k]

(3.6)

j∈Bk

for any Bk ⊂ Gρ , k ∈ Nd . This is true since the middle expression in (3.6) is always positive and in view of (1.8) and (3.3) has positive limit as k → ∞, k ∈ L. Now the desired result follows from Theorem 4 of [9] by standard Bernstein’s technique, analogously to Theorem 5 of that paper. √ Lemma 3.3. Under conditions (1.2), (1.3) and (1.7) for any K ∈ (0, 2αµ) one has 2 |Φ−1 (Fk (x)) − x| ≤ C[k]−αµ+K /2  if |x| ≤ K ln[k]. Here C may also depend on K. Proof. Follows the main lines of that given in [10] (Lemmas 2 and 3).

Lemma 3.4. If (1.2), (1.4) and (1.7) hold, then Ee2k ≤ C[k]α−ε0 for ε0 = αµδ/(2 + 2δ) where δ > 0 is the same as in Theorem 1.1 and µ is the same as in Lemma 3.2. Proof. In view of (2.2) our task is to show that E(ηk − ξk )2 ≤ C[k]−ε0 . To this end we take K from Lemma 3.3 and write   E(ηk − ξk )2 = E(ηk − ξk )2 I{|ξk | ≤ K ln[k]} + E(ηk − ξk )2 I{|ξk | > K ln[k]}  2 ≤ C[k]−αµ+K /2 + (E|ηk − ξk |2+δ )2/(2+δ) (P(|ξk | > K ln[k]))δ/(2+δ)  2 ≤ C[k]−αµ+K /2 + C(P(|ηk | > K ln[k]))δ/(2+δ)   + C|P(|ξk | > K ln[k]) − P(|ηk | > K ln[k])|δ/(2+δ) ≤ C([k]−αµ+K

2

/2

+ [k]−K

2

δ/2(2+δ)

+ [k]−αµδ/(2+δ) )

by Lemmas 3.2 and 3.3 and Theorem 1.1. The optimization in K yields the result. Lemma 3.5. If (1.2), (1.4), (1.7) hold and α − β ≤ ε0 /4, then sup |fk (x) − f (x)| ≤ C x∈R

2

where f (x) = (2π)−1/2 e−x Lemma 3.4.

/2

, fk appeared after (3.4) and ε0 is the same as in

Strong invariance principle

137

 2 Proof. Let ϕk (t) = E exp{itξk }, ϕ k (t) = E exp{ituk / σk2 + τk2 }, ϕ(t) = e−t /2 , where t ∈ R, i2 = −1. Note that   ϕk (t) = ϕ k (t) exp −τk2 t2 /2(σk2 + τk2 ) . By Lemma 3.4, for any t ∈ R

|ϕk (t) − ϕ(t)| ≤ E| exp{itξk } − exp{itηk }| ≤ |t|E|ξk − ηk | ≤ C|t|[k]−ε0 /2 . Therefore for any T > 0 and any x ∈ R, denoting νk = τk2 /(σk2 + τk2 ), one has  1 |fk (x) − f (x)| ≤ |ϕk (t) − ϕ(t)|dt 2π R     2 1 1 | ϕk (t)|e−t νk /2 + ϕ(t) dt |ϕk (t) − ϕ(t)|dt + ≤ 2π |t|≤T 2π |t|≥T  ∞  ∞ 2 −ε0 /2 −t2 /2 ≤ CT [k] +C e dt + C e−νk T t/2 dt T T    2 T νk 1 2 −ε0 /2 exp − ≤ C T [k] +1+ . T νk 2 The lemma follows if we take T = [k]α−β . Remark 4. Clearly the condition α − β ≤ ε0 /4 is equivalent to (α/β)(1 − µδ/8(1 + δ)) ≤ 1.

(3.7)

Note that in Lemmas 3.4, 3.5 and Lemma 3.6 below one can replace condition (1.4) on λ with that used in Theorem 1.1. Two lemmas which follow can be proven analogously to Lemmas 3.6 and 3.9 in [1]. Lemma 3.6. Suppose that (1.2), (1.4), (1.7) hold and α is so large that ε0 > 2. Then there exists ε1 > 0 such that  |ei | ≤ C(ω)[Nk ]1/2−ε1 a.s. i∈Lk

Lemma 3.7. Assume that D2 < ∞. If (1.3) holds and α − β > 6/ρ then, for some ε2 > 0,  (|vi | + |wi |) ≤ C(ω)[Nk ]1/2−ε2 a.s. i∈Lk

Lemma 3.8. Suppose that D2 < ∞. If (1.4), (1.7) hold and β > 6/ρ then, for some ε3 > 0,     σi2 + τi2 |Bi | σ − |ηi | ≤ C(ω)[Nk ]1/2−ε3 a.s. (3.8) |Bi | i∈Lk

Proof. With the help of an inequality     σi2 + τi2  1    var(ui + wi ) ≤ C[i]−βρ/4  ≤ σ −1 σ 2 − σ − |Bi | |Bi |

ensuing from Lemma 3.1, we come to (3.8) applying arguments analogous to those proving Lemma 3.8 in [1].

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Lemma 3.9. Suppose that X = {Xj , j ∈ Zd } is a weakly dependent random field and Y = {Yj , j ∈ Zd } is a field consisting of independent random variables and independent of X. Let I, J ⊂ Zd be disjoint finite sets and f : R|I| → R, g : R|J| → R be bounded Lipschitz functions. Then |cov(f (XI + YI ), g(XJ + YJ ))| ≤ Lip(f )Lip(g)(|I| ∧ |J|)θr

(3.9)

for r = dist(I, J), that is, such addition of Y does not alter the property (1.1). Proof. By a smoothing procedure we can reduce the general case to that of the random vector (YI , YJ ) with a density q(t1 , t2 ) = qI (t1 )qJ (t2 ), here t1 ∈ R|I| , t2 ∈ R|J| . Evidently (3.9) is true for a field Y consisting of some constants. Thus by independence hypothesis, the Fubini theorem and (1.1) we have |cov(f (XI + YI ), g(XJ + YJ ))|      cov(f (XI + yI ), g(XJ + yJ ))q(yI , yJ )dyI dyJ  =  ≤ Lip(f )Lip(g)(|I| ∧ |J|)θr q(yI , yJ )dyI dyJ = Lip(f )Lip(g)(|I| ∧ |J|)θr where the double integral is taken over R|I| × R|J| . Lemma 3.10 (Lemma 4.3 in [1]). There exists a bijection ψ : Z+ → L with the following properties: l < m ⇒ ∃s∗ = s∗ (l, m) ∈ {1, . . . , d} such that ψ(l)s∗ ≤ ψ(m)s∗ , ∃m0 ∈ Z+ such that m ≤ [ψ(m)]γ0 ∀m ≥ m0 , for any γ0 > (1 + 1/ρ)(1 − 1/d). Set Ym = ηψ(m) , m ∈ Z+ . Lemma 3.11. If (1.2), (1.4), (1.7) and (3.7) hold, then for every m ∈ N, m > 1, and all t = (t1 , . . . , tm ) ∈ Rm one has m   

m−1

     tl Yl E exp itm Ym  tl Yl − E exp i E exp i l=1

l=1

γ

2

≤ Cm∆ , i = −1,

(3.10)

where m



[ψ(m)]α  2 m−1 ∆= tl , r = dist Hψ(m) , ∪l=1 θr Hψ(l) , γ = m l=1

 1, 1/3,

∆ > 1, ∆≤1 (3.11)

and the cubes Hk were defined in (3.2). Proof. Let M > 0 be a number to be specified later and GM (t) be the function defined in (2.5). We set Yj,M = GM (ηψ(j) ), j ∈ Z+ . Note that | exp{itYj,M } − exp{itYj }| ≤ 2I{|Yj | > M }, t ∈ R.

Strong invariance principle

139

Therefore, we have   

m−1     tl Yl , exp −itm Ym  cov exp i l=1

 

m−1      ≤ cov exp i tl Yl,M , exp −itm Ym,M  l=1

+4

m 

P{|Yl | > M }.

(3.12)

l=1

2

Every summand in (3.12) except for the first one is not greater than Ce−M /2 . To estimate the first summand in the right hand side of (3.12), we notice that, for any k ∈ Nd , the random variable ηk,M is a Lipschitz function of ξk . Indeed, Lip(Fk ) ≤ C by Lemma 3.5 and ηk,M = hM (Fk (ξk )) where 

hM (x) = |Φ−1 (x)| ∧ M sign(Φ−1 (x)), x ∈ R.

√ 2 Clearly Lip(hM ) ≤ 2πeM /2 . By Lemma 3.9, one can estimate the covariance in the same way as if the normal variables wk , k ∈ Nd were constants, and with the help of (3.6) we obtain m   

m−1    !α   M2 tl Yl,M , exp −itm Ym,M  ≤ Cθr e ψ(m) t2l . cov exp i l=1

(3.13)

l=1

Thus, from (3.12)—(3.13) we see that m  

m−1         2 −M 2 /2 M2 α cov exp i , exp −it Y t Y t ≤ C me + e [ψ(m)] θ   m m l l r l . l=1

l=1

The result follows now by optimization in M. Lemma 3.12. Suppose that (1.2), (1.4), (1.7), (3.7) hold and β > 2γ0 /ρ where γ0 appears in the formulation of Lemma 3.10. Then we can redefine the random field X, without changing its distribution, on a new probability space together with a d−parameter Wiener process W = {Wt , t ∈ [0, ∞)d }, such that for some ε4 > 0       W (B )  i  (3.14) σ |Bi | ηi −   ≤ C(ω)[Nk ]1/2−ε4 a.s.   |B | i i∈Lk

Proof. Arguing as in the proof of Lemma 4.4 of [1], we see that by Berkes-Philipp strong approximation result ([3]) it is enough to establish the following fact. There exist sequences κm > 0 and zm > 104 m2 (m ∈ N, m > 1) such that m 2 (1) for any m ∈ N, m > 1, and all t = (t1 , . . . , tm ) ∈ Rm with l=1 t2l ≤ zm one has m 

  

m−1     tl Yl E exp itm Ym  ≤ κm , tl Yl − E exp i E exp i l=1

l=1

−1/4

(2) zm

1/2

1/4

1/2 m+1/4

ln zm + exp{−3zm /16}m1/2 zm + κm zm

= O(m−2 ), m → ∞.

A. Bulinski and A. Shashkin

140

We take zm = 104 mq , q > 8. Then it suffices to prove that one can take κm = O(exp{−AmR }), m → ∞, for some A > 0, R > 1. The distance r between Hm and any of the blocks H1 , . . . , Hm−1 is by construction not less than mins=1,...,d (ψ(m)s − 1)β . But since ψ(m) ∈ L, by Lemma 3.10 for m > m0 we have r ≥ C min ψ(m)βs ≥ C[ψ(m)]ρβ/2 ≥ Cmρβ/2γ0 .

(3.15)

s=1,...,d

From Lemma 3.10 one can also easily see that for large enough m [ψ(m)] ≤ C min ψ(m)2/ρ ≤ Cm2/ρ . s

(3.16)

s=1,...,d

Obviously, ∆1 := Cm2q+2α/ρ exp{−Cλmρβ/2γ0 } < 1 for all m large enough. Therefore, for such m, by Lemma 3.11, (1.4), (3.15) and (3.16), one has m 

 

m−1      1/3 tl Yl − E exp i tl Yl E exp itm Ym  ≤ Cm∆γ ≤ Cm∆1 E exp i l=1

l=1

where ∆ and γ are defined in (3.11). Thus one can take A = 1, R ∈ (1, ρβ/2γ0 ). The lemma is proved. Now we estimate the terms S((0, Nk ] \ Rk ), σW ((0, Nk ] \ Rk ), SN − S((0, Nk ]), σWN − σW ((0, Nk ]) when N ∈ Gτ and Nk < N ≤ Nk+1 . Here the relation a < b (a ≤ b) for a, b ∈ Zd is defined in the usual way. All the terms involving the Wiener process can be considered as sums of independent identically distributed normal random variables, therefore forming a weakly dependent field, we will proceed only with the partial sums generated by X. Obviously one can write |S((0, Nk ] \ Rk )| ≤

d 

(J)

2d−s Ms (Nk ), (0, N ] \ (0, Nk ] = ∪J Ik

s=1

" " (J) where Ms (N ) = maxn≤N (s) |Sn |, Ik = s∈J (nks , Ns ]× s∈J (0, nks ] and the union is taken over all non-empty subsets J of {1, . . . , d}. Furthermore,  (J) (J) (J) max |SN − S((0, Nk ])| ≤ Mk where Mk = sup |S(Ik )|, Nk
J

here the supremum is taken over all N such that nks < Ns ≤ nks +1 , s ∈ J. Lemma 3.13. Suppose that conditions of Theorem 1.1 are satisfied. Then there exists δ > 0 such that for any x > 0 and any block V,  P(M (V ) ≥ x |V |) ≤ Cx−2−δ . Proof. Follows from the second assertion of Theorem 1.1 and the Markov inequality. Lemma 3.14. Let conditions of Theorem 1.1 hold. If (1.7) is true then there exists γ1 > 0 such that for any block V = (m, m + n] with n ∈ Gρ and m ∈ Zd+ P(M (V ) ≥ |V |1/2 (ln |V |)d+1 ) ≤ C|V |−γ1 , where C does not depend on m and n.

Strong invariance principle

141

Proof. The proof is the same as that for the second inequality of Lemma 7 in [2]; the  needed Berry-Esseen type estimate for P(S(V ) ≥ x |V |), x > 0, can be obtained from the results mentioned in the proof of Lemma 3.2. The next two lemmas are proved analogously to Lemmas 6 and 9 established in [2] for mixing random fields. Lemma 3.15. Assume that conditions of Theorem 1.1 are satisfied and α > 8/(3τ ) − 1. Then max Ms (Nk ) ≤ C(ω)[Nk ]1/2−ε5 a.s. s=1,...,d

for some ε5 > 0 and every Nk ∈ Gτ . Lemma 3.16. Suppose that conditions of Theorem 1.1 hold and (1.7) is true. Let γ1 be the constant given by Lemma 3.14. If α > 2/γ1 then one has (J)

max Mk J

≤ C(ω)[Nk ]1/2−ε6 a.s.

for some ε6 > 0 and every Nk ∈ Gρ . Now, if we take γ0 , α, β to satisfy conditions      1 1 α µδ 6 γ0 > 1 + 1− , 1− < 1, β > , ρ d β 8(1 + δ) ρ α−β >

6 2γ0 8 ,β> ,α> − 1, αγ1 > 2, ρ ρ 3τ

then Theorem 1.2 follows from (3.5), Lemmas 3.6, 3.7, 3.8, 3.12, 3.15 and 3.16 with ε = mini=1,...,6 εi . The proof of Theorem 1.2 is completed.
4. The law of the iterated logarithm In order to provide applications of Theorem 1.2, we state now the law of the iterated logarithm for weakly dependent random fields. This is the first result of such type for the fields with dependence condition (1.1); it generalizes the laws of the iterated logarithm known for positively and negatively associated random variables, see, e.g., [5]. For x > 0, set Log x = log(x ∨ e). Theorem 4.1. Suppose that X is a random field satisfying all the conditions of Theorem 1.2. Then, for any τ ∈ (0, 1/(d − 1)), one has, almost surely, SN SN = 1 and lim inf  = −1, lim sup  2 2 2dσ [N ]LogLog[N ] 2dσ [N ]LogLog[N ]

as N → ∞, N ∈ Gτ .

 Proof. We have lim sup WN / 2dσ 2 [N ]LogLog[N ] ≤ 1, due to the LIL for d−parameter Wiener process [23]. The fact that this upper limit is not less than 1 as N → ∞, N ∈ Gτ , can be proved in the same way as the lower bound in classical law of the iterated logarithm for the Wiener process. Thus, Theorem follows from Theorem 1.2.

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142

Final remarks It should be noted that initially the authors placed on top of the first page the following dedication: To Professor M. S. Keane on the occasion of his anniversary.

To unify the style of the volume such inscriptions were omitted. We hope that Mike Keane will read this paper to discover that phrase, as well as our congratulations and best wishes. Acknowledgments A.Bulinski is grateful to the Department of Probability Theory and Stochastic Models of the University Paris-VI for hospitality. References [1] Balan, R. (2005). A strong invariance principle for associated random fields. Ann. Probab. 33 823–840. MR2123212 [2] Berkes, I. and Morrow, G. (1981). Strong invariance principle for mixing random fields. Z. Wahrsch. verw. Gebiete 57 15–37. MR623453 [3] Berkes, I. and Philipp, W. (1979). Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 29–54. MR515811 [4] Bulinski, A. V. (1993). Inequalities for the moments of sums of associated multi-indexed variables. Theory Probab. Appl. 38 342–349. MR1317986 [5] Bulinski, A. V. (1995). A functional law of the iterated logarithm for associated random fields. Fundam. Appl. Math. 1 623–639. MR1788546 [6] Bulinski, A. V. and Keane, M. S. (1996). Invariance principle for associated random fields. J. Math. Sci. 81 2905–2911. MR1420893 [7] Bulinski, A. V. and Suquet, Ch. (2001). Normal approximation for quasiassociated random fields. Statist. Probab. Lett. 54 215–226. MR1858636 [8] Bulinski, A. V. and Shabanovich, E. (1998). Asymptotical behaviour of some functionals of positively and negatively dependent random fields. Fundam. Appl. Math. 4 479–492. MR1801168 [9] Bulinski, A. V. and Shashkin, A. P. (2004). Rates in the central limit theorem for sums of dependent multiindexed random vectors. J. Math. Sci. 122 3343–3358. MR2078752 ¨ rgo ˝ , M. and Re ´ve ´sz, P. (1975). A new method to prove Strassen [10] Cso type laws of invariance principle I. Z. Wahrsch. verw. Gebiete 31 255–260. MR375411 [11] Doukhan, P. and Louhichi, S. (1998). A new weak dependence condition and application to moment inequalities. Stoch. Proc. Appl. 84 323–342. MR1719345 [12] Doukhan, P. and Lang, G. (2002). Rates in the empirical central limit theorem for stationary weakly dependent random fields. Statis. Inf. for Stoch. Proc. 5 199–228. MR1917292 [13] Esary, J., Proschan, F. and Walkup, D.(1967). Association of random variables with applications. Ann. Math. Statist. 38 1466–1474. MR217826 [14] Joag-Dev, K. and Proschan, F. (1983). Negative association of random variables, with applications. Ann. Statist. 11 286–295. MR684886

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´ mlos, J., Major, P. and Tusnady, G. (1975). An approximation of [15] Ko partial sums of independent random variables and the sample distribution function I. Z. Wahrsch. verw. Gebiete 32 111–131. MR375412 ´ mlos, J., Major, P. and Tusnady, G. (1976). An approximation of [16] Ko partial sums of independent random variables and the sample distribution function II. Z. Wahrsch. verw. Gebiete 34 33–58. MR402883 [17] Moricz, F. (1983). A general moment inequality for the maximum of the rectangular partial sums of multiple series. Acta Math. Hung. 41 337–346. MR703745 [18] Newman, C. M. (1980). Normal fluctuations and the FKG inequalities. Commun. Math. Phys. 74 119–128. MR576267 [19] Shashkin, A. P. (2004). A maximal inequality for a weakly dependent random field. Math. Notes 75 717–725. MR2085739 [20] Shashkin, A. P. (2004). A weak dependence property of a spin system. Transactions of XXIV Int. Sem. on Stability Problems for Stoch. Models. Yurmala, Latvia. 30–35. [21] Strassen, V. (1964). An invariance principle for the law of the iterated logarithm. Z. Wahrsch. verw. Gebiete 3 211–226. MR175194 [22] Yu, H. (1996). A strong invariance principle for associated random variables. Ann. Probab. 24 2079–2097. MR1415242 [23] Zimmerman, G. J. (1972). Some sample function properties of the twoparameter Gaussian process. Ann. Math. Statist. 43 1235–1246. MR317401

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 144–153 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000176

Incoherent boundary conditions and metastates Aernout C. D. van Enter1 , Karel Netoˇ cn´ y2 and Hendrikjan G. Schaap1 University of Groningen, Academy of Sciences of the Czech Republic Abstract. In this contribution we discuss the role which incoherent boundary conditions can play in the study of phase transitions. This is a question of particular relevance for the analysis of disordered systems, and in particular of spin glasses. For the moment our mathematical results only apply to ferromagnetic models which have an exact symmetry between low-temperature phases. We give a survey of these results and discuss possibilities to extend them to some situations where many pure states can coexist. An idea of the proofs as well as the reformulation of our results in the language of Newman-Stein metastates are also presented.

1. Introduction In the theory of Edwards-Anderson (short-range, independent-bond Ising-spin) spin-glass models, a long-running controversy exists about the nature of the spinglass phase, and in particular about the possibility of infinitely pure states coexisting. Whereas on the one hand there is a school which, inspired by Parisi’s [22, 24] famous (and now rigorously justified [34, 35]) solution of the SherringtonKirkpatrick equivalent-neighbour model, predicts that infinitely many pure states (= extremal Gibbs measures, we will use both terms interchangably) can coexist, the other extreme, the droplet model of Fisher and Huse, predicts that only two pure states can exist at low temperature, in any dimension [18]. An intermediate, and mathematically more responsible, position was developed by Newman and Stein, who have analyzed a number of properties which a situation with infinitely many pure states should imply [25–30]. One aspect which is of particular relevance for interpreting numerical work is the fact that the commonly used periodic or antiperiodic boundary conditions might prefer different pairs of pure states. Which one then would depend on the disorder realization and on the (realizationdependent) volume, a scenario described by them as “chaotic pairs” (see also [7]). This is a particular example of their notion of “chaotic size-dependence”, the phenomenon that the Gibbs measures on an increasing sequence of volumes may fail to converge almost surely (in the weak topology) to a thermodynamic limit. Such a situation may occur when the boundary conditions are not biased towards a particular phase (or a particular set of phases), that is they are not coherent. If such pointwise convergence fails, a weaker type of probabilistic convergence, e.g. convergence in distribution, may still be possible. Such a convergence has as its limit objects 1 Centre for Theoretical Physics, Rijksuniversiteit Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands, e-mail: [email protected]; [email protected] 2 Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 182 21 Prague 8, Czech Republic, e-mail: [email protected] AMS 2000 subject classifications: primary 82B20, 82B44; secondary 60K35, 60F99. Keywords and phrases: chaotic size dependence, metastates, random boundary conditions, Ising model, local limit behaviour.

144

Incoherent boundary conditions and metastates

145

“metastates”, distributions on the set of all possible Gibbs measures. (These objects are measures on all possible Gibbs measures, including the non-extremal ones. The metastate approach should be distinguished from the more commonly known fact that all Gibbs measures are convex combinations -mixtures- of extremal Gibbs measures.) For a recent description of the theory of metastates see also [2, 3]. Although our understanding of spin glasses is still not sufficient to have many specific results, the metastate theory has been worked out for a number of models, mostly of mean-field type [4, 5, 11, 19, 20]. Recently we have developed it for the simple case of the ferromagnetic Ising model with random boundary conditions [9, 10, 33]. This analysis is in contrast to most studies of phase transitions of lattice models which consider special boundary conditions, such as pure, free or (anti-)periodic ones. In this paper we review these results as well as discuss some possible implications for more complex and hopefully more realistic situations of disordered systems, compare [13]. We feel especially encouraged to do so by the recent advice from a prominent theoretical physicist that: “ Nitpickers ... should be encouraged in this field.... [12]”. 2. Notation and background For general background on the theory of Gibbs measures we refer to [8, 16]. We will here always consider Ising spin models, living on a finite-dimensional lattice Zd . The spins will take the values 1 and −1 and we will use small Greek letters σ, η, . . . to denote spin configurations in finite or infinite sets of sites. The nearest-neighbour Hamiltonians in a finite volume Λ ⊂ Zd will be given by

J  (i, j) σi ηj . (1) H Λ (σ, η) = J(i, j) σi σj +
i,j⊂Λ


i,j i∈Λ,j∈Λc

Fixing the boundary condition which is denoted by the η-variable, these are functions on the configuration spaces ΩΛ = {−1, 1}Λ . We allow sometimes that the boundary bonds J  take a different value (or are drawn from a different distribution) from the bulk bonds J. Our results are based on considering the ferromagnetic situation with random boundary conditions, where the J and the J  are constants (J < 0 for ferromagnets), and the disorder is only in the η-variables which are chosen to be symmetric i.i.d. Associated to these random Hamiltonians H Λ (σ, η) are random Gibbs measures µΛ η (σ) =

1 exp[−H Λ (σ, η)]. Z Λ (η)

(2)

We analyze the limit behaviour of such random Gibbs measures at low temperatures (i.e. |J|  1), in the case of dimension at least two, so that the set of extremal infinite-volume Gibbs measures contains more than one element. It is well known that in two dimensions there exist exactly two pure states, the plus state µ+ and the minus state µ− . In more than two dimensions also translationally non-invariant Gibbs states (e.g. Dobrushin interfaces) exist. Intuitively, one might expect that a random boundary condition favours a randomly chosen pure translation-invariant state. Such behaviour in fact is expected to hold in considerable generality, including the case where an infinite number of “similar” extremal Gibbs states coexist. In our ferromagnetic examples all these pure states are related by a (e.g. spinflip) symmetry of the interaction and the distribution of the boundary condition is

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also invariant under this symmetry. In more general cases one might have “homogeneous” pure phases not related by a symmetry. In such a case one might need to consider non-symmetric, and possibly even volume-dependent distributions for the boundary conditions to obtain chaotic size-dependence, but for not specially chosen distributions one expects to obtain a single one of these pure states. We will consider here only situations where a spin flip symmetry is present, at least at the level of the disorder distribution. Note further, that in some of these more general situations, in particular situations with many pure states, other types of boundary conditions (e.g. the periodic and the free ones, in contrast to what we are used to) may be not coherent, and they would pick out a random -chaotic- pair of Gibbs states, linked by the spin-flip symmetry. The physical intuition for the prevalence of the boundary conditions picking out an extremal Gibbs measure is to some extent supported by the result of [15], stating that for any Gibbs measure µ –pure or not–, µ-almost all boundary conditions will give rise to a pure state. However, choosing some prescribed Gibbs measure to weigh the boundary conditions has some built-in bias, and a fairer question would be to ask what happens if the boundary conditions are symmetric random i.i.d. We will see that the above intuition, that then both interfaces and mixtures are being suppressed, is essentially correct, but that the precise statements are somewhat weaker than one might naively expect. As a side remark we mention that biased, e.g. asymmetric, random boundary conditions are known to prefer pure states. This is discussed in [6, 17] for some examples. A similar behaviour pertains for boundary conditions interpolating between pure and free, as is for example shown for some particular cases in [1, 21]. 3. Results on Ising models with random boundary conditions In this section we describe our results on the chaotic size behaviour of the Ising model under random boundary conditions in more detail. We consider the sequences of finite-volume Gibbs states µΛ η along a sequence of concentric cubes ΛN with linear d size N , for any configuration η ∈ Ω = {−1, 1}Z sampled from the symmetric i.i.d. distribution with the marginals Prob(ηi = −1) = Prob(ηi = 1) = 21 . To any such sequence of states we assign the collection of its weak-topology limit points, which can in general be non-trivial and η-dependent. However, in our simple situation we show that the set of limit points has a simple structure: with probability 1, it contains exactly two elements – the Ising pure states µ+ and µ− . This is proven to be true, provided that a sufficiently sparse (depending on the dimension) sequence of cubes is taken and for sufficiently low temperatures (−J  1), at least in certain regions in the (J, J  )-plane which will next be described. Although one gets an identical picture in all the cases under consideration, these substantially differ in the complexity of the analysis required in the proof. Ground state with finite-temperature boundary conditions As a warm-up problem, following [9], we consider the case where −J = ∞ and J  is finite. Then all spins inside Λ take the same spin value, either plus or minus. Our choice of coupling parameters has excluded interface configurations. The total energy for either the plus or the minus configuration of the system in a cube ΛN ,
H N (±, η) = ±J  ηi (3) i: d(i,ΛN )=1

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is a sum of O(N d−1 ) 2-valued random variables, which are i.i.d. and of zero mean. Obviously,  −1 N µN (4) η (±) = 1 + exp[±2H (+, η)]

and the possible limit states are the plus configuration, the minus configuration, or a statistical mixture of the two, depending on the limit behaviour of the energy H N (+, η). According to the local limit theorem, the probability of this energy d−1 being in some finite interval decays as N − 2 . Summing this over N gives a finite answer if either d is at least 4, or if one chooses a sufficiently sparse sequence of increasing volumes ΛNk in d = 2 or d = 3. A Borel-Cantelli argument then implies that almost surely, that is for almost all boundary conditions, the only possible limit points are the plus configuration and the minus configuration. On the other hand, without the sparsity assumption on the growing volumes in dimension 2 and 3, again via a Borel-Cantelli argument, a countably infinite number of statistical mixtures is seen to occur as limit points. However, as these mixtures occur with decreasing probabilities when the volume increases, the metastate of our system is concentrated only on the plus and minus configuration, and the mixtures do not show up (they are null-recurrent). We expect a similar distinction between almost sure and metastate behaviour in various other situations.

Finite low temperatures with weak boundary conditions In the case where J and J  are both finite, but |J  |  −J, the spins inside Λ are no more frozen and thermal fluctuations have to be taken into account. Yet, one can expect that for the bulk bond −J being large enough, the behaviour does not change dramatically and the model can be analyzed as a small perturbation around the −J = ∞ model of the last section. A difference will be that the frozen plus and minus configurations are to be replaced with suitable plus and minus ensembles, and the energies with the free energies of these ensembles. A technical complication is that these free energies can no longer be written as sums of independent terms. This prevents us from having a precise local limit theorem. However, physically the situation should be rather similar, and one can indeed prove a related, but weaker result. To describe the perturbation method in more detail, we need to go to a contour description. Every pair of spin configurations related by a spin-flip corresponds to a contour configuration. We will distinguish the ensemble of plus-configurations ΩΛ + in which the spins outside of the exterior contours are plus and similarly the ensemble of minus-configurations ΩΛ − . When a contour ends at the boundary and separates one corner from the rest, this corner is defined to be in the interior, and for contours separating at least two corners from at least two other corners (these are interfaces of some sort) we can make a consistent choice for what we call the interior, see [9, 10]. It will turn out that our results do not depend on our precise choice. We consider the measures restricted to the plus and minus ensembles: µΛ η,+ (σ) =

1 Z Λ (η, +)

exp[−H Λ (σ, η)] 1[σ∈ΩΛ+ ]

(5)

and similarly for the minus ensemble. Under the weakness assumption |J  |  −J, the Gibbs probability of any interface in these ensembles is damped exponentially in the system size N uniformly for all η. Actually, we can prove the asymptotic triviality of both ensembles in the following strong form:

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Proposition 3.1. Let −J ≥ max{J ∗ , ∆|J  |} with large enough constants J ∗ , ∆ > N ± N 0. Then µΛ in the weak topology. (The convergence is exponentially fast η,± −→ µ uniformly in η.) ˜  | > −J for some Remark. Note that for Dobrushin boundary conditions, if ∆|J ˜ these boundary conditions favour an interface. However, such sufficiently small ∆, boundary conditions are exceptional, and in the next section we will prove a weaker statement, based on this fact, in d = 2. The convergence properties of the full, non-restricted measures are based on an estimate for the random free energies: F±Λ (η) = log Z Λ (η, ±)

(6)

namely, we prove the following weak local limit type upper bound: Proposition 3.2. Under the assumptions of Proposition 3.1, the inequality Prob(|F+ΛN (η) − F−ΛN (η)| ≤ N ε ) ≤ const N −(

d−1 2 −ε)

holds for any ε > 0 and N large enough. The proofs of both propositions are based on convergent cluster expansions for the measures µΛ η,+ and the characteristic function of the random free energy difference, respectively. Combining Proposition 3.2 with a Borel-Cantelli argument we get (7) lim |F+Λ (η) − F−Λ (η)| = ∞ Λ

provided that the limit is taken along a sparse enough sequence of cubes (unless d ≥ 4). By symmetry, this reads that the random free energy difference has +∞ and −∞ as the only limit points. Since the full Gibbs state is a convex combination of the plus and the minus ensembles with the weights related to the random free energy difference, µΛ η

Λ −1 Λ −1   Zη,+ Zη,− Λ µη,+ + 1 + Λ µΛ = 1+ Λ η,− Zη,+ Zη,−

(8)

this immediately yields the spectrum of the limit Gibbs states [9]. Finite low temperatures in d = 2, with strong boundary conditions In the case −J = |J  |, due to exceptional (e.g. Dobrushin-like, all spins left minus, all spins right plus) boundary conditions, we are to expect no uniform control anymore over the convergence of the cluster expansion. Indeed, one checks that for the contours touching the boundary the uniform lower bounds on their energies cease holding true. In order to control the contributions from these contours, we need to perform a multiscale analysis, along the lines of [14], with some large deviation estimates on the probability of these exceptional boundary conditions. We obtain in this way the following Propositions, corresponding to Propositions 3.1-3.2 [10]: Proposition 3.3. Let d = 2 and |J  | = −J ≥ J ∗∗ , with the constant J ∗∗ > 0 being N ± N large enough. Then µΛ η,± −→ µ weakly for almost all η.

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Proposition 3.4. Under the same assumptions as in Proposition 3.3, 1

Prob(|F+Λ (η) − F−Λ (η)| ≤ τ ) ≤ const(τ ) N −( 2 −ε)

(9)

for any τ, ε > 0 and N large enough. Observe that the limit statement of Proposition 3.3 holds true only on a probability 1 set of the boundary conditions (essentially the ones not favouring interfaces). The construction of this set is based on a Borel-Cantelli argument. These two Propositions imply the following Theorems on the almost sure behaviour and the metastate behaviour respectively: Theorem 3.1. Let the conditions of either Proposition 3.1 or Proposition 3.3 be satisfied, and take a sequence of increasing cubic volumes, which in d = 2 and d = 3 is chosen sufficiently sparse. Then for almost all boundary conditions η it holds that the weak limit points of the sequence of finite-volume Gibbs states are the plus and minus Ising states. Almost surely any open set in the set of Gibbs states, from which the extremal Gibbs measures have been removed, will not contain any limit points. Theorem 3.2. Let the conditions of either Proposition 3.1 or Proposition 3.3 be satisfied, and take a sequence of increasing cubic volumes. Then the metastate equals the mixture of two delta-distributions: 12 (δµ+ + δµ− ). Remark 3.1. This metastate is a different one from the one obtained with free or periodic boundary conditions which would be δ 12 (µ+ +µ− ) . In simulations it would mean that for a fixed realization and a fixed finite volume one typically sees the same state (either plus or minus, which one depending on the volume), the other one being invisible. For periodic or free boundary conditions both plus and minus states are accessible for any fixed volume. Remark 3.2. We expect that, just as in the ground state situation, for a non-sparse increasing sequence of volumes, mixtures will be null-recurrent in dimensions 2 and 3. In this case, any mixture, not only a countable number of them, could be a nullrecurrent limit point. However, as our Propositions only provide upper bounds, and no lower bounds, on the probabilities of small (and not even finite) free energy differences, such a result is out of our mathematical reach. Note that although the metastate description of Theorem 2 gives less detailed information than the almost sure statement of Theorem 1, it encapsulates the physical intuition actually better. Remark 3.3. It also follows from our arguments that, for almost all boundary conditions η, large (proportional to N d−1 ) contours are suppressed for large systems, so neither rigid, nor fluctuating interfaces will appear anywhere in the system. Fluctuating interfaces also would produce mixed states, so even in 2 dimensions, where interface states do not occur, this is a non-trivial result. Remark 3.4. Similar results can also be obtained for the case of very strong boundary conditions, |J  |  −J  1, [33]. Our method can easily be adapted to get the same result for all |J  | ≤ −J, and we believe in fact the result to be true for all J  = 0, −J  1.

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4. Spin glasses, Fisher-White type models, many states, space versus time Although our original motivation was due to our interest in the spin-glass model, our results for spin-glass models are still very modest. We note, however, that by a gauge transformation the ferromagnet with random boundary conditions is equivalent to a nearest-neighbour Mattis [23] spin-glass (= one-state Hopfield) model with fixed boundary conditions, the behaviour of which we can thus analyze by our methods. In a recent paper by Fisher and White [13], they discuss possible scenarios of infinitely many states, mostly based on constructions of stacking lower-dimensional Ising models. The motivation is to investigate what might happen in the putative infinite-state spin-glass scenario, for some (high-dimensional and/or long-range) Edwards-Anderson type model. One of their simplest models is based on “stacking” lower-dimensional Ising models, and we can derive some properties for such models from our results in the last section. For example one can consider N 2-dimensional Ising models in squares of size N by N in horizontal planes, decoupled in the vertical direction. There are now 2N ground states for free boundary conditions, corresponding to the choice of plus or minus in each plane. To avoid reducing the problem to studying the symmetry group of the interaction, one can randomize the problem some more, as follows: By choosing in every plane the bonds across a single line through the origin (in that plane) randomly plus or minus, one obtains that in every plane in the thermodynamic limit four ground states appear, (plus-plus, plus-minus, minus-plus and minus-minus). By imposing random boundary conditions, independently in each plane, one now expects an independent choice out of these four in each plane. By similar considerations this is true in √ as in the last section, we find that although 1 most planes, in O( N ) of them (that is a fraction of O( √N ) of them) mixtures will appear. If our boundary conditions are finite-temperature, this happens if the difference between the number of pluses and minuses stays bounded, otherwise this happens when there is a tie, and at the same time not such a strong spatial fluctuation that an interface ground state appears. The same remains true if one connects the planes by a sufficiently weak (one-dimensional) random coupling, such as for example having a random choice for all the vertical bonds on lines, each line connecting a pair of adjacent planes. If one would choose the values of all vertical bonds randomly from some symmetric distribution, we would obtain something like a collection of two-dimensional random field Ising models. Indeed, the effect of one plane on the next one would be like that of a random field, which would prevent a phase transition in any plane, and our arguments break down. For further discussion see [13]. Similar properties hold for models in which on one periodic sublattice one stacks in a horizontal direction, and in another one in a vertical direction. As the topology of weak convergence is a topology of convergence of local observables, a statement along the lines of Theorem 1 still holds, however the interpretation that one almost surely avoids mixtures now is incorrect. In fact, in these circumstances, for almost all boundary conditions, the system will be in a mixed state. Note that, although such mixed states are less stable than interface states (one can change the weights in the mixture by a finite-energy perturbation), they are much more likely to occur than interface states. As the set of sites which are influenced by the measure being a mixture has a density which approaches zero, in the thermodynamic limit the Parisi overlap distribution [24] will be trivially

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concentrated at 1. Similarly to Theorem 2, also the metastate still will be concentrated on the symmetric mixture of the delta-distributions on the pure states in each plane. We obtain similar results at sufficiently low temperatures, although again, now we cannot prove the occurrence of mixtures. It has been conjectured that similar to the chaotic size-dependence we have discussed, for random (incoherent) initial conditions a related phenomenon of chaotic time-dependence (non-convergence) might occur in disordered systems. If even the Cesaro average fails to converge, this phenomenon also has been called “historic” behaviour [31]. A similar distinction in the spatial problem also occurs. Although for ferromagnets with random boundary conditions the Cesaro average of the magnetisation at the origin, taken over a sequence of linearly increasing volumes, will exist, one does not expect this for the random field Ising model with free or periodic boundary conditions, essentially for the same reason as in the mean-field version of the problem [19, 20]. One would only expect a convergence in distribution to the metastate 1 + − 2 (δµ + δµ ). One could both scenarios (chaotic size dependence and chaotic time dependence) describe as examples of what Ruelle [32] calls “messy” behaviour. However, chaotic time dependence for stochastic dynamics of for example Glauber type requires the system to be infinitely large. Otherwise, for finite systems either one has a finite-state Markov chain with a unique invariant measure (for positive temperature Glauber dynamics), while for zero-temperature dynamics the system would get trapped, up to zero-energy spin flips. Concluding, we have illustrated how the notion of chaotic size-dependence naturally occurs, already in the simple ferromagnetic Ising model, once one allows for incoherent boundary conditions. As in many situations the choice of incoherent boundary conditions seems a physically realistic one, our results may be helpful in explaining why in general one expects experimentally to observe pure phases. Acknowledgments This research has been supported by FOM-GBE and by the project AVOZ10100520 of the Academy of Sciences of the Czech Republic. We thank Christof K¨ ulske for helpful advice on the manuscript. A. C. D. v. E. and K. N. thank Igor Medved’ for their related earlier collaboration which led to the results in [9]. A. C. D. v. E. thanks Pierre Picco and Veronique Gayrard for useful dicussions on metastate versus almost sure properties. He would like to thank Mike Keane for all the probabilistic inspiration he has provided to Dutch mathematical physics over the years. References ´, R., and Medved’, I. (2002). Finite-size effects for [1] Borgs, C., Kotecky the Potts model with weak boundary conditions. J. Stat. Phys., 109, 67–131. MR19279915 [2] Bovier, A. (2001). Statistical Mechanics of Disordered Systems. MaPhySto Lectures 10, Aarhus. [3] Bovier, A. (2006). Statistical Mechanics of Disordered Systems, a Mathematical Perspective. Cambridge Series in Statistical and Probabilistic Mathematics 18. Cambridge University Press.

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[4] Bovier, A., van Enter, A. C. D., and Niederhauser, B. (1999). Stochastic symmetry-breaking in a Gaussian Hopfield model. J. Stat. Phys. 95, 1-2, 181–213. MR1705585 [5] Bovier, A. and Gayrard, V. (1998). Hopfield models as generalized random mean field models. In Mathematical Aspects of Spin Glasses and Neural Networks. Progr. Probab., Vol. 41. Birkh¨ auser Boston, Boston, MA, 3–89. MR1601727 [6] Campanino, M. and van Enter, A. C. D. (1995). Weak versus strong uniqueness of Gibbs measures: a regular short-range example. J. Phys. A 28, 2, L45–L47. MR1323597 [7] van Enter, A. C. D. (1990). Stiffness exponent, number of pure states, and Almeida-Thouless line in spin-glasses. J. Stat. Phys. 60, 1-2, 275–279. MR1063219 ´ndez, R., and Sokal, A. D. (1993). Regular[8] van Enter, A. C. D., Ferna ity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory. J. Stat. Phys. 72, 5–6, 879–1167. MR1241537 ˇny ´, K. (2002). Chaotic [9] van Enter, A. C. D., Medved’, I., and Netoc size dependence in the Ising model with random boundary conditions. Markov Proc. Rel. Fields., 8, 479–508. MR1935655 ˇny ´, K., and Schaap, H.G. (2005). On the [10] van Enter, A. C. D., Netoc Ising model with random boundary condition. J. Stat. Phys. 118, 997-1057. [11] van Enter, A. C. D. and Schaap, H. G. (2002). Infinitely many states and stochastic symmetry in a Gaussian Potts-Hopfield model. J. Phys. A 35, 11, 2581–2592. MR1909236 [12] Fisher, D. S. (2004). Equilibrium states and dynamics of equilibration: General issues and open questions. Les Houches Summer School 2002, Slow Relaxations and Nonequilibrium Dynamics in Condensed Matter, Editors J.-L. Barrat, M. Feigelman, J. Kurchan et al, 523–553, Springer. [13] Fisher, D. S., and White, O. L. (2004) Spin glass models with infinitely many states. Phys. Rev. Lett. To appear. ¨ hlich, J. and Imbrie, J. Z. (1984). Improved perturbation expansion for [14] Fro disordered systems: beating Griffiths singularities. Comm. Math. Phys. 96, 2, 145–180. MR768253 [15] Georgii, H.-O. (1973). Two remarks on extremal equilibrium states. Comm. Math. Phys. 32, 107–118. MR426763 [16] Georgii, H.-O. (1988). Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics, Vol. 9. Walter de Gruyter & Co., Berlin. MR956646 [17] Higuchi, Y. and Yoshida, N. (1997). Slow relaxation of 2-D stochastic Ising models with random and non-random boundary conditions. In New Trends in Stochastic Analysis (Charingworth, 1994). World Sci. Publishing, River Edge, NJ, 153–167. MR1654356 [18] Huse, D. A. and Fisher, D. S. (1987). Pure states in spin glasses. J. Phys. A 20, 15, L997–L1003. MR914687 ¨lske, C. (1997). Metastates in disordered mean-field models: random field [19] Ku and Hopfield models. J. Stat. Phys. 88, 5-6, 1257–1293. MR1478069 ¨lske, C. (1998). Metastates in disordered mean-field models. II. The su[20] Ku perstates. J. Stat. Phys. 91, 1-2, 155–176. MR1632557 [21] Lebowitz, J. L. (1976). Thermodynamic limit of the free energy and correlation functions of spin systems. In Quantum Dynamics: Models and Mathematics (Proc. Sympos., Centre for Interdisciplinary Res., Bielefeld Univ., Biele-

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DOI: 10.1214/074921706000000185

Markovianity in space and time∗ M. N. M. van Lieshout1 Centre for Mathematics and Computer Science, Amsterdam Abstract. Markov chains in time, such as simple random walks, are at the heart of probability. In space, due to the absence of an obvious definition of past and future, a range of definitions of Markovianity have been proposed. In this paper, after a brief review, we introduce a new concept of Markovianity that aims to combine spatial and temporal conditional independence.

1. From Markov chain to Markov point process, and beyond This paper is devoted to the fundamental concept of Markovianity. Although its precise definition depends on the context, common ingredients are conditional independence and factorisation formulae that allow to break up complex, or high dimensional, probabilities into manageable, lower dimensional components. Thus, computations can be greatly simplified, sometimes to the point that a detailed probabilistic analysis is possible. If that cannot be done, feasible, efficient simulation algorithms that exploit the relatively simple building blocks may usually be designed instead. 1.1. Markov chains The family of Markov chains is one of the most fundamental and intensively studied classes of stochastic processes, see e.g. [2]. If we restrict ourselves to a bounded time horizon, say 0, 1, . . . , N for some N ∈ N, a discrete Markov chain is a random vector (X0 , . . . , XN ) with values in some denumerable set L = ∅ for which P(Xi = xi | X0 = x0 ; . . . ; Xi−1 = xi−1 ) = P(Xi = xi | Xi−1 = xi−1 )

(1)

for all 1 ≤ i ≤ N and all xj ∈ L, j = 0, . . . , i. In words, the Markov property (1) means that the probabilistic behaviour of the chain at some time i given knowledge of its complete past depends only on its state at the immediate past i−1, regardless of how it got to xi−1 . The right hand side of equation (1) is referred to as the transition probability at time i from xi−1 to xi . If P(Xi = xi | Xi−1 = xi−1 ) = p(xi−1 , xi ) does not depend on i, the Markov chain is said to be stationary. By the product rule, the joint distribution can be factorised as P(X0 = x0 ; . . . ; XN = xN ) = P(X0 = x0 )

N


p(xi−1 , xi ).

(2)

i=1 * This research is supported by the Technology Foundation STW, applied science division of NWO, and the technology programme of the Ministry of Economic Affairs (project CWI.6156 ‘Markov sequential point processes for image analysis and statistical physics’). 1 CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands, e-mail: [email protected] AMS 2000 subject classifications: primary 60G55, 60D05; secondary 62M30. Keywords and phrases: Hammersley–Clifford factorisation, marked point process, Markov chain, Monte Carlo sampling, neighbour relation, pairwise interaction, random sequential adsorption, sequential spatial process.

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P(Xi = xi | Xj = xj , j = i) = P(Xi = xi | Xi−1 = xi−1 , Xi+1 = xi+1 )

(3)

Consequently, for all i = 1, . . . , N − 1, and all xj ∈ L, j = 0, . . . , N (with obvious modifications at the extremes i = 0, N of the time interval of interest). Thus, the conditional distribution of the state at a single point in time depends only on the states at the immediate past and future. To tie in with a more general concept of Markovianity to be discussed below, the time slots i − 1, i + 1 (if within the finite horizon) may be called neighbours of i, i ∈ {0, . . . , N }. As a simple example, consider the celebrated simple random walk on the two dimensional lattice L = Z2 . The dynamics are as follows: A particle currently sitting at site x ∈ L moves to each of the neighbouring sites with equal probability. More precisely, p(x, y) = 1/4 if x and y are horizontally or vertically adjacent in L, that is, ||x − y|| = 1, and zero otherwise [25]. 1.2. Markov random fields The concept of Markovianity plays an important role in space as well as in time, especially in image analysis and statistical physics. Since, in contrast to the setting in the previous subsection, space does not allow a natural order, a symmetric, reflexive neighbourhood relation ∼ must be defined on the domain of definition, which we assume to be a finite set I. The generic example is a rectangular grid I ⊆ Z2 with i ∼ j if and only if ||i − j|| ≤ 1. Then, a discrete Markov random field with respect to ∼ is a random vector X = (Xi )i∈I , with values of the components Xi in some denumerable set L = ∅, that satisfies the local Markov property P(Xi = xi | Xj = xj , j = i) = P(Xi = xi | Xj = xj , j ∈ ∂(i)) where ∂(i) = {j ∈ I \ {i} : j ∼ i} is the set of neighbours of i. The expression should be compared to (3); indeed a Markov chain is a Markov random field on I = {0, 1, . . . , N } with i ∼ j if and only if |i − j| ≤ 1, i, j ∈ I. To state the analogue of (2), define a clique to be a set of sites C ⊆ I such that for all i, j ∈ C these sites are neighbours, that is i ∼ j. By default, the empty set is a clique. If we assume for simplicity that the joint probability mass function of the random vector X has no zeroes, it defines a Markov random field if and only if it can be factorised as
ϕC (xj , j ∈ C) (4) P(X0 = x0 ; . . . ; XN = xN ) = cliques C for some clique interaction functions ϕC (·) > 0. Hence, the clique interaction functions are the spatial analogues of the transition probabilities of a Markov chain. A two-dimensional example is the Ising model for spins in a magnetic field [15]. In this model, each node site of a finite subset of the lattice is assigned a spin value from the set L = {−1, 1}. Interaction occurs between spins at horizontally or vertically adjacent sites, so that cliques consist of at most two points. A particular spin value may be preferred due to the presence of an external magnetic field. More precisely, the Ising model is defined by (4) with ϕ{i} (xi ) = eα xi for singletons, ϕ{i,j} (xi , xj ) = eβ xi xj , and ϕ∅ set to the unique constant for which the right hand side of (4) is a probability mass function. The constant α reflects the external magnetic field and influences the frequencies of the two spin types. For β > 0, neighbouring sites tend to agree in spin, for β < 0, they tend to have different spins. For further details, the reader is referred to [11].

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1.3. Markov point processes The next step is to leave discrete domains and move into Euclidean space. Thus, let D ⊂ Rd be a disc, rectangle or other bounded set of positive volume. A realisation of a Markov point process on D is a subset x = {x1 , . . . , xn } of D with random, finite n ≥ 0. Again define a reflexive, symmetric neighbourhood relation ∼, for instance by x ∼ y if and only if ||x − y|| ≤ R for some R > 0. Note that each point has uncountably many potential neighbours, in contrast to the discrete case. Moreover, the probability of finding a point at any particular location a ∈ D will usually be 0, so densities rather than probability mass functions are needed. A suitable dominating measure is the homogeneous Poisson process (see e.g. [17]), because of its lack of spatial interaction. Indeed, given such a Poisson process places n points in D, the locations are i.i.d. and uniformly distributed over D. If desirable, one could easily attach marks to the points, for example a measurement or type label of an object located at the point. Doing so, the domain becomes D × M , where M is the mark set, and each xi can be written as (di , mi ). In this set-up, a Markov point process X is defined by a density f (·) that is hereditary in the sense that f (x) > 0 ⇒ f (y) > 0 for all configurations y ⊆ x and satisfies [18, 26] the local Markov condition, that is, whenever f (x) > 0, u ∈ x, the ratio f (x ∪ {u}) (5) λ(u | x) := f (x) depends only on u and {xi : u ∼ xi }. The function λ(· | ·) is usually referred to as conditional intensity. For the null set where f (x) = 0, λ(· | x) may be defined arbitrarily. Note that since realisations x of X are sets of (marked) points, f (·) is symmetric, i.e. invariant under permutations of the xi that constitute x. If f (x ∪ {u}) ≤ βf (x) for some β > 0 uniformly in x and u, the density f (·) is said to be locally stable. Note that local stability implies that f (·) is hereditary, and that the conditional intensity (5) is uniformly bounded in both its arguments. A factorisation is provided by the Hammersley–Clifford theorem which states [1, 26] that a marked point process with density f (·) is Markov if and only if
ϕ(y) (6) f (x) = cliques y⊆x for some non-negative, measurable interaction functions ϕ(·). The resemblance to (4) is obvious. An example of a locally stable Markov point process is the hard core model , a homogeneous Poisson process conditioned to contain no pair of points that are closer than R to one another. For the hard core model, ϕ({x, y}) = 1 {||x − y|| > R}, ϕ(y) ≡ 1 for singletons and configurations y containing three or more points. The interaction function for the empty set acts as normalising constant to make sure that f (·) integrates to unity. Although Markov point processes are useful modelling tools [18], the assumption of permutation invariance may be too restrictive. For instance in image interpretation, occlusion of one object by another may be dealt with by an ordering of the objects in terms of proximity to the camera [20], or in scene modelling various kinds of alignment of the parts that make up an objects may be modelled

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by means of non-symmetric neighbour relations [22]. Non-symmetric densities also arise naturally when local scale and intensity is taken into account by transforming the conditional intensity of a homogeneous template process, see [14]. In stark contrast to the vast literature on spatial point processes, sequential patterns have not been studied much. Indeed, an analogue to the theory of Markov point processes does not exist in the non-symmetrical setting. The present paper fills this gap. 2. Definitions and notation This paper is concerned with finite sequential spatial processes. Realisations of such processes at least consist of a finite sequence x = (x1 , . . . , xn ),

n ∈ N0

of points in some bounded subset D of the plane. Additionally, to each point xi , a mark mi in some complete, separable metric space, say M , may be attached. The mark may be a discrete type label, a real valued measurement taken at each location, or a vector of shape parameters to represent an object located at xi . Thus, we may write y = (y1 , . . . , yn ) = ((x1 , m1 ), . . . , (xn , mn )),

n ∈ N0 ,

for the configuration of marked points, and shall denote the family of all such configurations by N f . As an aside, the plane R2 may be replaced by Rd or any other complete separable metric space, equipped with the Borel σ-algebra and a finite diffuse Borel measure. The distribution of a finite sequential spatial process may be defined as follows. Given a finite diffuse Borel measure µ(·) on (D, BD ) so that µ(D) > 0, usually Lebesgue measure, and a mark probability measure µM (·) on M equipped with its Borel σ-algebra M, specify (1) a probability mass function qn , n ∈ N0 , for the number of points in D; (2) for each n, a Borel measurable and (µ × µM )n -integrable joint probability density pn (y1 , . . . , yn ) for the sequence of marked points y1 , . . . , yn ∈ D × M , given it has length n. Alternatively, a probability density f (·) may be specified directly on N f = ∪∞ n=0 (D× n M ) , the space of finite point configurations in D with marks in M , with respect to the reference measure ν(·) defined by ν(F ) equal to   ∞  e−µ(D) ··· 1 {(y1 , . . . , yn ) ∈ F } dµ × µM (y1 ) · · · dµ × µM (yn ) n! D×M D×M n=0 for F in the σ-algebra on finite marked point sequences generated by the Borel product σ-fields on (D × M )n . In words, ν(·) corresponds to a random sequence of Poisson length with independent components distributed according to the normalised reference measure µ(·) µM (·)/µ(D). It is readily observed that q0 = exp(−µ(D)) f (∅), and   e−µ(D) ··· f (y1 , . . . , yn ) dµ × µM (y1 ) · · · dµ × µM (yn ); qn = n! D×M D×M pn (y) =

e−µ(D) f (y) n! qn

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M. N. M. van Lieshout

for each n ∈ N and y ∈ (D × M )n . Reversely, if the length n(y) of y is n, f (y) = eµ(D) n! qn pn (y). Note that neither f (·) nor the pn (·, . . . , ·) are required to be symmetric [3, Ch. 5]. We are now ready to define and analyse a Markov concept for random sequences in the plane. To do so, we begin by defining a sequential conditional intensity λi ((x, m) | y) :=

f (si (y, (x, m))) (n + 1) f (y)

if f (y) > 0

(7)

for inserting (x, m) ∈ y at position i ∈ {1, . . . , n + 1} of y = (y1 , . . . , yn ). Here si (y, (x, m)) = (y1 , . . . , yi−1 , (x, m), yi , . . . , yn ). On the null set {y ∈ N f : f (y) = 0}, the sequential conditional intensity may be defined arbitrarily. The overall conditional probability of finding a marked point at du = dµ × µM (u) in any position in the vector given that the remainder of the sequence equals y is given by n+1 

λi (du | y).

i=1

The expression should be compared to its classic counterpart (5). As for Markov chains, we are mostly interested in λn+1 (· | y), but all λi (· | ·) are needed for the reversibility of the dynamic representation to be considered in Section 4. Note that provided f (·) is hereditary in the sense that f (y) > 0 implies f (z) > 0 for all subsequences z of y, then f (y1 , . . . , yn ) ∝ n!

n


λi (yi | y
(8)

i=1

where y 0 implies f (z) > 0 for all subsequences z of y; • for all sequences y for which f (y) > 0, the ratio f ((y, u))/f (y) depends only on u and its directed neighbours {yi ∈ y : u ∼ yi } in y.

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In particular, λn(y)+1 (· | y) is invariant under permutations of y. In the sequel, we shall sometimes abuse notation somewhat and write λn(y)+1 (· | y), where y = {yi ∈ y}, and n(y) is the cardinality of the set y. Due to the lack of symmetry, sequential spatial processes are particularly useful for modelling inhomogeneous space-time processes. For instance, to obtain a complete packing of non-intersecting particles [28], a typical algorithm keeps adding particles randomly until this becomes impossible due to violation of the condition of no overlap, see e.g. the recent review papers [6, 27]. A related soft-core model for landslides was proposed by Fiksel and Stoyan [9]. Example 1 (Simple sequential inhibition). Let π(·) be a probability density with respect to Lebesgue measure on a bounded planar Borel set D of strictly positive area. Suppose a population of animals arrives in the region D to set up nests, and would like to pick locations according to π(·). In other words, those subregions that have high π-mass are deemed the most suitable. However, as each animal needs its space, a newly arriving animal may not build its nest closer than some r > 0 to an existing nest location [4, 5]. Thus, if we write d(z, x) = min{||z − x|| : x ∈ x} for the minimal distance between z ∈ D and the components of x, pn (x1 , . . . , xn ) π(x2 ) 1 {d(x2 , x1 ) > r} π(xn ) 1 {d(xn , x r} ···  π(z) 1 {d(z, x1 ) > r} dz π(z) 1 {d(z, x r} dz D D

∝ π(x1 ) 

(9)

for xi ∈ D, i = 1, . . . , n. Some care has to be taken about division by zero. If n is larger than the packing number for balls of radius r in D, or qn = 0 for some other reason, pn (·, . . . , ·) may be chosen to be any probability density on Dn . For smaller n, or if qn > 0, set pn (x1 , . . . , xn ) = 0 whenever some term in the denominator of (9) is zero, and renormalise pn (·, . . . , ·) to integrate to unity. The total number of animals in D is either fixed – as in the formulation by [4] – or random according to some probability mass function qn , n ≤ np , the packing number, as in Random Sequential Adsorption [6, 27]. If n is fixed, that is qn = 1 {n = n0 } for some n0 ∈ N, f (x) is not hereditary, therefore not Markovian in the sense of Definition 1. If qn > 0 for n ≤ np , and np q = 1, the density is hereditary. In that case, provided f (x1 , . . . , xn ) > 0, n=0 n λn+1 (u | (x1 , . . . , xn )) =

π(u) 1 {d(u, x) > r} cn+1 qn+1  c n qn π(z) 1 {d(z, x) > r} dz D

(with 0/0 = 0), where cn is the normalising constant of pn (·, . . . , ·). The likelihood ratio is invariant under permutations of the components of x. However, it may depend on the sequence length through cn+1 qn+1 /(cn qn ), and, moreover, π(z) 1 {d(z, x) > r} dz may depend on the geometry of x, i.e. on the whole seD quence. Hence, in general, f (·) is Markovian only with respect to the trivial relation in which each pair of points is related. Example 2 (Sequential soft core). Further to Example 1, suppose the animals have no locational preferences, but claim territory within a certain radius. The radius can depend deterministically on the location [14] (in fertile regions, less space is needed than in poorer ones), be random and captured by a mark, or a combination of both 1 . To be specific, suppose an animal settling at x ∈ D claims 1 Assume the mark space (0, ∞), for concreteness, is equipped with a density g(·). By the transformation h(c1 ,c2 ) (x, m) = (c1 x, c2 m) for c1 , c2 > 0, the Lebesgue intensity measure µ(·) is scaled

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the region within a (stochastic) radius m > 0 of x for itself, and that newcomers are persuaded to avoid this region. An appropriate density with respect to ν(·) could be n( y) 
1{||xi −xj ||≤mj } n( y) f (y) ∝ β γ j
y  yi = (xi , mi ) ∈ D ×(0, ∞). Here β > 0 is an intensity parameter, and 0 ≤ γ < 1 reflects the strength of persuasion, the smaller γ, the stronger the inhibition. Indeed, for γ = 0, no new arrival is allowed to enter the territory of well-settled animals. Alternatively, if invaders demand space according to their own mark, replace mj by mi in the exponent of γ. Note that n(
y) 1{||x−xj ||≤mj } (n(y) + 1) λn(y)+1 ((x, m) | y) = β γ j=1 depends only on those yj = (xj , mj ) for which ||x−xj || ≤ mj . Hence f (·) is Markov with respect to the relation (x, r) ∼ (y, s) ⇔ ||x − y|| ≤ s. 3. Hammersley–Clifford factorisation The goal of this section is to formulate and prove a factorisation theorem for sequential spatial processes in analogy with (2) and (6). To do so, we need the notion of a directed clique interaction function. Indeed, for a marked point z ∈ D × M , define the sequence y to be a z-clique with respect to a reflexive relation ∼ on D × M if y either has length zero or all its components y ∈ y satisfy z ∼ y. The definition is z-directed but otherwise permutation invariant, so f , the family of unordered finite marked point we may map y onto the set y ∈ N configurations, by ignoring the permutation. Our main theorem is the following. Theorem 1. A sequential spatial process with density f (·) is Markov with respect to ∼ if and only if it can be factorised as f (y1 , . . . , yn ) = f (∅)

n



ϕ(yi , z)

(10)

i=1 z⊆y
f → for some non-negative, jointly measurable interaction function ϕ : (D × M ) × N [0, ∞) that vanishes except on cliques (i.e. ϕ(u, z) = 1 if z is no u-clique with respect to ∼). Here y
by λYc ((x, m) | y) := λ ( c

x

1 (x,m)

,

m ) c2 (x,m)

| h−1 (y) /(n(y) + 1). Provided f (·) is loc(x,m)

cally stable, the sequential density fYc (·) defined by λYc (· | ·) as in (8) is well-defined. For example, the marked hard core density f (y) ∝ 1 {||xi − xj || > mi + mj } with λ((x, m) | y) = 1 {||x − xi || > m + mi  for all (xi , mi ) ∈ y} is transformed into fYc ( y) ∝



i<j

1

||xi − xj || >

c1 (xj ,mj ) (mi c2 (xj ,mj )

+ mj ) . Intuitively, the resulting process Yc looks like a scal-

ing of Y at (x, m) by c(x, m). Indeed, if Y is Markov respect to

with

∼, then Yc is Markov with r s v u respect to (u, r) ∼c (v, s) ⇔ c (u,r) , c (u,r) ∼ c (u,r) , c (u,r) and the ∼c -neighbourhood 1

2

1

2

∂c (x, m) = hc(x,m) (∂(h−1 (x, m))) inherits from ∂(·) geometric properties such as convexity c(x,m) that are invariant under rescaling.

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Proof. To show that any density of the form (10) is Markovian, suppose f (y) > 0 but f (z) = 0 for some subsequence z of y. Then, again with the notation n(z) for the length of z, n( z)

ϕ(zi , x) = 0. f (∅) i=1 x⊆z
Hence either f (∅) = 0 or some term ϕ(zi , x) = 0, but, as both feature in the factorisation (10) of f (y) too, f (y) must be 0 as well, in contradiction with the assumption. Assume that f (y) > 0. Now, f ((y, u))/f (y) = z⊆y ϕ(u, z) depends only on u and its directed neighbours in y, as ϕ(u, z) = 1 whenever z is no u-clique. Reversely, set ϕ(y1 , ∅) := f (y1 )/f (∅) = λ1 (y1 |∅), and define putative interaction functions ϕ(·, ·) recursively as follows. If {y1 , . . . , yn } is no yn+1 -clique, ϕ(yn+1 , {y1 , . . . , yn }) := 1; else ϕ(yn+1 , {y1 , . . . , yn }) :=

f (y1 , . . . , yn+1 ) f (y1 , . . . , yn ) z ϕ(yn+1 , z)

(11)

where the product ranges over all strict subsets {y1 , . . . , yn } =  z ⊂ {y1 , . . . , yn }. To deal with any zeroes, we use the convention 0/0 = 0. We first show that ϕ(·, ·) is a well-defined interaction function. By the Markov assumption, f (y1 , . . . , yn+1 )/f (y1 , . . . , yn ) is invariant under permutations of y1 , . . . , yn , so a simple induction argument yields the permutation invariance of ϕ(yn+1 , {y1 , . . . , yn }) in its second argument. By definition, ϕ(·, ·) vanishes except on cliques, hence it is an interaction function. To show that if the denominator of (11) is zero, so is the numerator, note that if f (∅) = 0, by the assumption that the process is hereditary, necessarily f ≡ 0 which contradicts the fact that f (·) is a probability density. Therefore f (∅) > 0 and ϕ(y1 , ∅) is well-defined. Suppose ϕ(·, ·) is well-defined for sets of cardinality at most n − 1 ≥ 0 as its second argument. Let {y1 , . . . , yn } be an yn+1 -clique. The Markov assumption implies that if f (y1 , . . . , yn ) = 0, also f (y1 , . . . , yn+1 ) is zero. Furthermore, if f (y1 , . . . , yn ) > 0 but ϕ(yn+1 , z) = 0 for some strict subset z ⊂ {y1 , . . . , yn }, by the induction assumption, f ((z, yn+1 )) = 0, where the sequence z is obtained from z by the permutation induced by (y1 , . . . , yn ). A fortiori f (y1 , . . . , yn+1 ) = 0. We conclude that (11) is well-defined by induction. It remains to show that ϕ(·, ·) satisfies the desired factorisation. To do so, we again proceed by induction. By definition, f (y1 ) = f (∅) ϕ(y1 , ∅) for any y1 ∈ D×M , so the factorisation holds for sequences of length at most 1. Suppose (10) holds for all sequences that are at most n ≥ 1 long, and consider any sequence y = (y1 , . . . , yn+1 ) with components in D × M . If f (y1 , . . . , yn ) = 0, by the assumption on hereditariness, f (y1 , . . . , yn+1 ) = 0. By the induction hypothesis, f (y1 , . . . , yn ) = n n+1 f (∅) i=1 z⊆y
f (y1 , . . . , yn+1 ) =

(12)

z⊆y
because of the induction hypothesis. We shall distinguish several cases. Firstly, suppose that {y1 , . . . , yn } is an yn+1 -clique and recall that if the denominator in the right hand side of (11) is zero, then so are ϕ(yn+1 , {y1 , . . . , yn })

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and f (y1 , . . . , yn+1 ), so the desired factorisation holds. If, on the other hand, the denominator is strictly positive, by definition (11), f (y1 , . . . , yn+1 ) = f (y1 , . . . , yn )




ϕ(yn+1 , z)

(13)

z⊆{y1 ,...,yn }

and (10) holds by substitution of (13) in (12). Secondly, assume {y1 , . . . , yn } is no yn+1 -clique. Then, yn+1 ∼ yi for some yi , i = 1, . . . , n. Now, by the Markov assumption, f (y1 , . . . , yi−1 , yi+1 , . . . , yn ) > 0, and λn+1 (yn+1 | {y1 , . . . , yn }) = λn+1 (yn+1 | {y1 , . . . , yi−1 , yi+1 , . . . , yn }) 1 f (y1 , . . . , yi−1 , yi+1 , . . . , yn+1 ) = n + 1 f (y1 , . . . , yi−1 , yi+1 , . . . , yn )
1 ϕ(yn+1 , z) = n+1 z⊆{y1 ,...,yi−1 ,yi+1 ,...,yn }

=




1 n+1

ϕ(yn+1 , z)

z⊆{y1 ,...,yn }

where the last two equations follow from the induction hypothesis and the fact that ϕ(yn+1 , z) = 1 whenever z contains yi . We conclude that (13) holds, hence (10). The factorisation (10) is similar to that of the joint distribution of a Markov chain, cf. (2). Indeed, the product of transition probabilities p(x i−1 , xi ) from state xi−1 to xi is replaced by a product of interaction functions z⊆y 0 for n ≤ np and qn = 0 for n > np in order to be sure that the simple sequential inhibition model (9) is hereditary. Under this assumption, the interaction functions can be computed iteratively by (11). Set rn := ncn qn /(cn−1 qn−1 ) for n = 1, 2, . . . , np , 0 otherwise, and I(x) :=  π(z) 1 {d(z, x) > r} dz for any finite set x of points in D, with I(∅) = 1. Then D ϕ(x, ∅) = r1 π(x);



ϕ(x, y) = 1 {d(x, y) > r} exp 



(−1)n(y\z) log

z⊆y



  rn(z)+1  I(z)

for non-empty configurations y with I(y) > 0. Otherwise, ϕ(x, y) = 0. In order to verify the above expressions, note that by equation (8), it is sufficient to verify that the sequential conditional intensity, or equivalently the likelihood ratio for adding a point at the end of a given sequence, has the desired form. Indeed, for x = (x1 , . . . , xn ), n ≥ 1, such that I(x) and f (x) are strictly positive,       rn(z)+1 f (sn+1 (x, xn+1 ))  (−1)n(y\z) log = r1 π(xn+1 ) exp  f (x) I(z) ∅=y⊆x z⊆y

=

rn+1 π(xn+1 ) I(x)

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provided d(xn+1 , {x1 , . . . , xn }) > r, which implies d(xn+1 , y) > r for any subset y of x = {x1 , . . . , xn }, and zero otherwise. The last identity is a consequence of Newton’s binomium. Example 2 (ctd). For the sequential soft core model, ϕ(y, ∅) ϕ((x1 , m1 ), (x2 , m2 ))

= β; = γ 1{||x1 −x2 ||≤m2 } ,

and 1 otherwise 2 . Example 3. Pairwise interaction models of the form   n

ϕ(yi , ∅) f (∅) ϕ(yi , yj ) i=1

j
are particularly convenient. Note that the sequential soft core model discussed in the previous example exhibits pairwise interactions only. It has a constant penalty term γ ∈ [0, 1) for each pair yi ∼ yj . It may be more natural to let the interaction vary with distance, for example quadratically as in 2 2 ϕ((x, r), (y, s)) = 1 − (1 − ||x − y||2 /Rr,s )

for ||x − y|| ≤ Rr,s . Here the range of interaction Rr,s may depend on the marks. A sufficient condition for the model to be integrable is that ϕ(y, ∅) is uniformly bounded in y and the pairwise interaction function ϕ(yi , yj ) is bounded by 1. Thus, pairwise interaction models are particularly suitable for modelling repulsion between marked points. 4. Dynamic representation Markov sequential spatial processes arise naturally as the limit distribution of a jump process [7] with transitions that add or remove one component of the sequence at a time. From a statistical point of view, it is then possible to obtain samples from some sequential spatial model of interest by running the pure jump process into equilibrium. Standard Markov chain Monte Carlo ideas [10] can then be applied to estimate model parameters by the maximum likelihood principle, to perform likelihood ratio tests to assess the goodness-of-fit of the model, to compute confidence regions, profile likelihoods and so on. Moreover, at least in principle, it is possible to determine exactly when equilibrium is reached, by applying the coupling ideas of Propp and Wilson [24] along the lines proposed in [8, 16, 19] and the references therein. Below, we first propose a birth-and-death jump process, then develop a discrete time Metropolis–Hastings style sampler. 2 The

fYc (∅)

density

n(y) i=1

of

z⊆y
the 

ϕ

scaled

h−1 (z) c(yi )

∪

Yc of xi i {( c (y ) , c m )} 1 i 2 (yi ) process

of a template Markov marked point process Y .

 Example 2 can be factorised as if ϕ(·) denotes the interaction function

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164

4.1. Spatial birth-and-death processes A spatial birth-and-death process is a continuous time Markov process with state space N f . Its only transitions are the insertion of a marked point in the current sequence (a birth), or the deletion of a component (a death). Suppose the current state is y, and write bi (y, u) dµ × µM (u) for the birth rate of a new marked point in du, u ∈ D × M , to be inserted at position i of y, and di (y) for the death rate of the reverse transition. Then the detailed balance equations are given by e−µ(D) e−µ(D) f (y) bi (y, u) = f (si (y, u)) di (si (y, u)). n! (n + 1)! If we take unit death rate for each marked point, bi (y, u) =

f (si (y, u)) (n + 1) f (y)

(14)

again with the convention 0/0 = 0. Note that if the jump process starts in the set {y ∈ N f : f (y) > 0}, it will almost surely never leave it. Clearly, for this choice of birth and death rates, f (·) is an invariant probability density. Since our process evolves in continuous time, some care has to be taken to avoid explosion, that is, infinitely many transitions in finite time. Set Bn := sup

n+1 

n( y)=n i=1

bi (y, u) dµ × µM (u) D×M

for the upper bound on the total birth rate from sequences of length n ≥ 0, and note that n  di (y) = n. δn := inf n( y)=n

i=1

By [23, Prop. 5.1, Thm. 7.1], sufficient conditions for the existence of a unique jump process with birth rates (14) and unit death rates with a unique invariant probability measure to which it converges in distribution regardless of the initial state are that δn > 0 for all n ≥ 1 and one of the following holds: • Bn = 0 for all sufficiently large n ≥ n0 ≥ 0; • Bn > 0 for all n ≥ 1 and ∞  δ1 · · · δn = ∞; B · · · Bn n=1 1

∞  B1 · · · Bn−1 < ∞. δ1 · · · δn n=2

For densities f (·), such as the sequential soft core model of Example 2, that satisfy f (si (y, u)) ≤ βf (y) (15) for some β > 0 and all y ∈ N f , u ∈ D×M , i = 1, . . . , n(y)+1, the total birth rate is bounded by β µ(D), and it is easily verified that the above Preston conditions hold. Moreover, from a practical point of view, the jump process may be implemented by thinning the transitions of a process with unit death rate and constant birth rate ˜bi (y, ·) ≡ β/(n(y) + 1), which avoids having to compute explicitly the parameter n( y)+1 

 i=1

D×M

bi (y, u) dµ × µM (u) + n(y)

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165

of the exponentially distributed waiting times in between jumps. The retention probability of a transition from y to si (y, u) is given by n( y)

f (si (y, u)) 1
ϕ(yj , z ∪ {u}). ϕ(u, z) = β f (y) β j=i z⊆y
z⊆y<j

The first term depends on the directed neighbours yj of u (u ∼ yj ) in y 0, the chain will almost surely never leave the set H := {y ∈ N f : f (y) > 0}. We shall restrict the Markov chain to H from now on. In order to show that the Metropolis–Hastings chain Yn , n ∈ N0 , converges to f (·) in total variation from any initial state in H, we need to establish aperiodicity and Harris recurrence, that is Harris ergodicity [21]. Aperiodicity immediately follows from the fact that self-transitions occur. A sufficient condition for Harris recurrence (in fact for the even stronger property of geometric ergodicity) is that there exist a function V : N f ∩ H → [1, ∞), constants b < ∞ and γ < 1, and a measurable small set C ⊂ N f ∩ H such that P V (y) := E [V (Xn+1 )|Xn = y] ≤ γ V (y) + b 1 {y ∈ C} .

(16)

Recall that a set C is small if there exists a non-zero measure φ(·) and an integer n such that the probability that the Metropolis–Hastings chain reaches any measurable subset B of N f from any y ∈ C in n steps is at least as large as ν(B). See [21] for further details.

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From now on, assume that (15) holds. Then, the drift condition (16) can be verified by an adaptation of the proof of [12, Prop. 3.3] with V (y) = An(y) for some A > 1. Indeed, note that the acceptance probability for inserting u ∈ D × M at position i in y is bounded by βµ(D)/(n(y) + 1), which does not exceed a prefixed constant > 0 if n(y) is sufficiently large. Similarly, the acceptance probability for removing a component from the sequence y reduces to 1 if y is long enough. Now, n( y)+1   A−1 1 α(y, si (y, u)) dµ × µM (u) P V (y) = V (y) 2 y) + 1) D×M µ(D) (n( i=1 n( y)

 A−1 − 1 1 α(y, y(−i) ) + V (y). + V (y) 2 n( y ) i=1

(17)

Since for sufficiently long sequences of marked points, say n(y) > N , deaths are always accepted and α(y, si (y, u)) ≤ uniformly in its arguments, (17) is less  than or equal to 12 (A − 1) + 21 (A−1 − 1) + 1 V (y). This and the uniform lower bound 1/(β µ(D)) on the acceptance probability for deaths yield the desired result by the same arguments as in the proof of Propositions 3.2–3.3 in [12] with C = n  ∪N n=0 (D × M ) ∩ H. 4.3. Reversible jump Markov chains The Metropolis–Hastings algorithm of the previous subsection is a special case of a so-called reversible jump Markov chain [13]. This is a proposal–acceptance/rejection scheme for moves between subspaces of different dimension. Suppose a target equilibrium distribution and a proposal probability for each move type are given. The aim is to define acceptance probabilities in such a way that the resulting Markov chain is well-defined and the detailed balance equations hold. In order to avoid singularities due to the different dimensions, for each move type, a symmetric dominating measure ξ(·, ·) on the product of the state space with itself is needed. It turns out [13] that acceptance probabilities that are ratios of the joint density of the target and proposal distributions with respect to ξ(·, ·) evaluated at the proposed and current state do the trick. In the context of Section 4.2, define, for measurable subsets A and B of N f ,  1 ξ(A × B) = 1 {y ∈ A} n(y) + 1 Nf n( y)+1   × 1 {si (y, u) ∈ B} dµ × µM (u) dν(y) i=1 D×M

+



Nf

n( y)

1 {y ∈ A}

   1 y(−i) ∈ B dν(y). i=1

The measure is symmetric by the form of the reference measure ν(·). The joint bi-variate density of the product of the distribution to sample from and the birth proposal distribution with respect to ξ(·, ·) is given by f (y, si (y, u)) = f (y)/(2 µ(D)). Similarly for death proposals we have f (y, y(−i) ) = f (y)/(2n(y)). Hence, the acceptance probabilities for births and deaths, as given in the previous section, follow as ratios of joint bi-variate densities truncated at 1. Note that there is no need to worry about division by zero, as we had restricted the Metropolis– Hastings chain to the family of sequences having strictly positive density.

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Acknowledgments The author would like to thank Mike Keane for piloting CWI’s spatial statistics research safely through a re-organisation, for luring her back to The Netherlands, and for a pleasant collaboration on the annual Lunteren Meeting. She acknowledges A. Baddeley, U. Hahn, E. Vedel Jensen, J. Møller, L. Nielsen, Y. Ogata, T. Schreiber, R. Stoica, D. Stoyan, and the participants in STW project CWI.6156 for discussions and preprints. References [1] Baddeley, A. J. and Møller, J. (1989). Nearest-neighbour Markov point processes and random sets. International Statistical Review , 57, 89–121. [2] Chung, K. L. (1967). Markov Chains with Stationary Transition Probabilities. Second edition. Die Grundlehren der mathematischen Wissenschaften, Band 104. Springer-Verlag New York, Inc., New York. MR217872 [3] Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer Series in Statistics. Springer-Verlag, New York. MR950166 [4] Diggle, P. J. (1983). Statistical Analysis of Spatial Point Patterns. Mathematics in Biology. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London. MR743593 [5] Diggle, P. J., Besag, J., and Gleaves., J. T. (1976). Statistical analysis of spatial point patterns by means of distance methods. Biometrics, 32, 659– 667. [6] Evans, J. W. (1993). Random and cooperative sequential adsorption. Reviews of Modern Physics, 65, 1281–1329. [7] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II. Second edition. John Wiley & Sons Inc., New York. MR270403 ´ndez, R., and Garcia, N. L. (2002). Perfect simula[8] Ferrari, P. A., Ferna tion for interacting point processes, loss networks and Ising models. Stochastic Process. Appl., 102, 1, 63–88. MR1934155 [9] Fiksel, Th. and Stoyan, D. (1983). Mathematisch-statistische Bestimmung von Gef˝ ardungsgebieten bei Erdfallprozessen. Zeitschrift f˝ ur angewandte Geologie, 29, 455–459. [10] Gilks, W. R., Richardson, S., and Spiegelhalter, D.J. (1996). Markov Chain Monte Carlo in Practice. Interdisciplinary Statistics. Chapman & Hall, London. MR1397966 ´ [11] Geman, D. (1990). Random fields and inverse problems in imaging. In Ecole d’´et´e de Probabilit´es de Saint-Flour XVIII—1988. Lecture Notes in Math., Vol. 1427. Springer, Berlin, 113–193. MR1100283 [12] Geyer, C. (1999). Likelihood inference for spatial point processes. In Stochastic Geometry (Toulouse, 1996). Monogr. Statist. Appl. Probab., Vol. 80. Chapman & Hall/CRC, Boca Raton, FL, 79–140. MR1673118 [13] Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82, 4, 711–732. MR1380810 [14] Hahn, U., Jensen, E. B. V., van Lieshout, M.-C., and Nielsen, L. S. (2003). Inhomogeneous spatial point processes by location-dependent scaling. Adv. in Appl. Probab., 35, 2, 319–336. MR1970475 [15] Ising, E. (1925). Beitrag zur Theorie des Ferromagnetismus. Zeitschrift f˝ ur Physik , 31, 253–258.

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[16] Kendall, W. S. and Møller, J. (2000). Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Adv. in Appl. Probab., 32, 3, 844–865. MR1788098 [17] Kingman, J. F. C. (1993). Poisson Processes. Oxford Studies in Probability, Vol. 3. The Clarendon Press Oxford University Press, New York. MR1207584 [18] van Lieshout, M.N.M. (2000). Markov Point Processes and Their Applications. Imperial College Press, London. MR1789230 [19] van Lieshout, M. N. M. and Stoica, R. S. (2005). Perfect simulation for marked point processes. Computional Statistics & Data Analysis. Published online, March 23, 2006, at http://www.sciencedirect.com [20] Mardia, K.V., Qian, W., Shah, D., and de Souza, K. (1997). Deformable template recognition of multiple occluded objects. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19, 1036–1042. [21] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Communications and Control Engineering Series. Springer-Verlag London Ltd., London. MR1287609 [22] Ortner, M. (2004). Processus ponctuels marqu´es pour l’extraction automatique de caricatures de bˆatiments `a partir de mod`eles num´eriques d’el´evation. PhD thesis, University of Nice – Sophia Antipolis. [23] Preston, C. (1975). Spatial birth-and-death processes. Bull. Inst. Internat. Statist., 46, 2, 371–391, 405–408. MR474532 [24] Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9, 1–2, 223–252. MR1611693 ´ve ´sz, P. (1990). Random Walk in Random and Nonrandom Environments. [25] Re World Scientific Publishing Co. Inc., Teaneck, NJ. MR1082348 [26] Ripley, B. D. and Kelly, F. P. (1977). Markov point processes. J. London Math. Soc. (2), 15, 1, 188–192. MR436387 [27] Stoyan, D. (1998). Random sets: Models and statistics. International Statistical Review , 66, 1–27. [28] Tanemura, M. (1979). On random complete packing by discs. Annals of the Institute of Statistical Mathematics, 31, 351–365.

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 169–175 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000194

Mixing and tight polyhedra Thomas Ward1,∗ University of East Anglia Abstract: Actions of Zd by automorphisms of compact zero-dimensional groups exhibit a range of mixing behaviour. Schmidt introduced the notion of mixing shapes for these systems, and proved that non-mixing shapes can only arise non-trivially for actions on zero-dimensional groups. Masser has shown that the failure of higher-order mixing is always witnessed by nonmixing shapes. Here we show how valuations can be used to understand the (non-)mixing behaviour of a certain family of examples. The sharpest information arises for systems corresponding to tight polyhedra.

1. Introduction Let α be a Zd -action by invertible measure-preserving transformations of a prob(j) (j) (j) ability space (X, B, µ). A sequence of vectors (n1 , n2 , . . . , nr )j
1 in (Zd )r that are moving apart in the sense that (j)

n(j) s − nt

−→ ∞ as j −→ ∞ for any s = t

is called mixing for α if for any measurable sets A1 , . . . , Ar , 
(j) (j) µ α−n1 (A1 ) ∩ · · · ∩ α−nr (Ar ) −→ µ(A1 ) · · · µ(Ar ) as j −→ ∞.

(1)

(2)

If (1) guarantees (2), then α is r-mixing or mixing of order r. Mixing of order 2 is called simply mixing. The maximum value of r for which (1) implies (2) is the order of mixing M(α) of α (if there is no maximum then α is mixing of all orders, and we write M(α) = ∞). For single transformations (the case d = 1) it is not known if mixing implies mixing of all orders. For Z2 -actions, Ledrappier’s example [4] shows that mixing does not imply 3-mixing. Motivated by the way in which Ledrappier’s example fails to be 3-mixing, Schmidt introduced the following notion: A finite set {n1 , . . . , nr } of integer vectors is called a mixing shape for α if   (3) µ α−kn1 (A1 ) ∩ · · · ∩ α−knr (Ar ) −→ µ(A1 ) · · · µ(Ar ) as k −→ ∞. The maximum value of r for which (3) holds for all shapes of cardinality r is the shape order of mixing S(α). Clearly M(α)
S(α), but in general there are no other relations; the following is shown in [9]. Lemma 1. For any s, 1
s
∞, there is a measure-preserving Z2 -action with M(α) = 1 and S(α) = s. ∗ The

author thanks Manfred Einsiedler for discussions leading to this result. of Mathematics, University of East Anglia, Norwich, United Kingdom, e-mail: [email protected] AMS 2000 subject classifications: primary 22D40, 22D40; secondary 52B11. Keywords and phrases: mixing, polyhedra. 1 School

169

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T. Ward

For algebraic systems — those in which X is a compact abelian group, µ is Haar measure, and αn is an automorphism of X for each n ∈ Zd — if all shapes are mixing, then the system is mixing of all orders (see [6], [8]). Whether the quantitative version of this relationship might hold was asked by Schmidt [7, Problem 2.11]: If all shapes with r elements are mixing, is an algebraic dynamical system r-mixing? For r = 2 this means that the individual elements of an algebraic Zd -action are mixing transformations if and only if the whole action is mixing, which is proved in [6, Theorem 1.6]. For d = 2 and r = 3 this was shown in [2]. Finally, Masser proved this in complete generality [5]. Theorem 2 (Masser). For any algebraic dynamical system (X, α) on a zerodimensional group X, M(α) = S(α). In conjunction with (4) and the algebraic characterization (5), Theorem 2 shows that M(α) = S(α) for any algebraic dynamical system α. The problem of determining the exact order of mixing for a given system remains: By [6, Chap. VIII], there is — in principle — an algorithm that works from a presentation of the module defining an algebraic Zd -action and determines all the non–mixing shapes, which by Masser’s result [5] then determines the exact order of mixing. By [2], all possible orders of mixing arise: for any m  1 and d  2, there is an algebraic Zd -action with M(α) = m Our purpose here is to show how the methods from [2] extend to d > 2. This gives sharp information about mixing properties for a distinguished class of examples associated to tight polyhedra. 2. Inequalities for order of mixing By [8], for an algebraic dynamical system α on a connected group, M(α) > 1 ⇒ M(α) = ∞,

(4)

so in particular M(α) = S(α) in this case. Thus finite order of mixing for mixing systems can only arise on groups that are not connected. Following [6], any algebraic Zd -action α on a compact abelian group X is associated via duality to a ±1 module M = MX over the ring Rd = Z[u±1 1 , . . . , ud ] (multiplication by ui is dual to ei the automorphism α for i = 1, . . . , d). Conversely, any Rd -module M determines an algebraic Zd -action αM on the compact abelian group XM . Approximating the indicator functions of the sets appearing in (2) by finite trigonometric polynomials shows that (2) for αM is equivalent to the property that for any elements a1 , . . . , ar of M , not all zero, (j) (j) a1 un1 + · · · + ar unr = 0M (5) can only hold for finitely many values of j, where un = un1 1 · · · und d is the monomial corresponding to the position n ∈ Zd . This algebraic formulation of mixing may be used to show that (2) holds for αM if and only if it holds for all the systems αRd /p for prime ideals p associated to M (see [8] for example). The group XRd /p is connected if and only if p ∩ Z = {0}, so these two remarks together mean that it is enough to study systems associated to modules of the form Rd /p where p is a prime ideal containing a rational prime p.

Mixing and tight polyhedra

171

The (dramatic) simplifying assumption made here concerns the shape of the prime ideal p: from now on, we assume that p = p, f˜ for some polynomial f˜ ∈ Rd . The degree to which this assumption is restrictive depends on d: For d = 2, any mixing system can be reduced to this case. For d > 2, the ideal p could take the form p, f˜1 , . . . , f˜s  for any s = 1, . . . , d − 1. In the language of [1], our assumption amounts to requiring that the system be of entropy rank (d − 1). Once the prime p is fixed, the systems we study are therefore parameterized by a single polynomial f˜ ∈ Rd which is only defined modulo p. Since p is fixed, we ±1 write Rd,p = Fp [u±1 1 , . . . , ud ], and think of the defining polynomial as f ∈ Rd,p . Thus the dynamical system we study corresponds to the module Rd,p /f  ∼ = Rd /p, f˜

(6)

where f˜ is any element of Rd with f˜ ≡ f (mod p) and the isomorphism in (6) is an isomorphism of Rd -modules. Write the polynomial f as a finite sum  cf,n un , cf,n ∈ Fp . f (u) = n∈Zd

The support of f is the finite set S(f ) = {n ∈ Zd | cf,n = 0}; denote the convex hull of the support by N (f ). Theorem 2 would follow at once if we knew that a non-mixing sequence of order r (that is, a witness to the statement that M(α) < r) was somehow forced to be, or to nearly be, a non-mixing shape of order r (a witness to the statement that S(α) < r). The full picture is much more complicated, in part because the presence of the Frobenius automorphism of Fp leads to many families of solutions to the underlying equations – see [5]. Here we show that in a special setting the simple arguments from [2] do indeed force a non-mixing sequence to approximate a non-mixing shape, giving an elementary approach to Theorem 2 for this very special setting. Let P be a convex polyhedron in Rd . A parallel redrawing of P is another polyhedron Q with the property that every edge of Q is parallel to an edge of P . Figure 1 shows a parallel redrawing of a pentagon. Definition 3. A convex polyhedron P in Rd is tight if any parallel redrawing of P is homothetic to P . For example, in R2 , the only tight polyhedra are triangles. In R3 there are infinitely many combinatorially distinct tight convex polyhedra. Among the Platonic solids, the tetrahedron, octahedron and icosahedron are tight, while the dodecahedron and cube are not. Tightness can be studied via the dimension of the space of parallel redrawings of a polyhedron; see papers of Whiteley [10], [11].  

   

 




 

   



 






Fig 1. A parallel redrawing of a pentagon.

T. Ward

172

±1 Theorem 4. Let f be an irreducible polynomial in Rd,p = Fp [u±1 1 , . . . , ud ], and d let α = αRd,p /f  be the algebraic Z -action associated to the Rd -module Rd,p /f . Let v be the number of vertices in N (f ). Then

(1) any non-mixing sequence for α along some subsequence contains, with uniform error, a parallel redrawing of N (f ); (2) hence v − 1
M(α)
S(α)
|S(f )| − 1. Corollary 5. If N (f ) is tight, then S(α) = M(α). 3. Proofs Throughout we use the characterisation (5) of mixing. (j)

(j)

Lemma 6. Let (n1 , . . . , nr )j
1 be a sequence of r-tuples of vectors in Zd with the property that there are non-zero elements a1 , . . . , ar ∈ Rd,p /f  with (j)

(j)

a1 un1 + · · · + ar unr = 0 in Rd,p /f  for all j  1.

(7)

Then there is a constant K with the property that for every edge e of N (f ) there is (j) (j) (j) (j) an edge e of the convex hull of the set {n1 , . . . , nr }, joining ns to nt say, for (j)  s with which there is a point n (j)

(j)

(1)  ns − ns 
K; (j) (j)  s − nt is parallel to e. (2) the line through n (j)

(j)

For large j, the points n1 , . . . , nr are widely separated, so Lemma 6 means the edges of the convex hull of these points approximate in direction the edges of N (f ) more and more accurately as j goes to infinity. Proof of Lemma 6. Pick an edge e of N (f ). Choose a primitive integer vector v1 orthogonal to e which points outward from N (f ) (that is, with the property that for any points x ∈ N (f ) and y ∈ e, the scalar product (x − y) · v1 is negative). Also choose an ultrametric valuation | · |v1 on Rd,p /f  with the property that the vector t (log |u1 |v1 , . . . , log |ud |v1 ) is a vector of unit length parallel to v1 that also points outward from N (f ). This valuation may be found by extending the vector v1 to a set of primitive integer vec(2) (d) (j) tors {v1 , v1 , . . . , v1 } with v1 · v1 < 0 for j  2 that generates Zd , as illustrated in Figure 2, and then thinking of Rd,p /f  as (2)

(d)

Fp [uv1 ][uv1 , . . . , uv1 ]/f . Let K1 = 2 max {| log |mi |v1 |}. i=1,...,r

Now for fixed j  1 choose t with the property that (j)

(j)

|unt |v1  |uns |v1 for all s, 1
s
r. Then the ultrametric inequality for | · |v1 and the relation (7) show that there must (j) be (at least) one other vertex ns which is no further than K1 from the hyperplane (j) orthogonal to v1 through nt .

Mixing and tight polyhedra

  

173



 v1   (3)  v1   

 N (f ) e

(2)  v1 

Fig 2. Extending v1 to a basis.

Now choose finitely many vectors v2 , . . . , vk and a constant K < ∞ (depending on the choice of the vectors) with the following property. For each , 2

k, repeat the construction above corresponding to v1 and let K
= 2 max {| log |mi |v
|}. i=1,...,r

The (purely geometrical) property sought is that any vector k ∈ Zd with the property that k is no further than distance K
from the hyperplane orthogonal to v
through k for all , 1

k, must be within distance K of k . Now apply the k different ultrametrics |·|v1 , . . . , |·|vk to the relation (7) to deduce (j) (j) that there must be a pair of vertices ns and nt (the parameter j is still fixed; (j) all other quantities including s and t depend on it) with the property that ns lies (j) within distance K
of the hyperplane orthogonal to v
through nt for 1

k. Since all the vectors v
are orthogonal to the edge e, this proves the lemma. (j)

(j)

Proof of Theorem 4. Let (n1 , . . . , nr )j
1 be a non-mixing sequence for α. Thus by (5) there are non-zero elements a1 , . . . , ar ∈ Rd,p /f  with (j)

(j)

a1 un1 + · · · + ar unr = 0 in Rd,p /f  for all j  1.

(8)

Pick a vertex v1 of N (f ) and an edge e1 starting at v1 of N (f ), and relabel the non-mixing sequence so that the edge e1 is approximated in direction (in the sense (j) (j) of Lemma 6) by the pair n1 , n2 for all j  1. By Lemma 6, for each j there (j) (j) (j) (j) (j) is a vector m1 with n2 − m1 
K such that the line joining n1 to m1 is parallel to e1 for all j. Since the set of integer vectors v with v
K is finite, we may find an infinite set S1 ⊂ N with (j)

(j)

n1 − m1 = k1 , a constant, for all j ∈ S1 . This gives an improved version of the relation (8), (j)

(j)

(j)

a1 un1 + a2 um1 + · · · + ar unr = 0 in Rd,p /f  for all j ∈ S1

(9)

where a2 = a2 uk1 . Now select another edge e2 of N (f ) starting at v1 whose approximating pair (j) (j) is n1 and (after relabelling) n3 . We now need to allow 2K of movement in n3 to give m3 . This gives an infinite set S2 ⊂ S1 ⊂ N and a modified version of (9) (j)

(j)

(j)

(j)

a1 un1 + a2 um1 + a3 um2 + · · · + ar unr = 0 in Rd,p /f  for all j ∈ S2

(10)

T. Ward

174



        (j)       n4 
     m(j) 
 4 
  
  (j) n1
 Fig 3. Approximating a loop in a parallel redrawing of N (f ).

(j)

(j)

(j)

(j)

in which n1 − m1 is parallel to e1 and m1 − m2 is parallel to e2 . Continue this process of relabelling, passing to a subsequence and adjusting the coefficients in (10) to exhaust all the edges along some path from v1 . The type of situation (j) (j) that may emerge is shown in Figure 3, where n1 is fixed, n2 has been moved (j) (j) no further than K, n3 a distance no more than 2K and n4 a distance no more than 3K to give edges parallel to edges of N (f ). By Lemma 6 there may be an edge (j) of N (f ) for which n4 is the approximating partner, and we have already chosen (j) (j) to adjust n4 to m4 . It is difficult to control what loops may arise: for example the Herschel graph [3] shows that a convex polyhedron need not be Hamiltonian as a graph. Nonetheless, the bold path in Figure 3 is, to within a uniformly bounded error, a parallel redrawing of that loop in N (f ). This process may be continued to modify all the (j) points ns by uniformly bounded amounts to end up with an infinite set S∗ ⊂ N and a relation (j)

(j)

(j)

(j)

a1 un1 + a2 um1 + a3 um2 + · · · + ar umr−1 = 0 in Rd,p /f  for all j ∈ S∗ (11) with the property that every edge of N (f ) is parallel to within a uniformly bounded (j) (j) (j) error to an edge in the convex hull of the set {n1 , m1 , . . . , mr−1 } for all j ∈ S∗ , proving part 1. In particular, r  v, so M(α) < r implies r  v, hence M(α)  v−1. This proves one of the inequalities in part 2. All that remains is to prove the other inequality in part 2. If  cf,n un , cf,n ∈ Fp f (u) = n∈S(f )

then the relation



cf,n un = 0 in Rd,p /f 

n∈S(f )

implies that


n∈S(f )

cf,n u

n

pk

=



cf,n un = 0 in Rd,p /f  for all k  1,

n∈pk ·S(f )

so S(f ) is a non-mixing shape, and S(α)
|S(f )| − 1. Proof of Corollary 5. If N (f ) is tight, then (11) may be improved further: multiply each of the coefficients by a monomial chosen to shift the vertices by a uniformly bounded amount to lie on an integer multiple of N (f ). The resulting se(j) (j) (j) ˜ 1 ,...,m ˜ r−1 } is homothetic to N (f ) and so is a non-mixing shape. quence {˜ n1 , m Thus M(α) < r implies that S(α) < r, so S(α)
M(α).

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175

References [1] Einsiedler, M., Lind, D., Miles, R., and Ward, T. (2001). Expansive subdynamics for algebraic Zd -actions. Ergodic Theory Dynam. Systems 21, 6, 1695–1729. MR1869066 [2] Einsiedler, M. and Ward, T. (2003). Asymptotic geometry of non-mixing sequences. Ergodic Theory Dynam. Systems 23, 1, 75–85. MR1971197 [3] Herschel, A. S. (1862). Sir Wm. Hamilton’s icosian game. Quart. J. Pure Applied Math. 5, 305. [4] Ledrappier, F. (1978). Un champ markovien peut ˆetre d’entropie nulle et m´elangeant. C. R. Acad. Sci. Paris S´er. A-B 287, 7, A561–A563. MR512106 [5] Masser, D. W. (2004). Mixing and linear equations over groups in positive characteristic. Israel J. Math. 142, 189–204. MR2085715 [6] Schmidt, K. (1995). Dynamical Systems of Algebraic Origin. Progress in Mathematics, Vol. 128. Birkh¨ auser Verlag, Basel. MR1345152 [7] Schmidt, K. (2001). The dynamics of algebraic Zd -actions. In European Congress of Mathematics, Vol. I (Barcelona, 2000). Progr. Math., Vol. 201. Birkh¨ auser, Basel, 543–553. MR1905342 [8] Schmidt, K. and Ward, T. (1993). Mixing automorphisms of compact groups and a theorem of Schlickewei. Invent. Math. 111, 1, 69–76. MR1193598 [9] Ward, T. (1997/98). Three results on mixing shapes. New York J. Math. 3A, Proceedings of the New York Journal of Mathematics Conference, June 9–13, 1997, 1–10 (electronic). MR1604565 [10] Whiteley, W. (1986). Parallel redrawings of configurations in 3-space. Preprint. [11] Whiteley, W. (1989). A matroid on hypergraphs, with applications in scene analysis and geometry. Discrete Comput. Geom. 4, 1, 75–95. MR964145

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 176–188 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000202

Entropy quotients and correct digits in number-theoretic expansions Wieb Bosma1 , Karma Dajani2 and Cor Kraaikamp3 Radboud University Nijmegen, Utrecht University, and University of Technology Delft Abstract: Expansions that furnish increasingly good approximations to real numbers are usually related to dynamical systems. Although comparing dynamical systems seems difficult in general, Lochs was able in 1964 to relate the relative speed of approximation of decimal and regular continued fraction expansions (almost everywhere) to the quotient of the entropies of their dynamical systems. He used detailed knowledge of the continued fraction operator. In 2001, a generalization of Lochs’ result was given by Dajani and Fieldsteel in [7], describing the rate at which the digits of one number-theoretic expansion determine those of another. Their proofs are based on covering arguments and not on the dynamics of specific maps. In this paper we give a dynamical proof for certain classes of transformations, and we describe explicitly the distribution of the number of digits determined when comparing two expansions in integer bases. Finally, using this generalization of Lochs’ result, we estimate the unknown entropy of certain number theoretic expansions by comparing the speed of convergence with that of an expansion with known entropy.

1. Introduction The goal of this paper is to compare the number of digits determined in one expansion of a real number as a function of the number of digits given in some other expansion. For the moment the regular continued fraction (RCF) expansion and the decimal expansion will serve as examples, the digits then being partial quotients and decimal digits. It may seem difficult at first sight to compare the decimal expansion with the continued fraction expansion, since the dynamics of these expansions are very different. However, in the early nineteen-sixties G. Lochs [11] obtained a surprising and beautiful result that will serve as a prototype for the results we are after. In it, the number m(n) of partial quotients (or continued fraction digits) determined by the first n decimal digits is compared with n. Thus knowledge of the first n decimal digits of the real number x determines completely the value of exactly m(n) partial quotients; to know unambiguously more partial quotients requires more decimal digits. Of course m(n) = m(x, n) will depend on x. But asymptotically the following holds (in this paper ‘almost all’ statements are with respect to the Lebesgue measure λ). Theorem 1.1 (Lochs). For almost all x: m(n) 6 log 2 log 10 = = 0.97027014 · · · . n→∞ n π2 lim

1 Mathematisch

Instituut, Radboud Universiteit Nijmegen, Postbus 9010, 6500 GL Nijmegen, The Netherlands, e-mail: [email protected], url: http://www.math.ru.nl/∼bosma/ 2 Fac. Wiskunde en Informatica, Universiteit Utrecht, Postbus 80.000 3508 TA Utrecht, The Netherlands, e-mail: [email protected], url: http://www.math.uu.nl/people/dajani/ 3 EWI (DIAM), Technische Universiteit Delft, Mekelweg 4, 2628 CD Delft, The Netherlands, e-mail: [email protected], url: http://ssor.twi.tudelft.nl/∼cork/cor.htm AMS 2000 subject classifications: primary 11K55, 28D20; secondary 11K50, 11K16. Keywords and phrases: digits, expansion, continued fraction, radix, entropy. 176

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Roughly speaking, this theorem tells us that usually around 97 partial quotients are determined by 100 decimal digits. By way of example Lochs calculated (on an early main-frame computer [12]) that the first 1000 decimals of π determine 968 partial quotients. Examining the function m(n) more closely, one can formulate Lochs’ result as follows. Let Bn (x) be the decimal cylinder of order n containing x, and Cm (x) the continued fraction cylinder of order m containing x, then m(n) is the largest index for which Bn (x) ⊆ Cm (x). Denoting by S the decimal map S(x) = 10x mod 1, and by by T the continued fraction map T (x) = x1 −  x1 , then Lochs’ theorem says that m(x) h(S) lim = (a.e.). n→∞ n h(T ) In 2001, the second author and Fieldsteel generalized in [7] Lochs’ result to any two sequences of interval partitions satisfying the result of Shannon–McMillan–Breiman theorem, (see [3], p. 129), that can be formulated as follows. Theorem 1.2 (Shannon–McMillan–Breiman). Let T be an ergodic measure preserving transformation on a probability space (X, B, µ) and let P be a finite or countably infinite generating partition for T for which Hµ (P ) < ∞, where Hµ (P ) denotes the entropy of the partition P . Then for µ-almost every x: lim

n→∞

− log µ (Pn (x)) = hµ (T ) . n

Here h (T ) denotes the entropy of T and Pn (x) denotes the element of the partition
n−1 µ−i P containing x. i=0 T

The proof in [7] is based on general measure-theoretic covering arguments, and not on the dynamics of specific maps. However, with the technique in [7] it is quite difficult to have a grip on the distribution of m(n). For that, one needs the dynamics of the underlying transformations as reflected in the way the partitions are refined under iterations. In this paper we give a dynamical proof for certain classes of transformations, and we describe explicitly the distribution of m(n) when comparing two expansions in integer bases. We end the paper with some numerical experiments estimating unknown entropies of certain number theoretical expansions by comparing them with expansions of known entropy. 2. Fibred systems In order to generalize Lochs’ result we start with some notations and definitions that place maps like the continued fraction transformation in a more general framework; see also [15], from which the following is taken with a slight modification. Definitions 2.1. Let B be an interval in R, and let T : B → B be a surjective map. We call the pair (B, T ) a fibred system if the following conditions are satisfied: (a) there is a finite or countably infinite set D (called the digit set), and (b) there is a map k : B → D such that the sets B(i) = k−1 {i} = {x ∈ B : k(x) = i} form a partition of B; we assume that the sets B(i) are intervals, and that P = {B(i) : i ∈ D} is a generating partition. Moreover, we require that

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(c) the restriction of T to any B(i) is an injective continuous map. The cylinder of rank n determined by the digits k1 , k2 , . . . , kn ∈ D is the set B(k1 , k2 , . . . , kn ) = B(k1 ) ∩ T −1 B(k2 ) ∩ . . . ∩ T −n+1 B(kn ); by definition B is the cylinder of rank 0. Two cylinder sets Bn , Bn∗ of rank n are called adjacent if they are contained in the same cylinder set Bn−1 of rank n − 1 and their closures have non-empty intersection: Bn ∩ Bn∗ = ∅. For any fibred system (B, T ) the map T can be viewed as a shift map in the following way. Consider the following correspondence Ψ : x → (k1 , k2 , . . . , kn , . . .), where ki = j if and only if T i−1 x ∈ B(j), then T : Ψ(x) → (k2 , k3 , . . . , kn , . . .). We write Bn (x) for the cylinder of rank n which contains x, i.e., Bn (x) = B(k1 , k2 , . . . , kn )

⇐⇒

Ψ(x) = (k1 , k2 , . . . , kn , . . .).

A sequence (k1 , . . . , kn ) ∈ Dn is called admissible if there exists x ∈ B such that Ψ(x) = (k1 , . . . , kn , . . .). It will be clear that Ψ associates to any x ∈ B the ‘expansion’ (k1 , k2 , . . .), and that the expansion is determined by T and the digit map k. In order to arrive at the desired type of result for the fibred system (B, T ), we need to impose some restrictions on T . Definition 2.2. A fibred system (B, T ) is called a number theoretic fibred system and the map T a number theoretic fibred map if T satisfies the additional conditions: (d) T has an invariant probability measure µT which is equivalent to λ, i.e., there exist c1 and c2 , with 0 < c1 < c2 < ∞, such that c1 λ(E) ≤ µT (E) ≤ c2 λ(E), for every Borel set E ⊂ B, and (e) T is ergodic with respect to µT (and hence ergodic with respect to the Lebesgue measure λ). Examples 2.3. The following examples are used throughout the text. (i) The first example is the regular continued fraction. Here B = [0, 1) ⊂ R, the 1 , 1i ], map T is given by T (x) = x1 − k(x), and k(x) =  x1 . Hence B(i) = ( i+1 for all i ≥ 1. It is a standard result that the map is ergodic with the Gaussmeasure as invariant measure. The entropy of T equals h(T ) = π 2 /(6 log 2) = 2.373 · · · . (ii) The second standard example is the g-adic expansion, for an integer g ≥ 2. Here the map is Tg (x) = g · x − k(x), and k(x) = g · x. Hence Tg (x) is the representative of g · x mod 1 in B = [0, 1), and B(i) = [ gi , i+1 g ), for 0 ≤ i < g. Again this is a number theoretic fibred map, with invariant measure λ. The entropy of Tg equals h(Tg ) = log g.

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1 , 1i ) for (iii) A third example is given by the L¨ uroth expansion. Now B(i) = [ i+1 i ≥ 1. If T is taken as an increasing linear map onto [0, 1) on each B(i), the L¨ uroth series expansion is obtained; if T is taken to be decreasing linear onto [0, 1) it generates the alternating L¨ uroth series expansion of a number in [0, 1). See [1] and [10], where it was shown that the invariant measure is the Lebesgue measure and that the entropy equals

h(T ) =

∞  log(k(k + 1))

k=1

k(k + 1)

= 2.046 · · · .

(iv) As a fourth example we take Bolyai’s expansion; again B = [0, 1), and now T : [0, 1) → [0, 1) is defined by T (x) = (x + 1)2 − 1 − ε1 (x), where

√  1);  0, if x ∈ B(0) = [0, √ 2−√ ε1 (x) = 1, if x ∈ B(1) = [√2 − 1, 3 − 1);  2, if x ∈ B(2) = [ 3 − 1, 1).

Setting εn = εn (x) = ε1 (T n−1 (x)), n ≥ 1, one has   √ x = −1 + ε1 + ε2 + ε3 + · · ·.

It is shown in [14] that T is a number theoretic fibred map. Neither the invariant measure µ nor the exact value of the entropy h(T ) is known; compare Experiment (6.3) and the reference to [9] below. (v) As a final example, let β > 1 satisfy β 2 = β + 1, i.e., β is the golden mean. We describe the β-continued fraction expansion of x. It is an interesting combination of regular continued fractions and the β-adic expansion. A β-integer is a real number of the form an β n + an−1 β n−1 + · · · + a0 where an , an−1 , . . . , a0 is a finite sequence of 0’s and 1’s without consecutive 1’s, i.e., ai · ai−1 = 1 for i = 1, 2, . . . , n. In fact, β-integers can be defined for any β > 1; see e.g. [6] or [8]. For x ∈ (0, 1), let x1 β denote the largest β-integer ≤ x1 , and consider the  transformation T : [0, 1) → [0, 1), defined by T (x) = x1 − x1 β for x = 0, and T (0) = 0. Iteration of T generates continued fraction expansions of the form 1

x=

1

b1 + b2 +

,

1 b3 + . . .

where the bi are β-integers. This expansion is called the β-continued fraction of x. Clearly, this continued fraction expansion is not an f -expansion (cf. [15]) in the classical sense, but it has many of the essential properties of f -expansions,

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and thus R´enyi’s approach in [14] suggests that T has a finite invariant measure equivalent to the Lebesgue measure, with a density bounded away from zero and infinity. Very little seems to be known about such continued fractions. Recently, Bernat showed in [2] that if x ∈ Q(β), then x has a finite β-expansion. We have the following lemma. Lemma 2.4. Let T be a number theoretic fibred map on B, then for almost all x: log λ(Bn (x)) = 1. n→∞ log µT (Bn (x)) lim

Proof. Since c1 λ(Bn (x)) ≤ µT (Bn (x)) ≤ c2 λ(Bn (x)), it follows that − log c2 + log µT (Bn (x)) log λ(Bn (x)) − log c1 + log µT (Bn (x)) ≤ ≤ . log µT (Bn (x)) log µT (Bn (x)) log µT (Bn (x)) Taking limits the desired result follows, since P is a generating partition and therefore limn→∞ µT (Bn (x)) = 0 almost surely. The following formalizes a property that will turn out to be useful in comparing number theoretic maps. Definition 2.5. Let T be a number theoretic fibred map, and let I ⊂ B be an interval. Let m = m(I) ≥ 0 be the largest integer for which we can find an admissible sequence of digits a1 , a2 , . . . , am such that I ⊂ B(a1 , a2 , . . . , am ). We say that T is r-regular for r ∈ N, if for some constant L ≥ 1 the following hold: (i) for every pair B, B ∗ of adjacent cylinders of rank n: 1 λ(Bn ) ≤ ≤ L, L λ(Bn∗ ) (ii) for almost every x ∈ I there exists a positive integer j ≤ r such that either ∗ ∗ Bm+j (x) ⊂ I or Bm+j ⊂ I for an adjacent cylinder Bm+j of Bm+j (x). Thus r-regularity expresses, loosely speaking, that if the expansion T agrees to the first m digits in both endpoints of a given interval I, then for a.e x ∈ I there exists a cylinder of rank at most m + r with the property that it or an adjacent cylinder contains x, and is contained entirely in I; moreover, the size of two adjacent cylinders differs by not more than the constant factor L. Examples 2.6. We give three examples. (i) Let I ⊂ [0, 1) be a subinterval of positive length, and let m = m(I) be such that Bm is the smallest RCF-cylinder containing I, i.e., there exists a vector (a1 , . . . , am ) ∈ Nm such that Bm = Bm (a1 , . . . , am ) ⊃ I and λ(I ∩ ([0, 1) \ Bm+1 (a1 , . . . , am , a))) > 0 for all a ∈ N. One can easily check that r = 3 in case of the RCF. Furthermore, it is well-known, see e.g. (4.10) in [3], p. 43, that λ(Bm (a1 , . . . , am )) =

1 , Qm (Qm + Qm−1 )

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which yields, together with the well-known recurrence relations for the partial fraction denominator sequence (Qm )m≥0 (see [3], (4.2), p. 41): Q−1 := 0;

Q0 := 1;

Qm = am Qm−1 + Qm−2 ,

m ≥ 1,

that λ(Bm (a1 , . . . , am )) ≤ 3λ(Bm (a1 , . . . , am + 1)), for all (a1 , . . . , am ) ∈ Nm , i.e., the continued fraction map T is 3-regular, with L = 3. (ii) The g-adic expansion is not r-regular for any r. Although adjacent rank n cylinders are of the same size, so L = 1 can be taken in (2.5)(i), property (2.5)(ii) does not hold. This can be seen by taking x in a very small interval I around 1/g. (iii) The alternating L¨ uroth map is 3-regular with L = 2. 3. A comparison result We would like to compare two expansions. In this section we will therefore assume that S and T are both number theoretic fibred maps on B. We denote the cylinders of rank n of S by An , and those of T by Bn . We let m(x, n) for x ∈ B and n ≥ 1 be the number of T -digits determined by n digits with respect to S, so m(x, n) is the largest positive integer m such that An (x) ⊂ Bm (x). Yet another way of putting this, is that both endpoints of An (x) agree to exactly the first m digits with respect to T . We have the following general theorem. Theorem 3.1. Suppose that T is r-regular. Then for almost all x ∈ B lim

n→∞

h(S) m(x, n) = . n h(T )

Proof. For x ∈ B let the first n S-digits be given; then by definition of m = m(x, n) one has that An (x) ⊂ Bm (x). Since T is r-regular, there exists 1 ≤ j ≤ r such that Bm+j ⊂ An (x) ⊂ Bm (x), where Bm+j is either Bm+j (x), or adjacent to it of the same rank. By regularity again, for some L ≥ 1, 1 1 λ(Bm+r (x)) ≤ λ(Bm+j (x)) ≤ λ(Bm+j ), L L so

1 1 1 (− log L + log λ(Bm+r (x))) ≤ log λ(An (x)) ≤ log λ(Bm (x)), n n n and the result follows from Lemma (2.4) (applied to both T and S), and the Theorem of Shannon-McMillan-Breiman (1.2). Example 3.2. Let mRCF g (x, n) be the number of partial quotients of x determined by the first n digits of x in its g-adic expansion. Then for almost all x: mRCF 6 log 2 log g g (x, n) = lim . n→∞ n π2

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This generalizes Lochs’ theorem to arbitrary g-adic expansions. Example 3.3. Let mALE uroth digits of x g (x, n) be the number of alternating L¨ determined by the first n digits of x in its g-adic expansion. Then for almost all x: mALE log g g (x, n) = ∞ log k(k+1) . n→∞ n lim

k=1

k(k+1)

Note that the conditions in Theorem 3.1 are not symmetric in S and T . Since the regularity condition is not satisfied by h-adic expansions, Theorem 3.1 is of no use in comparing g-adic and h-adic expansions, nor in proving that the first n regular partial quotients determine usually π 2 /(6 log 2 log g) digits in base g. 4. Comparing radix expansions In this section we compare g-adic and h-adic expansions, for integers g, h ≥ 2, by explicitly studying the distribution of m(n). Given a positive integer n, there exists a unique positive integer  = (n), such that h−(+1) ≤ g −n ≤ h− . (4.1) Thus the measure λ(An ) of a g-cylinder of rank n is comparable to that of an h-cylinder of rank (n). It follows that (n) log g (n) + 1 ≤ ≤ lim inf , lim sup n→∞ n log h n n→∞ i.e., (n) log g lim = , (4.2) n→∞ n log h which is the ratio of the entropies of the maps Tg and Th introduced in (2.3)(ii). Of course, one expects the following to hold: (h)

log g mg (x, n) = lim n→∞ n log h

(a.e.).

(4.3) (h)

which is the analog of Lochs’ result for the maps Tg and Th ; here mg (x, n) is defined as before: it is the largest positive integer m such that An (x) is contained in the h-adic cylinder Bm (x). One should realize that, in general, (n) has no obvious relation to m(n); in fact equation (4.2) is merely a statement about the relative speed with which g-adic and h-adic cylinders shrink. Let x ∈ [0, 1) be a generic number for S = Tg for which we are given the first n digits t1 , t2 , . . . , tn of its g-adic expansion. These digits define the g-adic cylinder An (x) = An (t1 , t2 , . . . , tn ). (h)

Let m(n) = mg (n) and (n) be as defined above; note that m(n) ≤ (n),

for all n ≥ 1.

(4.4)

The sequence (m(n))∞ n=1 is non-decreasing, but may remain constant some time; this means that it ‘hangs’ for a while, so m(n + t) = · · · = m(n + 1) = m(n), after which it ‘jumps’ to a larger value, so m(n + t + 1) > m(n + t). Let (nk )k≥1 be the subsequence of n for which m(n) ‘jumps’, that is, for which m(nk ) > m(nk − 1).

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Lemma 4.5. For almost all x, (h)

mg (nk ) log g = . k→∞ nk log h lim

Proof. By definition, Bm(nk ) (x) ⊃ Ank (x). The cylinder Bm(nk ) (x) consists of h cylinders of rank m(nk ) + 1, and Ank (x) intersects (at least) two of these, since otherwise some Bm(nk )+1 ⊃ Ank (x), contradicting maximality of m(nk ), and thus it contains an endpoint e of some Bm(nk )+1 lying in the interior of Bm(nk ) (x). On the other hand, by definition of nk , we know that m(nk ) > m(nk − 1), so Bm(nk ) (x) ⊃ Ank −1 (x), and therefore Ank −1 (x) contains an endpoint f of Bm(nk ) (x) as well as e. Now e and f are at least λ(Bm(nk )+1 ) = h−(m(nk )+1) apart. Therefore h−(m(nk )+1) ≤ λ(Ank −1 (x)) = g · λ(Ank (x)) = g · g −nk ≤ g · h−(nk ) , by (4.1). Hence

h−(m(nk )+1) ≤ g · h−(nk ) ,

which, in combination with (4.4) implies (nk ) − 1 −

log g ≤ m(nk ) ≤ (nk ), log h

k ≥ 1.

(4.6)

But from (4.6) the Lemma follows immediately. Next we would like to show that (4.3) follows from (4.6); this is easy in case h = 2. Compare [7]. Corollary 4.7. With notations as above, for any g ∈ N, g ≥ 2 (2)

log g mg (x, n) = n→∞ n log 2 lim

(a.e.).

Proof. If h = 2 the mid-point ξ of Bm(n) lies somewhere in An . Now m(n + 1) = m(n) if this mid-point ξ is located in An+1 (t1 , t2 , . . . , tn , tn+1 ), that is, if it is located in the same g-adic cylinder of order n + 1 as x. Notice that this happens with probability g1 , and that the randomness here is determined by the g-adic digit tn+1 . To be more precise, let H denote the event that we will ‘hang’ at time n, i.e., the event that m(n) = m(n + 1), and let D be a random variable with realizations t ∈ {0, 1, . . . , g − 1}, defined by ξ ∈ An+1 (t1 , . . . , tn , D). Now P(H) =

g−1 

P(H|D = i) · P(D = i),

i=0

and from P(H|D = i) = 1/g for 0 ≤ i ≤ g − 1 it then follows that P(H) = 1/g; due to the discrete uniform distribution of the digit tn+1 of x we do not have to know the probabilities P(D = i). Thus we see that for each k ≥ 1 (and with n0 = 0) the random variable vk = nk − nk−1 is geometrically distributed with parameter p = 1/g. Furthermore, the vk ’s are independent, and, since nk ≥ k 1≤

nk + nk+1 − nk 1 1 nk+1 = =1+ vk+1 ≤ 1 + vk+1 . nk nk nk k

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Because v1 , v2 , . . . are independent and identically distributed with finite expectation, it follows from the Lemma of Borel-Cantelli that limk→∞ vk+1 k = 0 (a.e.), and therefore nk+1 lim = 1 (a.e.). k→∞ nk Given any n ≥ 1, there exists k = k(n) ≥ 0 such that nk < n ≤ nk+1 . Since m(nk ) ≤ m(n) ≤ m(nk+1 ) one has m(nk(n)+1 ) m(nk(n) ) m(n) ≤ ≤ , nk(n)+1 n nk(n) But then the result follows from (4.6) with h = 2. If h ≥ 3 the situation is more complicated; there might be more than one midpoint ξi ‘hitting’ An . In case only one ξi ‘hits’ An (as is always the case when h = 2), we speak of a type 1 situation. In case there is more than one ‘hit’ we speak of a type 2 situation. Now change the sequence of jump-times (nk )k≥1 by adding those n for which we are in a type 2 situation, and remove all nk ’s for which we are in a type 1 situation. Denote this sequence by (n∗k )k≥1 . If this sequence is finite, we were originally in a type 1 situation for n sufficiently large. In case (n∗k )k≥1 is an infinite sequence, notice that for every n for which there exists a k such that n = n∗k one has 1-regularity, i.e., Bm(n∗k )+1 ⊂ An∗k (x) ⊂ Bm(n∗k ) (x) , where Bm(n∗k )+1 is either Bm(n∗k )+1 (x) or an adjacent interval. Following the proof of Theorem 3.1 one has m(n∗k ) log g = ∗ k→∞ nk log h lim

(a.e.).

Let (ˆ nk )k≥1 be the sequence we get by merging (nk )k≥1 and (n∗k )k≥1 . Notice that log g m(ˆ nk ) = k→∞ n ˆk log h lim

(a.e.).

(4.8)

If (˜ nk )k≥1 is N \ (ˆ nk )k≥1 , we are left to show that log g m(˜ nk ) = k→∞ n ˜k log h lim

(a.e.).

Let n = n ˜ j for some j ≥ 1. Then there exist unique k = k(j) and h = h(j) such that nk+1 = n ˆ h+1 , nk ≤ n ˆ h and n ˆ h < n < nk+1 . Since nk+1 − n ˆh = n ˆ h+1 − n ˆ h = vh+1 is geometrically distributed with parameter 1/g, and since v1 , v2 , . . . are independent with finite expectation, it follows as in the case h = 2, that 1 1 n ˆ h+1 =1+ vh+1 ≤ 1 + vh+1 → 1 n ˆh n ˆh h

as j → ∞

But then (4.3) follows from (4.8) and from m(ˆ nh+1 ) m(ˆ nh ) m(n) ≤ ≤ . n ˆ h+1 n n ˆh We have proved the following theorem; again, see also [7].

(a.e.).

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Theorem 4.9. Let g, h ∈ N≥2 . Then for almost all x (h)

mg (x, n) log g = . lim n→∞ n log h 5. Remarks on generalizations At first sight one might think that the proof of Theorem 4.9 can easily be extended to more general number theoretic fibered maps, by understanding the distribution of m(n). However, closer examination of the proof of Theorem 4.9 shows that there are two points that make generalizations of the proof hard, if not impossible. The first is the observation, that the ‘hanging time’ vk (in a type 1 situation) is geometrically distributed, and that the vk ’s are independently identically distributed. Recall that this observation follows from the fact that the digits given by the g-adic map Sg are independently identically distributed and have a discrete uniform distribution. As soon as this last property no longer holds (e.g., if S = Tγ , the expansion with respect to some non-integer γ > 1), we need to know P(D = i) for each i ∈ {0, 1, . . . , γ − 1}, and this might be difficult. Even if we assume S to be the g-adic map, another problem arises when T is not the h-adic map, but, for example, T = Tγ . In that case all the ingredients of the proof of Theorem 4.9 seem to work with the exception that at the ‘jump-times’ (nk )k we might not be able to show that (nk ) − C ≤ m(nk ) ≤ (nk ),

(5.1)

with C ≤ 1 some fixed constant. Notice that from (5.1) it would follow that log g m(nk ) = . k→∞ nk log γ lim

For one class of γ ∈ (1, 2) one can show that (5.1) still holds. These γ’s are the so-called ‘pseudo-golden mean’ numbers; γ > 1 is a ‘pseudo-golden mean’ number if γ is the positive root of X k − X k−1 − · ·√ · − X − 1 = 0, for some k ∈ N, k ≥ 2 1 (in case k = 2 one has that γ = β = 2 (1 + 5), which is the golden mean). These ‘pseudo-golden mean’ numbers γ are all Pisot numbers, and satisfy 1=

1 1 1 + 2 + ··· + k. γ γ γ

Theorem 5.2. With notations as before, let S = Tg and T = Tγ , with g ∈ N≥2 , and γ > 1 a ‘pseudo golden number’. Then (γ)

log g mg (n) = , n→∞ n log γ lim

(a.e.).

This result is immediate from [7]. Here we present a different proof, based on the distribution of the number of correct digits, m(n). Proof of Theorem 5.2. In case k = 2, a digit 1 is always followed by a zero, and (γ) only the γ-cylinders Bm corresponding to a sequence of γ-digits ending with 0 are (γ) refined. Thus from the definition of m(n) it follows that the last digit of Bm(n) (x) is always 0 (if it were 1, the choice was wrong). Note that in this case (γ)

λ(Bm(n) (x)) = γ −m(n) .

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Let (n) be defined as before, and notice that at a ‘jump-time’ n = nk one has γ −(m(n)+2) ≤ g · g −n ≤ g · γ −(n) ,

(5.3)

from which

log g ≤ m(n). log γ Since m(n) ≤ (n) we see that (5.1) follows in case γ equals the golden mean. In case k ≥ 3 the situation is slightly more complicated; let us consider here k = 3. Now any sequence of γ-digits ending with two consecutive 1’s must be followed by a zero, and therefore − by the definition of m(n) − the last digit of (γ) Bm(n) (x) is either 0, or is 1 which is preceded by 0. One can easily convince oneself then that  γ −m(n) if dm(n) (x) = 0, (γ) λ(Bm(n) (x)) = −(m(n)+1) −(m(n)+2) γ +γ if dm(n) (x) = 1. (n) − 2 −

Here dm(n) (x) is the m(n)th γ-digit of x. (γ) Now let n = nk be a ‘jump-time’. If dm(n) (x) = 0, then Bm(n) (x) consists of two γ-cylinders, one of length γ −(m(n)+1) and one of length γ −(m(n)+2) + γ −(m(n)+3) . One of these two cylinders is contained in An−1 (x) (due to the fact that n is a ‘jump-time’), and (5.3) is therefore satisfied. Of course one has m(n) ≤ (n) in this (γ) case. In case dm(n) (x) = 1 we see that Bm(n) (x) consists of two γ-cylinders, one of length γ −(m(n)+1) and one of length γ −(m(n)+2) . Now (γ)

λ(Bm(n) (x)) = γ −(m(n)+1) + γ −(m(n)+2) ≤ γ −m(n) , and by definition of (n) one has m(n) ≤ (n). Again one of these two sub-cylinders (γ) of Bm(n) (x) is contained in An−1 (x), and one has γ −(m(n)+2) ≤ g · g −n ≤ g · γ −(n) . We see that (5.3) is again satisfied. For k ≥ 4 the proof is similar; one only needs to consider more cases. 6. Estimation of (unknown) entropies In this section we report on some numerical experiments. Our experiments were carried out using the computer algebra system Magma, see [4]. The general set-up of the experiments was as follows. We choose n random digits for the expansion of a number x in [0, 1) with respect to a number theoretic fibred map S, and compute mTS (x, n), the number of digits of x with respect to the number theoretic fibred map T , determined completely by the first n digits of x with respect to S. This is done by comparing the T -expansions of both endpoints of the S-cylinder An (x). Experiment 6.1. The first experiment was designed to test the set-up. It aimed to verify Lochs’ result: we let S = T10 , the decimal expansion map, and T = T RCF for the regular continued fraction map. For N = 1000 random real numbers of n = 1000 decimal digits, we computed the number of partial quotients determined by the endpoints of the decimal cylinder An (x). We ran this experiment twice; the averages (over 1000 runs) were 970.534 and 969.178 (with standard deviations around 24). Compare these averages to the value 970.270 · · · predicted by Theorem 1.1.

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Experiment 6.2. In this experiment we compared the relative speed of approximation of g-adic and h-adic expansions: h→ g↓ 2 7 10

2 pred 2.807 3.322

7 observed

2.805 3.320

2.805 3.320

pred 0.356 1.183

observed 0.355 0.355 1.182 1.182

pred 0.301 0.845 -

10 observed 0.300 0.300 0.844 0.844 -

In fact we compared binary, 7-adic and decimal expansions of N = 1000 random real numbers with each of the other two expansions, again by computing to how many h-digits both endpoints of the g-adic cylinder A1000 (x) agreed. The two tables list the values found in each of two rounds of this experiment and compares the result with the value found by Theorem 4.9. Note that by (4.4) always m(n) ≤ (n) and thus the observed ratio should approximate log g/ log h from below. By reversing the roles of g and h, it is thus possible in this case to approximate the entropy quotient by sandwiching! Experiment 6.3. By choosing random real numbers of 1000 decimal digits, we attempted to estimate the unknown entropy of the Bolyai map (cf. (2.3)(iv)). We found that on average 1000 digits determined 2178.3 Bolyai digits in the first run of N = 250 random reals, and 2178.0 in the second; the standard deviations were around 13. These values would correspond to an entropy of around 1.0570 or 1.0572. Choosing N random Bolyai expansions of length 1000 (by finding the Bolyai expansion to 1000 digits for random reals), we determined the number of decimal digits in one experiment, and the number of regular continued fraction partial quotients in another experiment. In the first run of N = 1000, we found that on average 458 decimal digits were determined, in the second run 457.8 on average (with standard deviations around 4). These values would correspond to an entropy of around 1.0546 for the first, and 1.0541 for the second run, according to Theorem 5 in [7]. In runs with N = 1000 of the second experiment we found that 444.36 and 444.37 partial quotients were determined on average, with standard deviations of around 17. This suggests an entropy of 1.0545 or 1.0546 according to Theorem 3.1. These entropy estimates for the Bolyai-map were reported in a preliminary version of this paper [5]. Using a different method, based on the fixed-points of the Bolyai-map, Jenkinson and Pollicott were able to find in [9] sharper estimates for the entropy. To be precise, they found 1.056313074, and showed that this is within 1.3 × 10−7 of the true entropy. Experiment 6.4. In our final experiment we attempted to estimate the entropy of the β-continued fraction map, see (2.3)(v). As before, we chose real numbers with 1000 random decimal digits and determined the number of β-integer partial quotients determined by them. In the first run, the 1000 decimals determined 877.922 partial quotients on average, in the second run 878.125 partial quotients were determined. These values correspond to an entropy of 2.021 and 2.022. References [1] Barrionuevo, J., Burton, R. M., Dajani, K. and Kraaikamp, C. (1996). Ergodic properties of generalized L¨ uroth series, Acta Arithm. 74 311–327. MR1378226

188

W. Bosma, K. Dajani and C. Kraaikamp

[2] Bernat, J. (2003). Continued fractions and numeration in Fibonacci basis. In Proceedings of WORDS’03, TUCS Gen. Publ. 27, Turku Cent. Comput. Sci., Turku, 2003, pp. 135–137. MR2081346 [3] Billingsley, P. (1965) Ergodic Theory and Information, New York–London– Sydney: John Wiley and Sons. MR192027 [4] Bosma, W., Cannon, J. and Playoust C. (1997). The Magma algebra system I: The user language, J. Symbolic Computation 24 235–265. MR1484478 [5] Bosma, W., Dajani, K. and Kraaikamp, C. (1999). Entropy and counting correct digits, Department of Math. Nijmegen University Report No. 9925. http://www.math.ru.nl/onderzoek/reports/reports1999.html ˘ Frougny, C., Gazeau, J.-P. and Krejcar, R. (1998). Beta[6] Burd´ik, C., integers as a group. In Dynamical Systems, (Luminy-Marseille, 1998), 125–136, River Edge, NJ: World Sci. Publishing 2000. MR1796153 [7] Dajani, K. and Fieldsteel, A. (2001). Equipartition of interval partitions and an application to number theory, Proc. Amer. Math. Soc. 129 3453–3460. MR1860476 [8] Frougny, C., Gazeau, J.-P. and Krejcar, R. (2003). Additive and multiplicative properties of point sets based on beta-integers, Th. Comp. Sci. 303 491–516. MR1990778 [9] Jenkinson, O. and Pollicott, M. (2000). Ergodic properties of the BolyaiR´enyi expansion, Indag. Math. (N.S.) 11 399-418. MR1813480 [10] Kalpazidou, S., Knopfmacher, A. and Knopfmacher, J. (1991). Metric properties of alternating L¨ uroth series, Portugaliae Math. 48 319–325. MR1127129 [11] Lochs, G. (1964). Vergleich der Genauigkeit von Dezimalbruch und Kettenbruch, Abh. Math. Sem. Hamburg 27 142–144. MR162753 [12] Lochs, G. (1963). Die ersten 968 Kettenbruchnenner von π, Monatsh. Math. 67 311–316. MR158507 [13] Nakada, H. (1981). Metrical theory for a class of continued fraction transformations and their natural extensions, Tokyo J. Math. 4 399–426 MR646050 ´nyi, A. (1957). Representations for real numbers and their ergodic proper[14] Re ties, Acta Math. Acad. Sci. Hung. 8 472–493. MR97374 [15] Schweiger, F. (1995). Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford: Oxford University Press. MR1419320

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 189–197 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000211

Mixing property and pseudo random sequences Makoto Mori1 Nihon University Abstract: We will give a summary about the relations between the spectra of the Perron–Frobenius operator and pseudo random sequences for 1-dimensional cases.

There are many difficulties to construct general theory of higher-dimensional cases. We will give several examples for these cases. 1. Perron–Frobenius operator on one-dimensional dynamical systems Let I = [0, 1] and F be a piecewise monotonic and C 2 transformations on it. We assume F is expanding in the following sense: ξ = lim inf ess inf n→∞

x∈I

1 log |F n  (x)| > 0. n

1

Definition 1.1. An operator P : L → L1 defined by
P f (x) = f (y)|F  (y)|−1 y : F (y)=x

is called the Perron–Frobenius operator associated with F . It is well known that the spectra of the Perron–Frobenius operator P associated with the transformation F determine the ergodic properties of the dynamical system (I, F ). Roughly speaking: 1. The dimension of the eigenspace associated with eigenvalue 1 equals the number of the ergodic components. 2. The eigenfunction ρ of P associated with the eigenvalue 1 for which ρ ≥ 0 and  ρ dx = 1 becomes the density function of the invariant probability measure. 3. If 1 is simple and there exists no eigenvalue modulus 1 except 1, then the dynamical system is mixing. 4. The second greatest eigenvalue determines the decay rate of correlation. To be more precise, the Perron–Frobenius operator satisfies: 1. P is contracting, and 1 is an eigenvalue. Hence, its spectrum radius equals 1. 2. P is positive. Hence there exists a nonnegative eigenfucntion associated with the eigenvalue 1. 1 Department

of Mathematics, College of Humanities and Sciences, Nihon University, Japan, e-mail: [email protected] AMS 2000 subject classifications: primary 37B10. Keywords and phrases: subshift, g-function. 189

M. Mori

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3. On th unit circle, the eigenvalues of P coincide with the eigenvalues of the unitary operator U f (x) = f (F (x)). 4. The essential spectrum radius of P : L1 → L1 equals 1, that is, it is of no use to consider decay rate for the functions in L1 . 5. When we restrict P to BV the set of functions with bounded variation, then its essential spectrum radius equals e−ξ . Hence, the decay rate for a function with bounded variation is at most e−ξ . Hereafter, we only consider P restricted to BV . Since P is not compact, it is not easy to calculate its spectra. Hofbauer and Keller ([1]) proved that they coincide with the singularities of the zeta function   ∞ n

z |F n  (p)|−1  . ζ(z) = exp  n n n=1 p : F (p)=p

The zeta function has the radius of convergence 1, and it has a meromorphic extension to eξ . But it is not easy to study the singularities in |z| > 1. Since F is piecewise monotone, there exists a finite set A which we call alphabet with the following properties. 1. For each a ∈ A, an interval a corresponds, and {a}a∈A forms a partition of I, 2. F is monotone on each a, and it has a C 2 extension to the closure of a. We define

 +1 if F is monotone increasing in a, sgn a = −1 otherwise.

We call F Markov if there exists an alphabet A, such that, if F (a)o ∩ b =  ∅ then F (a) ⊃ bo , where J o denotes the inner of a set J. We will summarize notations which we use in this article. 1. A sequence of symbols a1 · · · an (ai ∈ A) is called a word and (a) |w| = n (the length of a word), n (b) w = i=1 F −i+1 (ai ) (the interval corresponding to a word),

(c) We denote the empty word by , and define || = 0, sgn  = +1 and  = I,

(d) we call w is admissible if w =  ∅. We denote the set of admissible words with length n by Wn and W = ∪∞ n=0 Wn , n (e) sgn w = i=1 sgn ai .

2. For a point x ∈ I, ax1 ax2 · · · is called the expansion of x which is defined by F n (x) ∈ an+1 . We usually identify x and its expansion. 3. For a sequence of symbols s = a1 a2 · · · (ai ∈ A), (a) s[n, m] = an an+1 · · · am (b) s[n] = an .

(n ≤ m),

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191

4. For a word w = a1 · · · an and x ∈ I, wx is an infinite sequence of symbols defined by wx = a1 · · · an ax1 ax2 · · · . We call wx exists if there exists a point y whose expansion equals wx. We have a natural order on A, that is, a < b when x < y for any x ∈ a and y ∈ b. Then we define an order on words by 1. w < w if |w| < |w |, 2. w = a1 · · · an , w = b1 · · · bn and if ai = bi for k < i ≤ n and ak < bk , then w < w w > w

if sgn ak+1 · · · an = +1, otherwise.

2. Pseudo random sequences We call a sequence {xn }∞ n=1 uniformly distributed if for any interval J ⊂ I #{xn ∈ J : n ≤ N } = |J|, N →∞ N lim

where |J| is the Lebesgue measure of an interval J. Uniformly distributed sequece {xn }∞ n=1 is called of low discrepancy if





#{xn ∈ J : n ≤ N } log N



DN = sup

− |J| = O . N N J

It is well known that this is best possible and low discrepancy sequences play important role in numerical integrations. We can construct a low discrepancy sequence using dynamical system.

Definition 2.1. The set {wx}w∈W for which wx exists and arranged in the order of words is called a van der Corput sequence defined by F . When F (x) = 2x (mod 1), A has two elements so we express A = {0, 1}, and every words are admissible. Thus our van der Corput sequence is x, 0x, 1x, 00x, 10x, 01x, 11x, 000x, . . . , and original van der Corput sequence is the case when x = 12 . Our first goal is the following theorem: Theorem 2.2 ([4]). Let F be a Markov and topologically transitive transformation with the same slope |F  | ≡ β > 1. Then, for any x ∈ I, the van der Corput sequence is of low discrepancy if and only if there exists no eigenvalue of the Perron– Frobenius operator in the annuls e−ξ < |z| ≤ 1 except z = 1. For Markov β–transformations, Ninomiya ([9]) proved the necessary and sufficient condition that the van der Corput sequence for x = 0 is of low discrepancy by direct calculation. He extended the results to non Markov β–transformations ([10]). Rough sketch of the proof is the following. For any interval J
P n 1J (x) = 1J (y)|F n  (y)|−1 y : F n (y)=x

= β −n




|w|=n

1J (wx).

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Note that |w|=n 1J (wx) is the number of hits to J in the van der Corput sequence corresponding to words with length n. On the other hand, as a rough expression, P n 1J (x) = |J|ρ(x) + η n , where η is the second greatest eigenvalue in modulus. Thus the number of wx which hits to J with |w| = n equals β n |J|ρ(x) + (βη)n . The discrepancy depends on the term (βη)n , and we already knew that |η| ≥ β −1 . Thus η = β −1 is the best possible case, and at that time, the number of wx which hit to J with |w| ≤ n equals β n+1 − 1 |J|ρ(x) + n. β−1 The number N of admissible wx with |w| ≤ n is of order β n . This says the discrepancy is of order logNN . 3. Fredholm matrix To make the proof of Theorem 2.2 rigorous, we define generating functions. Let F be a piecewise linear Markov transformation with the same slope β > 1. Let for an interval J ∞
z n P n 1J (x). sJ (z, x) = n=0

Let s(z, x) be a vector with coefficients sa (z, x) (a ∈ A). Then we get a renewal equation of the form: s(z, x) = (I − Φ(z))−1 χ(x), where Φ(z) is a A × A matrix and χ(x) is a A dimensional vector whose coefficients equal:  zβ −1 if F (a) ⊃ bo , Φ(z)a,b = 0 otherwise, χ(x)a = 1(a) (x). We call the matrix Φ(z) the Fredholm matrix associated with F . Theorem 3.1. det(I − Φ(z)) =

1 . ζ(z)

This theorem says that the eigenvalues of P is determined by the zeros of det(I − Φ(z)). We can express sJ (z, x) using the Fredholm matrix and a row vector whose coefficients are the functions with the radius of convergence β ([4]). From these, we can get the rigorous proof of Theorem 2.2. We can extend the above discussion to more general cases for 1-dimensional transformations. We use the signed symbolic dynamics defined by the orbits of the endpoints of a (a ∈ A). Let for x ∈ I x+ = lim ay1 ay2 · · · , y↑x

x = lim ay1 ay2 · · · . −

y↓x

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For an interval J, we denote (sup J)+ and (inf J)− by J + and J − , respectively. Especially, for a ∈ A, we denote by a+ and a− instead of a+ and a− . Let A˜ be the set of aσ (a ∈ A, σ = ±). We also define yσ

s (z, x) =

∞


n=0

z n β −n




σ(y σ , wx)δ[w[1] ⊃ ay1 , ∃θwx],

w∈Wn

where θ is the shift and  1 if L is true, δ[L] = 0 if L is false,  + 12 if y ≥σ x, σ(y σ , x) = − 12 if y <σ x,  x < y σ = +, x <σ y = x > y σ = −. Then

+

−

sJ (z, x) = sJ (z, x) + sJ (z, x).

We can also get the similar equation s(z, x) = (I − Φ(z))−1 χ(z, x), where σ

s(z, x) = (sa (z, x))a∈A,σ=± , σ

χ(z, x) = (χa (z, x))a∈A,σ=± ,  σ(y σ , x) yσ χ (z, x) = ∞ n −n σ(θn y σ , x) n=0 z β

y

if θy σ = (ay2 )σ sgn a1 otherwise,

Φ(z) = (φaσ ,bτ (z))a,b∈A,σ,τ =± ,  y −1  if bτ ≤σ θy σ , and θy σ = (ay2 )σ sgn a1 +zβ /2 y φyσ ,bτ (z) = −zβ −1 /2 if bτ >σ θy σ , and θy σ = (ay2 )σ sgn a1   ∞ n −n σ(θn y σ , bτ )(sgn y σ [1, n − 1]) otherwise. n=1 z β

Moreover, Theorem 3.1 also holds (cf. [3]). This leads to the general theory concerning the discrepancy of sequences generated by one-dimensional piecewise linear transformations. Definition 3.2. We call an endpoint of a a Markov endpoint if the image of this point by F n for some n ≥ 1coincides with some endpoint of b (b ∈ A). Theorem 3.3 ([2]). Let us denote by k the number of non-Markov endpoints of F. 1. Let ζn β −n be the n-th coefficient of (1 − z)ζ(z). Assume that ζn is bounded, then  (log N )k+2 DN = O . N

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2. Let Φ11 (z) be the minor of the Fredholm matrix Φ(z) corresponding to Markov endpoints. Assume that det(I − Φ11 (z)) = 0 in |z| < β and there exists no sigularity of ζ(z) in |z| < β except 1. Then the discrepancy satisfies  (log N )k+1 DN = O . N 4. Higher-dimensional cases We want to extend the result to higher-dimensional cases. Let I = [0, 1]d . We will call a product of intervals also by intervals. We can define the Perron–Frobenius operator P for higher-dimensional cases. Here we use Jacobian det F instead of derivative F  . For Markov cases, we can find the spectra of the Perron–Frobenius operator by the Fredholm matrix by almost the same discussion. Even for non Markov piecewise linear cases, instead of signed symbolic dynamics using the idea of screens we can construct Fredholm matrix and we can determine the spectra of the Perron–Frobenius operator ([6]).

We consider a space of functions for which there exists a decomposition f = w Cw 1w such that for any 0 < r < 1
|Cw |r|w| < ∞. w

This is a slight extention of the space of functions with bounded variation in 1dimensional cases. Here, note that, though the corresponding zeta function has meromorphic extention to |z| < eξ , even for the simplest Bernoulli transformation F (x, y) = (2x, 2y) (mod 1), the essential spectrum radius is greater than e−ξ . Here is a question. Are there any transformation whose spectrum radius equals e−ξ ?

If such a transformation for which the Jacobian is constant exists, we will be able to construct van der Corput sequences of low discrepancy. Here we call a sequence of low discrepancy if



 d

#{xi ∈ J : i ≤ N } (log N ) − |J|

= O sup

, N N J

where sup is taken over all intervals J ⊂ I. It is proved that this is best possible for d = 2, and this will be also true for d ≥ 3. We have not yet the answer, but we have constructed low discrepancy sequences for d = 2 using dynamical system. Also for d = 3 we have constructed low discrepancy sequences using a sequence of transformations F1 , F2 , . . . but we have not yet constructed a transformation for which Fn = F n . We have not suceeded to get the answer to our question, but these results suggests that the question will be positive. 4.1. Two-dimensional cases To construct low discrepancy sequences, we need a transformation not only expanding but also shuffling. For two-dimensional cases, let s0 be the infinite sequence of 0’s, and s1 = w1 w2 · · · ,

Mixing property and pseudo random sequences

195

where wn is the word with length 2n−1 and only the last symbol equals 1 and other symbols are 0, that is, s1 = 101000100000001 · · · . We consider the digitwise sum modulo 2 on the set of infinite sequences of 0 and 1. Let θ be a shift map to left. Then s0 , s1 , θs1 , . . . , θn−1 s1 generate all the words with length n. Let us define    s θx x + y1 , = F θy y sx1 where we identify x ∈ [0, 1) and its binary expansion x1 x2 · · · . Then we can prove for I = [α, α + 2−n+m ] × [β, β + 2−n−m ) for binary rationals α and β and m ≤ n,the image of I by F n does not overlap and F n (I) = [0, 1)2 . For example, I = [0, 1) × [0, 1/4) (n = m = 1), though θy belongs only to [0, 1/2), x1 takes both 0 and 1. Hence, θy expands all [0, 1) by adding s0 or s1 depending on where x belongs. Thus F (I) = [0, 1)2 . Theorem 4.1 ([7]). The van der Corput sequence generated by the above F is of low discrepancy. 4.2. Three-dimensional cases For three-dimensional cases, we have not yet proved to get low discrepancy sequences by one transformation. So we use a sequece of transformations F, F2 , F3 , . . .. Even for higher dimensional cases, we will be able to construct low discrepancy sequences in the same way. However, we have no proof yet. We denote     x x    Fn y = y   . z z Then expressing x , y  , z  in binary expansions, we define   xn       xn+1 x1  yn     y1   yn+1   zn       ..   z1   zn+1   .        x2  xn+2    = +M  x1  .  y2   yn+2   y1         z2   zn+2   z1        .. .. 0 . .   .. . Here, M is an infinite-dimensional matrix. One example of M is following.

M. Mori

196

x1 y1 z1 x2 y2 z2 x3 y3 z3 x4 y4 z4 x5 y5 z5 x6 y6 z6 x7 y7 z7 x8 y8 z8 x9 y9 z9 x10 y10 z10

x−1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

y−1 1 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

z−1 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

x−2 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

y−2 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

z−2 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

x−3 0 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0

y−3 1 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0

z−3 0 1 0 0 1 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0

x−4 0 1 0 0 0 1 0 0 1 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0

y−4 0 0 1 1 0 0 1 0 0 0 0 1 1 0 1 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0

z−4 1 0 0 0 1 0 0 1 0 1 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 0 0 0 0 0

This M has following properties: for any nonnegative integers n and m: the determinant of the minor matrix with coordinates {x1 , . . . , xn+m } × {y−1 , . . . , y−n , z−1 . . . , z−m } and the determinant of the minor matrix with coordinates {x1 , . . . , xn , y1 , . . . , ym } × {z−1 , . . . , z−n−m } do not vanish. M also has symmetry in the permutations x, y and z. This matrix corresponds to s0 and s1 in two-dimensional case, and this shuffles coordinates, and we get for k ≥ n + m I = [α, α + 2−k−n−m ) × [β, β + 2−k+n ) × [γ, γ + 2−k+m ) or then

I = [α, α + 2−k−n ) × [β, β + 2−k−m ) × [γ, γ + 2−k+n+m ), Fk (I) = [0, 1)3 ,

where α, β and γ are binary rationals. Theorem 4.2. The van der Corput sequence generated by F1 , F2 , . . . is of low discrepancy. References [1] Hofbauer, F. and Keller, G. (1984). Zeta-functions and transfer-operators for piecewise linear transformations. J. Reine Angew. Math. 352, 100–113. MR758696

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[2] Ichikawa, Y. and Mori, M. (2004). Discrepancy of van der Corput sequences generated by piecewise linear transformations. Monte Carlo Methods and Appl. 10, 12, 107–116. MR2096253 [3] Mori, M. (1990). Fredholm determinant for piecewise linear transformations. Osaka J. Math. 27, 81–116. MR1049827 [4] Mori, M. (1998). Low discrepancy sequences generated by piecewise linear maps. Monte Carlo Methods and Appl. 4, 1, 141–162. MR1637372 [5] Mori, M. (1999). Discrepancy of sequences generated by piecewise monotone maps. Monte Carlo Methods Appl. 5, 1, 55–68. MR1684993 [6] Mori, M. (1999). Fredholm determinant for higher-dimensional piecewise linear transformations. Japan. J. Math. (N.S.) 25, 2, 317–342. MR1735464 [7] Mori, M. (2002). Construction of two dimensional low discrepancy sequences, Monte Carlo methods and Appl. 8, 2, 159-170. MR1916915 [8] Mori, M. Construction of 3 dimensional low discrepancy sequences, submitted. [9] Ninomiya, S. (1998). Constructing a new class of low-discrepancy sequences by using the β-adic transformation. Math. Comput. Simul. 47, 405–420. MR1641375 [10] Ninomiya, S. (1998). On the discrepancy of the β-adic van der Corput sequence. J. Math. Sci. Univ. Tokyo 5, 2, 345–366. MR1633866

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 198–211 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000220

Numeration systems as dynamical systems – introduction Teturo Kamae1 Matsuyama University Abstract: A numeration system originally implies a digitization of real numbers, but in this paper it rather implies a compactification of real numbers as a result of the digitization. By definition, a numeration system with G, where G is a nontrivial closed multiplicative subgroup of R+ , is a nontrivial compact metrizable space Ω admitting a continuous (λω + t)-action of (λ, t) ∈ G × R to ω ∈ Ω, such that the (ω + t)-action is strictly ergodic with the unique invariant probability measure µΩ , which is the unique G-invariant probability measure attaining the topological entropy | log λ| of the transformation ω → λω for any λ = 1. We construct a class of numeration systems coming from weighted substitutions, which contains those coming from substitutions or β-expansions with algebraic β. It also contains those with G = R+ . We obtained an exact formula for the ζ-function of the numeration systems coming from weighted substitutions and studied the properties. We found a lot of applications of the numeration systems to the β-expansions, Fractal geometry or the deterministic self-similar processes which are seen in [10]. This paper is based on [9] changing the way of presentation. The complete version of this paper is in [10].

1. Numeration systems By a numeration system, we mean a compact metrizable space Ω with at least 2 elements as follows: (
1) There exists a nontrivial closed multiplicative subgroup G of R+ and a continuous action λω + t of (λ, t) ∈ G × R to ω ∈ Ω such that λ
(λω + t) + t
= λ
λω + λ
t + t
. (
2) The (ω + t)-action of t ∈ R to ω ∈ Ω is strictly ergodic with the unique invariant probability measure µΩ called the equilibrium measure on Ω. Consequently, it is invariant under the (λω + t)-action of (λ, t) ∈ G × R to ω ∈ Ω as well. (
3) For any fixed λ0 ∈ G, the transformation ω → λ0 ω on Ω has the | log λ0 |topological entropy. For any probability measure ν on Ω other than µΩ which is invariant under the λω-action of λ ∈ G to ω, and 1 = λ0 ∈ G, it holds that hν (λ0 ) < hµΩ (λ0 ) = | log λ0 |. The (ω + t)-action of t ∈ R to ω ∈ Ω is called the additive action or R-action, while the λω-action of λ ∈ G to ω ∈ Ω is called the multiplicative action or G-action. Note that if Ω is a numeration system, then Ω is a connected space with the continuum cardinality. Also, note that the multiplicative group G as above is either R+ or {λn ; n ∈ Z} for some λ > 1. Moreover, the additive action is faithful, that is, ω + t = ω implies t = 0 for any ω ∈ Ω and t ∈ R. 1 Matsuyama

University, 790-8578 Japan, e-mail: [email protected] AMS 2000 subject classifications: primary 37B10. Keywords and phrases: numeration system, weighted substitution, fractal function and set, self-similar process, ζ-function. 198

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199

This is because if there exist ω1 ∈ Ω and t1 = 0 such that ω1 + t1 = ω1 , then take a sequence λn in G such that λn → 0 and λn ω1 converges as n → ∞. Let ω∞ := limn→∞ λn ω1 . For any t ∈ R, let an be a sequence of integers such that an λn t1 → t as n → ∞. Then we have ω∞ + t = lim (λn ω1 + λn an t1 ) n→∞

= lim λn (ω1 + an t1 ) = lim λn ω1 = ω∞ . n→∞

n→∞

Thus, ω∞ becomes a fixed point of the (ω + t)-action of t ∈ R to ω ∈ Ω. Since this action is minimal, we have Ω = {ω∞ }, contradicting with that Ω has at least 2 elements. An example of a numeration system is the set {0, 1}Z with the product topology divided by the closed equivalence relation ∼ such that (. . . , α−2 , α−1 ; α0 , α1 , α2 , . . .) ∼ (. . . , β−2 , β−1 ; β0 , β1 , β2 , . . .) if and only if there exists N ∈ Z ∪ {±∞} satisfying that αn = βn (∀n > N ), αN = βN + 1 and αn = 0, βn = 1 (∀n < N ) or the same statement with α and β exchanged. Let Ω(2) := {0, 1}Z / ∼ and the
equivalence class containing ∞ Z (. . . , α−2 , α−1 ; α0 , α1 , α2 , . . .) ∈ {0, 1} is denoted by n=−∞ αn 2n ∈ Ω(2). Then, Ω(2) is an additive topological group with the addition as follows: ∞ 

αn 2n +

∞ 

βn 2n =

γn 2n

n=−∞

n=−∞

n=−∞

∞ 

if and only if there exists (. . . , η−2 , η−1 ; η0 , η1 , η2 , . . .) ∈ {0, 1}Z satisfying that 2ηn+1 + γn = αn + βn + ηn (∀n ∈ Z). This is isomorphic to the 2-adic solenoidal group which is by definition the projective limit of the projective system θ : R/Z → R/Z with θ(α) = 2α (α ∈ R/Z). Moreover, R is imbedded in Ω(2) continuously as a dense additive in
∞ subgroup n the way that a nonnegative real number α is identified with n=−∞ αn 2 such
N that α = n=−∞ αn 2n and αn = 0 (∀n > N ) for some
∞ N ∈ Z, whilena negative real number −α with α as above is identified with n=−∞ (1 − αn )2 . Then, R acts additively to Ω(2) by this addition. Furthermore, G := {2k ; k ∈ Z} acts multiplicatively to Ω(2) by 2

k

∞ 

n=−∞

n

αn 2 =

∞ 

αn−k 2n .

n=−∞

Thus, we have a group of actions on Ω(2) satisfying (
1), (
2)and (
3) with G := {2k ; k ∈ Z} and the equilibrium measure (1/2, 1/2)Z . Theorem 1.1. Ω(2) is a numeration system with G = {2n ; n ∈ Z}. We can express Ω(2) in the following different way. By a partition of the upper half plane H := {z = x + iy; y > 0}, we mean a disjoint family of open sets such that the union of their closures coincides with H. Let us consider the space Ω(2)
of partitions ω of H by open squares of the form (x1 , x2 ) × (y1 , y2 ) with x2 − x1 = y2 − y1 = y1 and y1 ∈ G such that (x1 , x2 ) × (y1 , y2 ) ∈ ω implies (x1 , (x1 + x2 )/2) × (y1 /2, y1 ) ∈ ω (type 0) and ((x1 + x2 )/2, x2 ) × (y1 /2, y1 ) ∈ ω (type 1).

(1)

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T. Kamae y

i




x Fig 1. The tiling corresponding to · · · 01.101 · · · .

An example of ω ∈ Ω(2)
is shown in Figure 1. For ω ∈ Ω(2)
, let (α0 , α1 , . . .) be the sequence of the types defined in (1) of the squares in ω intersecting with the half vertical line from +0 + i to +0 + i∞ and let (α−1 , α−2 , . . .) be the sequence of the types of the squares in
ω intersecting with the line segment from +0 + i to ∞ n +0.
∞Then, ω nis identified with n=−∞ αn 2 . Note that replacing +0 by −0, we get n=−∞ βn 2 such that (. . . , α−2 , α−1 ; α0 , α1 , . . .) ∼ (. . . , β−2 , β−1 ; β0 , β1 , . . .). The topology on Ω(2)
is defined so that ωn ∈ Ω(2)
converges to ω ∈ Ω(2)
as n → ∞ if for every R ∈ ω, there exist Rn ∈ ωn such that limn→∞ ρ(R, Rn ) = 0, where ρ is the Hausdorff metric between sets R, R
⊂ H ρ(R, R
) := max{sup
inf
|z − z
|, sup inf |z − z
|}. z∈R z ∈R

z
∈R
z∈R

(2)

For ω ∈ Ω(2)
, t ∈ R and λ ∈ {2n ; n ∈ R}, ω + t ∈ Ω(2)
and λω ∈ Ω(2)
are defined as the partitions ω + t := {(x1 − t, x2 − t) × (y1 , y2 ); (x1 , x2 ) × (y1 , y2 ) ∈ ω} and λω := {(λx1 , λx2 ) × (λy1 , λy2 ); (x1 , x2 ) × (y1 , y2 ) ∈ ω}. Let κ : Ω(2)
→ Ω(2) be the identification mapping defined above. Then, κ is a homeomorphism between Ω(2)
and Ω(2) such that κ(ω + t) = κ(ω) + t and κ(λω) = λκ(ω) for any ω ∈ Ω(2)
, t ∈ R and λ ∈ {2n ; n ∈ Z}. Thus, Ω(2)
is isomorphic to Ω(2) as a numeration system and will be identified with Ω(2).

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201

We generalize this construction. Let A be a nonempty finite set. An element in A is called a color. An open rectangle (x1 , x2 ) × (y1 , y2 ) in H is called an admissible tile if x2 − x1 = y1 (3) is satisfied (see Figure 2). In another word, an admissible tile is a rectangle (x1 , x2 )× (y1 , y2 ) in H such that the lower side has the hyperbolic length 1. Let R be the set of admissible tiles in H. A colored tiling ω is a subset of R × A such that (1) R ∩ R
= ∅ for any (R, a) and (R
, a
) in ω with (R, a) = (R
, a
), and (2) ∪a∈A ∪(R,a)∈ω R = H. An element in R × A is called a colored tile. We denote dom(ω) := {R; (R, a) ∈ ω for some a ∈ A}. For R ∈ dom(ω), there exists a unique a ∈ A such that (R, a) ∈ ω, which is denoted by ω(R) and is called the color of the tile R (in ω). Let R = (x1 , x2 ) × (y1 , y2 ). We call y2 /y1 the vertical size of the tile R which is denoted by S(R). Let Ω(A) be the set of colored tilings with colors in A. A topology is introduced on Ω(A) so that a net {ωn }n∈I ⊂ Ω(A) converges to ω ∈ Ω(A) if for every (R, a) ∈ ω, there exists (Rn , an ) ∈ ωn such that an = a for any sufficiently large n ∈ I and lim ρ(R, Rn ) = 0, n→∞

where ρ is the Hausdorff metric defined in (2). For an admissible tile R := (x1 , x2 ) × (y1 , y2 ), t ∈ R and λ ∈ R+ , we denote R + t := (x1 + t, x2 + t) × (y1 , y2 ) λR := (λx1 , λx2 ) × (λy1 , λy2 ). Note that they are also admissible tiles.

x Fig 2. Admissible tiles.

T. Kamae

202

For ω ∈ Ω(A), t ∈ R and λ ∈ R+ , we define ω + t ∈ Ω(A) and λω ∈ Ω(A) as follows: ω + t = {(R − t, a); (R, a) ∈ ω} λω = {(λR, a); (R, a) ∈ ω}. Thus, we define a continuous group action λω + t of (λ, t) ∈ R+ × R to ω ∈ Ω(A). We construct compact metrizable subspaces of Ω(A) corresponding to weighted substitutions which are numeration systems. Though
A ≥ 2 is assumed in [7], we consider the case
A = 1 as well. 2. Remarks on the notations In this paper, the notations are changed in a large scale from the previous papers [7], [8] and [9] of the author. The main changes are as follows: (1) Here, the colored tilings are defined on the upper half plane H, not on R2 as in the previous papers. The multiplicative action here agree with the multiplication on H, while it agree with the logarithmic version of the multiplication at one coodinate in the previous papers. Here, the tiles are open rectangles, not half open rectangles as in the previous papers. (2) Here, we simplified the proof in [9] for the space of colored tilings coming from weighted substitutions to be numeration systems by omitting the arguments on the topological entropy. (3) The roles of x-axis and y-axis for colored tilings are exchanged here and in [9] from those in [7] and [8]. (4) Here and in [9], the set of colors is denoted by A instead of Σ. Colors are denoted by a, a
, ai (etc.) instead of σ, σ
, σi (etc.). (5) Here and in [9], the weighted substitution is denoted by (σ, τ ) instead of (ϕ, η). (6) Here and in [9], admissible tiles are denoted by R, R
, Ri , Ri (etc.) instead of S, S
, Si , S i (etc.). (7) Here and in [9], the terminology “primitive” for substitutions is used instead of “mixing” in [7] and [8]. 3. Weighted substitutions ∞ A substitution σ on a set A is a mapping A → A+ , where A+ = =1 A . For ξ ∈ A+ , we denote |ξ| :=  if ξ ∈ A , and ξ with |ξ| =  is usually denoted by ξ0 ξ1 · · · ξ−1 with ξi ∈ A. We can extend σ to be a homomorphism A+ → A+ as follows: σ(ξ) := σ(ξ0 )σ(ξ1 ) · · · σ(ξ−1 ), where ξ ∈ A and the right-hand side is the concatenations of σ(ξi )’s. We can define σ 2 , σ 3 , . . . as the compositions of σ : A+ → A+ . A weighted substitution (σ, τ ) on A is a mapping A → A+ × (0, 1)+ such that
|σ(a)| = |τ (a)| and i<|τ (a)| τ (a)i = 1 for any a ∈ A. Note that σ is a substitution on A. We define τ n : A → (0, 1)+ (n = 2, 3, . . .) (depending on σ) inductively by τ n (a)k = τ (a)i τ n−1 (σ(a)i )j for any a ∈ A and i, j, k with 0 ≤ i < |σ(a)|, 0 ≤ j < |σ n−1 (σ(a)i )|, k =

 h
|σ n−1 (σ(a)h )| + j.

Numeration systems as dynamical systems

203

Then, (σ n , τ n ) is also a weighted substitution for n = 2, 3, . . .. A substitution σ on A is called primitive if there exists a positive integer n such that for any a, a
∈ A, σ n (a)i = a
holds for some i with 0 ≤ i < |σ n (a)|. For a weighted substitution (σ, τ ) on A, we always assume that the substitution σ is primitive.

(4)

We define the base set B(σ, τ ) as the closed, multiplicative subgroup of R+ generated by the set  n  τ (a)i ; a ∈ A, n = 0, 1, . . . and 0 ≤ i < |σ n (a)| . such that σ n (a)i = a Example 3.1. Let A = {+, −} and (σ, τ ) be a weighted substitution such that + → (+, 4/9)(−, 1/9)(+, 4/9) − → (−, 4/9)(+, 1/9)(−, 4/9), where we express a weighted substitution (σ, τ ) by a → (σ(a)0 , τ (a)0 )(σ(a)1 , τ (a)1 ) · · · (a ∈ A). Then, 4/9 ∈ B(σ, τ ) since σ(+)0 = + and τ (+)0 = 4/9. Moreover, 1/81 ∈ B(σ, τ ) since σ 2 (+)4 = + and τ 2 (+)4 = 1/81. Since 4/9 and 1/81 do not have a common multiplicative base, we have B(σ, τ ) = R+ . This weighted substitution is discussed in the following sections. The repetition of this weighted substitution starting at + is shown in Figure 3 by colored tiles. The substituted word of a color is represented as the sequence of colors of the connected tiles in below in order from left. The horizontal (additive) sizes of tiles are proportional to the weights and the vertical (multiplicative) sizes are the inverse of the weights. Let G := B(σ, τ ). Then, there exists a function g : A → R+ such that g(σ(a)i )G = g(a)τ (a)i G

(5)

for any a ∈ A and 0 ≤ i < |σ(a)|. Note that if G = R+ , then we can take g ≡ 1. In the other case, we can define g by g(a0 ) = 1 and g(a) := τ n (a0 )i for some n and i such that σ n (a0 )i = a, where a0 is any fixed element in A. Let (σ, τ ) be a weighted substitution satisfying (4). Let G = B(σ, τ ). Let g satisfy (5). Let Ω(σ, τ, g)
be the set of all elements ω in Ω(A) such that for any ((x1 , x2 ) × (y1 , y2 ), a) ∈ ω, we have (I) y1 ∈ g(a)G, and (II) (Ri , σ(a)i ) ∈ ω holds for i = 0, 1, . . . , |σ(a)| − 1, where i

R := (x1 + (x2 − x1 )

i−1  j=0

τ (a)j , x1 + (x2 − x1 )

i 

τ (a)j )

j=0

× (τ (a)i y1 , y1 ). A vertical line γ := {x} × (−∞, ∞) is called a separating line of ω ∈ Ω(σ, τ, g)
if for any (R, a) ∈ ω, R ∩ γ = ∅. Let Ω(σ, τ, g)

be the set of all ω ∈ Ω(σ, τ, g)
which do not have a separating line and Ω(σ, τ, g) be the closure of Ω(σ, τ, g)

. Then, the action of G × R on Ω(σ, τ, g) satisfies (
1). We usually denote Ω(σ, τ, 1) simply by Ω(σ, τ ).

T. Kamae

204

+ :

4/9

:

1/9

4/9

+

+ −

+

+

+

+

− +

+ -

− +

+

+

-

+ -

−

+

+

+ -

− x

Fig 3. The weighted substitution in Example 1.

Remark 3.2 ([7]). A nontrivial primitive substitution σ : A → A+ , where “non
trivial” means a∈A |σ(a)| ≥ 2, is considered as a weighted substitution in a canonical way. Let M := (
{0 ≤ i < |σ(a)|; σ(a)i = a
})a,a
∈A be the associate matrix. Let λ be the maximum eigen-value of M and ξ := (ξa )a∈A be a positive column vector such that M ξ = λξ. Define weight τ by τ (a)i =

ξσ(a)i , λξa

which is called the natural weight of σ. Thus, we get a weighted substitution (σ, τ ) which admits weight 1. We modify (σ, τ ) if necessary in the following way. If there exists a ∈ A with |σ(a)| = 1, so that a → (a
, 1) is a part of (σ, τ ), then we replace all the occurrences of a in the right hand side of “→” by a
and remove a from A together with the rule a → (a
, 1) from (σ, τ ). We continue this process until no a ∈ A satisfies |σ(a)| = 1. After that if there exist a, a
∈ A such that (σ(a), τ (a)) = (σ(a
), τ (a
)), then we identify them. For example, the 2-adic expansion substitution 1 → 12, 2 → 12 corresponds to the weighted substitution 1 → (1, 1/2)(1, 1/2). The Thue-Morse substitution

Numeration systems as dynamical systems

205

1 → 12, 2 → 21 corresponds to the weighted substitution 1 → (1, 1/2)(2, 1/2), 2 → (2, 1/2)(1, 1/2). The Fibonacci substitution 1 → 12, 2 →√1 corresponds to the weighted substitution 1 → (1, λ−1 )(1, λ−2 ), where λ = (1 + 5)/2. The weighted substitution (σ, τ ) obtained in this way satisfies that B(σ, τ ) = {λn ; n ∈ Z} and that g in (5) can be defined by g(a) = ξa (a ∈ A). Dynamical systems coming from substitutions are discussed by many authors (see [2], for example). Our weighted substitutions are a generalization of them. Let (σ, τ ) be a weighted substitution on A satisfying (4). Let g satisfy (5). Consider Ω(σ, τ, g). We call the tile Ri in (II) the i-th child of the tile (x1 , x2 ) × (y1 , y2 ), and (x1 , x2 ) × (y1 , y2 ) the mother of Ri . Note that the vertical size S(Ri ) of Ri coincides with the inverse of the weight τ (a)i . If Rj is a child of Rj+1 for j = 0, 1, . . . , k − 1. Then, the tile R0 is called a k-th descendant of the tile Rk . If R0 is the i-th tile among the set of the k-th descendants of Rk counting as 0, 1, 2, · · · from left, we call R0 the (k, i)-descendant of the tile Rk . In this case, we also say that Rk is the k-th ancestor of R0 . Theorem 3.3. The space Ω(σ, τ, g) is a numeration system with G = B(σ, τ ). Proof. We have already proved (
1) and (
2) in Theorem 3 of [7]. Here we prove (
3). Let Ω := Ω(σ, τ, g) and µΩ be the equilibrium measure. Since µΩ is the unique invariant probability measure under the additive action, it is also invariant under the multiplicative action. By Goodman [4], it is sufficient to prove that for any λ ∈ G with λ = 1, the transformation ω → λω on Ω has the metrical entropy | log λ| under µΩ , while it has the metrical entropy less than | log λ| under any other G-invariant probability measure. Lemma 3.4. Let Σ := {ω ∈ Ω; ω has a separating line} Σ0 := {ω ∈ Ω; y-axis is the separating line of ω}. Then, we have (i) Σ \ Σ0 is dissipative with respect to the G-action, so that ν(Σ \ Σ0 ) = 0 for any G-invariant probability measure ν on Ω. (ii) For any ω ∈ Σ0 , ω restricted to the right quarter plane (0, ∞) × (0, ∞) and to the left quarter plane (−∞, 0) × (0, ∞) are cyclic individually with respect to the G-action. Hence, Gω with respect to the G-action is either cyclic or conjugate to a 2-dimensional irrational rotation with a multiplicative time parameter. (iii) Σ0 is a finite union of minimal and equicontinuous sets with respect to the G-action. In fact, there is a mapping from the set of pairs a ∈ A and i with 0 ≤ i < i + 1 < |σ(a)| onto the set of minimal sets in Σ0 . Proof. (i) If the line x = u is the separating line of ω ∈ Ω, then x = λu is the separating line of λω. Hence, Σ \ Σ0 is dissipative. (ii) Let ω ∈ Σ0 . Denote by ω + the restriction of ω to the right quarter plane (0, ∞) × (0, ∞), while by ω − the restriction of ω to the left quarter plane (−∞, 0) × (0, ∞). Let (Ri± )i∈Z be the sequence of tiles in dom(ω) such that Ri± intersects ± with the upper half lines of x = ±0, and Ri± is a child of Ri+1 for any i ∈ Z ± ± ± (± respectively). Let ai := ω(Ri ) be the colors of Ri (± respectively). Define mappings σ± from A to A by σ+ (a) = σ(a)0 and σ− (a) = σ(a)|σ(a)|−1 . Since ± ± σ± (a± i ) = ai−1 (i ∈ Z) (± respectively), the sequence (ai )i∈Z is periodic, which

206

T. Kamae

also implies that the vertical sizes S(Ri± ) of Ri± , which coincide with the inverses of the weights τ (ai+1 )± , are also periodic in i ∈ Z with the period, say r± which + −1 + is is the minimum period of (a± := τ r (a+ 0 )0 i )i∈Z (± respectively). Then, λ + − r − − −1 the minimum (multiplicative) cycle of ω , while λ := τ (a0 )|σr− (a− )|−1 is the 0

minimum (multiplicative) cycle of ω − , that is, λω ± = ω ± holds for λ = λ± and λ± is the minimum among λ > 1 with this property (± respectively). Therefore, ω is cyclic with respect to the G-action if λ+ and λ− have a common multiplicative base. In this case, the minimum cycle of ω is the minimum positive number x such that x = (λ+ )n = (λ− )m holds for some positive integers n, m. Otherwise, the G-action to Gω is conjugate to an 2-dimensional irrational rotation with a multiplicative time parameter. (iii) We use the notations in the proof of (ii). Take any pair (a, i) with a ∈ A and 0 ≤ i < i + 1 < |σ(a)|. Take any ω
∈ Ω having a tile R ∈ dom(ω
) with ω
(R) = a such that the y-axis passes in between the i-th child of R and the i + 1-th child of R. Let ψ(a, i) be the set of limit points of λω
as λ ∈ G tends to ∞. Note that this does not depend on the choice of ω
. Then, ψ(a, i) is a closed G-invariant subset n n of Σ0 . Moreover, since the sequence (σ− (σ(a)i ), σ+ (σ(a)i+1 ))n=0,1,2,... enter into a cycle after some time, ψ(a, i) is minimal and equicontinuous with respect to the G-action. To prove that the mapping ψ is onto, take any ω ∈ Σ0 . There exists ωn ∈ Ω(σ, τ, g)

which converges to ω as n → ∞. We may assume that there exists a pair (a, i) such that for any n = 1, 2, · · · , there exists R ∈ dom(ωn ) with a = ωn (R) such that the y-axis separetes the i-th child of R and the i + 1-th child of R. Then, ω ∈ ψ(a, i), which proves that ψ is a mapping from the set of pairs (a, i) with a ∈ A and 0 ≤ i < i + 1 < |σ(a)| onto the set of minimal sets in Σ0 with respect to the G-action. Example 3.5. Let p with 0 < p < 1 satisfy that log p/ log(1 − p) is irrational. Let (σ, τ ) be a weighted substitution on A = {1} such that 1 → (1, p)(1, 1 − p). Then, B(σ, τ ) = R+ holds. Let Ω = Ω(σ, τ ). In this case, elements in Σ0 are not periodic, but almost periodic as shown in Figure 4. Then, the dynamical system (Σ0 , λ (λ ∈ R+ )) is isomorphic to ((R/Z)2 , Tλ (λ ∈ R+ )) with Tλ (x, y) = (x + log λ/ log(1/p), y + log λ/ log(1/(1 − p))). Lemma 3.6. It holds that hµΩ (λ) = | log λ| for any λ ∈ G. Let λ = 1 and ν be any other λ-invariant probability measure on Ω, then hν (λ) < | log λ|. Proof. To prove the lemma, it is sufficient to prove the statements for λ > 1. Take any G-invariant probability measure ν on Ω which attains the topological entropy of the multiplication by λ1 ∈ G with λ1 > 1, that is, hν (λ1 ) = log λ1 . We assume also that the G-action to Ω is ergodic with respect to ν. Then by Lemma 3.4, either ν(Σ0 ) = 1 or ν(Ω \ Σ) = 1. In the former case, hν (λ) = 0 holds for any λ ∈ G since the G-action on Σ0 is equicontinuous by Lemma 3.4, which contradicts with the assumption. Thus, we have ν(Ω \ Σ) = 1. For ω ∈ Ω, let R0 (ω) ∈ dom(ω) be such that R0 (ω) = (x1 , x2 ) × (y1 , y2 ) with x1 ≤ 0 < x2 and y1 ≤ 1 < y2 . Take a0 ∈ A such that ν({ω ∈ Ω; ω(R0 (ω)) = a0 }) > 0.

Numeration systems as dynamical systems

207

y

vertical sizes 1/(1 − p)

vertical sizes 1/p

x Fig 4. An element in Σ0 in Example 2.

Take b0 := max{b ≤ 1; b ∈ g(a0 )G} (see (5)). Let Ω1 := { ω ∈ Ω; the set {λ ∈ G; λω(R0 (λω)) = a0 } is unbounded at 0 and ∞ simultaneously } Ω0 := { ω ∈ Ω1 ; R0 (ω) = (x1 , x2 ) × (y1 , y2 ) with y1 = b0 and ω(R0 (ω)) = a0 }. For ω ∈ Ω0 , let λ0 (ω) be the smallest λ ∈ G with λ > 1 such that λω ∈ Ω0 . Define a mapping Λ : Ω0 → Ω0 by Λ(ω) := λ0 (ω)ω. For k = 0, 1, 2, · · · and i = 0, 1, · · · , |σ k (a0 )| − 1, let P (k, i) := { ω ∈ Ω0 ; λ0 (ω)−1 R0 (λ0 (ω)ω) is the (k, i)-descendant of R0 (ω)} (see Figure 5) and let P := { P (k, i); k = 1, 2, · · · , 0 ≤ i < |σ k (a0 )| } be a measurable partition of Ω0 . Note that λ0 (ω) = τ k (a0 )−1 if ω ∈ P (k, i). i Since ν(Ω1 ) = 1 by the ergodicity and   Ω1 = λP (k, i), P (k,i)∈P

1≤λ<τ k (a0 )

−1 i

λ∈G

there exists a unique Λ-invariant probability measure ν0 on Ω0 such that for any Borel set B ⊂ Ω, we have −1

ν(B) = C(ν)



P (k,i)∈P



b0 τ k (a0 )−1 i

b0

ν0 (λ−1 B ∩ P (k, i))dλ/λ

T. Kamae y

208

a0 b0



a0

x Fig 5. ω ∈ P (3, 4) with λ0 (ω) = 13/3

with



C(ν) :=

− log τ k (a0 )i ν0 (P (k, i)) < ∞

(6)

P (k,i)∈P

if G = R+ and ν(B) = C(ν)−1



P (k,i)∈P

with



C(ν) :=



ν0 (λ−1 B ∩ P (k, i))

λ∈G −1 b0 ≤λ
(− log τ k (a0 )i / log β) ν0 (P (k, i)) < ∞

(7)

P (k,i)∈P

if G = {β n ; n ∈ Z} with β > 1. Since  τ k (a0 )i = 1 and P (k,i)∈P



ν0 (P (k, i)) = 1,

P (k,i)∈P

we have Hν0 (P) := −



log ν0 (P (k, i)) · ν0 (P (k, i))



log τ k (a0 )i · ν0 (P (k, i))

P (k,i)∈P

≤ −

(8)

P (k,i)∈P

by the convexity of − log x. The equality in (8) holds if and only if ν0 (P (k, i)) = τ k (a0 )i (∀P (k, i) ∈ P).

(9)

Numeration systems as dynamical systems

209

By (6), (7), and (8), we have Hν0 (P) = −



log ν0 (P (k, i)) · ν0 (P (k, i)) < ∞.

P (k,i)∈P

For any ω, ω
∈ Ω0 such that Λk (ω) and Λk (ω
) belong to the same element in P for k = 0, 1, 2, · · · , the horizontal position of R0 (ω), say (x1 , x2 ), coincides with that of R0 (ω
). Therefore, ω and ω
restricted to (x1 , x2 ) × (0, b0 ) coincide. In the same way, if Λk (ω) and Λk (ω
) belong to the same element in P for any k ∈ Z, then R0 := R0 (ω) = R0 (ω
) holds and all the ancestors of R0 in ω and ω
coincide as well as their colors. Therefore, ω and ω
restricted to the region covered by the ancestors of R0 coincide. Hence, if ω or ω
does not have the separating lines, then ω = ω
holds. Since ν(Σ) = 0, we have ν0 (Σ ∩ Ω0 ) = 0. Hence, the above argument implies that P is a generator of the system (Ω0 , ν, Λ). Thus, hν0 (Λ) = hν0 (Λ, P). It follows from (8) that hν0 (Λ) = hν0 (Λ, P) ≤ Hν0 (P)  ≤− log τ k (a0 )i · ν0 (P (k, i)).

(10)

P (k,i)∈P

The equality in the above that 

hν0 (Λ) = −

log τ k (a0 )i · ν0 (P (k, i))

P (k,i)∈P

holds if and only if (Λn P)n∈Z is an independent sequence with respect to ν0 satisfying (9). Since hν (λ1 )/ log λ1 =  =

hν0 (Λ) λ (ω)dν0 (ω) Ω0 0

−




hν0 (Λ) , k P (k,i)∈P log τ (a0 )i · ν0 (P (k, i))

hν (λ1 ) ≤ log λ1 follows from (10), while the equality holds if and only if (Λn P)n∈Z is an independent sequence with respect to ν0 satisfying (9). Let this probability measure be µ. Then, it is not difficult to prove that µ is invariant under the additive action. Hence, the uniqueness of such measure ([7]) proves µ = µΩ , which completes the proof of Lemma 3.6 and Theorem 3.3. The following Theorem 3.7 follows from a known result about the spectrum of unitary operators corresponding to the affine action (Lemma 11.6 of [13], for example). Theorem 3.7 ([10]). Let Ω be a numeration system with G = R+ , that is, with the multiplicative R+ -action. Then, the additive action on the probability space Ω with respect to µΩ has a pure Lebesgue spectrum.

T. Kamae

210

4. ζ-function Here, we listed only the results on the ζ-functions. For the proof, refer [10]. Let Ω := Ω(σ, τ, g) satisfying (4) and (5). For α ∈ C, we define the associated matrices on the suffix set A × A as follows:     (11) τ (a)α Mα :=  i i;σ(a)i =a


a,a
∈A

 τ (a)α 0 a,a
∈A

Mα,+ := 1σ(a)0 =a
  Mα,− := 1σ(a)|σ(a)|−1 =a
τ (a)α |σ(a)|−1

a,a
∈A

Let Θ be the set of closed orbits of Ω with respect to the action of G. That is, Θ is the family of subsets ξ of Ω such that ξ = Gω for some ω ∈ Ω with λω = ω for some λ ∈ G with λ > 1. We call λ as above a multiplicative cycle of ξ. The minimum multiplicative cycle of ξ is denoted by c(ξ). Note that c(ξ) exists since λω = ω for any ω ∈ Ω and λ ∈ G with 1 < λ < min{τ (a)−1 i ; a ∈ A, 0 ≤ i < |τ (a)|}. We say that ξ ∈ Θ has a separating line if ω ∈ ξ has a separating line. Note that in this case, the separating line is necessarily the y-axis and is in common among ω ∈ ξ. Denote by Θ0 the set of ξ ∈ Θ with the separating line. Define the ζ-function of G-action to Ω by  (1 − c(ξ)−α )−1 , (12) ζΩ (α) := ξ∈Θ

where the infinite product converges for any α ∈ C with R(α) > 1. It is extended to the whole complex plane by the analytic extension. Theorem 4.1. We have ζΩ (α) =

det(I − Mα,+ ) det(I − Mα,− ) ζΣ0 (α), det(I − Mα )

where ζΣ0 (α) :=



(1 − c(ξ)−α )−1

ξ∈Θ0

is a finite product with respect to ξ ∈ Θ0 . Theorem 4.2. (i) ζΩ (α) = 0 if R(α) = 0. (ii) In the region R(α) = 0, α is a pole of ζΩ (α) with multiplicity k if and only if it is a zero of det(I − Mα ) with multiplicity k for any k = 1, 2, . . .. (iii) 1 is a simple pole of ζΩ (α). Theorem 4.3. For Ω = Ω(σ, η, g), if B(σ, τ ) = {λn ; n ∈ Z} with λ > 1, then there exist polynomials p, q ∈ Z[z] such that ζΩ (α) = p(λα )/q(λα ). Conversely, if ζΩ (α) = p(λα )/q(λα ) holds for some polynomials p, q ∈ Z[z] and λ > 1, then B(σ, τ ) = {λkn ; n ∈ Z} for some positive integer k. Theorem 4.4. If B(σ, τ ) = {λn ; n ∈ Z}, then λ is an algebraic number. Acknowledgment The author thanks his old friend, Prof. Mike Keane for his useful discussions and encouragements to develop this research for more than 10 years.

Numeration systems as dynamical systems

211

References [1] Arnoux, P. and Ito, S. (2001). Pisot substitutions and Rauzy fractals. Bull. Belg. Math. Soc. Simon Stevin 8, 2, 181–207. MR1838930 [2] Fogg, N. P. (2002). Substitutions in Dynamics, Arithmetics and Combinatorics. Lecture Notes in Mathematics, Vol. 1794. Springer-Verlag, Berlin. MR1970385 [3] Dunford, N. and Schwartz, J.T. (1963). Linear Operators II, Interscience Publishers John Wiley & Sons, New York–London. MR0188745 [4] Goodman, T. N. T. (1971). Relating topological entropy and measure entropy. Bull. London Math. Soc. 3, 176–180. MR289746 [5] Gjini, N. and Kamae, T. (1999). Coboundary on colored tiling space as Rauzy fractal. Indag. Math. (N.S.) 10, 3, 407–421. MR1819898 [6] Ito, S. and Takahashi, Y. (1974). Markov subshifts and realization of βexpansions. J. Math. Soc. Japan 26, 33–55. MR346134 [7] Kamae, T. (1998). Linear expansions, strictly ergodic homogeneous cocycles and fractals. Israel J. Math. 106, 313–337. MR1656897 [8] Kamae, T. (2001). Stochastic analysis based on deterministic Brownian motion. Israel J. Math. 125, 317–346. MR1853816 [9] Kamae, T. (2006). Numeration systems, fractals and stochastic processes, Israel J. Math. (to appear). [10] Kamae, T. (2005) Numeration systems as dynamical systems. Preprint, available at http://www14.plala.or.jp/kamae. [11] Petersen, K. (1983). Ergodic Theory. Cambridge Studies in Advanced Mathematics, Vol. 2. Cambridge University Press, Cambridge. MR833286 [12] Plessner, A. (1941). Spectral theory of linear operators I, Uspekhi Matem. Nauk 9, 3-125. MR0005798 [13] Starkov, A. N. (2000). Dynamical Systems on Homogeneous Spaces, Translations of Mathematical Monographs 190, Amer. Math. Soc. MR1746847 [14] Walters, P. (1975). Ergodic Theory − Introductory Lectures, Lecture Notes in Mathematics 458. Springer-Verlag, Berlin-New York. MR0480949

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 212–224 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000239

Hyperelliptic curves, continued fractions, and Somos sequences Alfred J. van der Poorten1,∗ Centre for Number Theory Research, Sydney Abstract: We detail the continued fraction expansion of the square root of a monic polynomials of even degree. We note that each step of the expansion corresponds to addition of the divisor at infinity, and interpret the data yielded by the general expansion. In the quartic and sextic cases we observe explicitly that the parameters appearing in the continued fraction expansion yield integer sequences defined by bilinear relations instancing sequences of Somos type.

The sequence . . . , 3, 2, 1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, . . . is produced by the recursive definition Bh+3 = (Bh−1 Bh+2 + Bh Bh+1 )/Bh−2

(1)

and consists entirely of integers. On this matter Don Zagier comments [26] that ‘the proof comes from the theory of elliptic curves, and can be expressed either in terms of the denominators of the co-ordinates of the multiples of a particular point on a particular elliptic curve, or in terms of special values of certain Jacobi theta functions.’ Below, I detail the continued fraction expansion of the square root of a monic polynomials of even degree. In the quartic and sextic cases I observe explicitly that the parameters appearing in the expansion yield integer sequences defined by relations including and generalising that of the example (1). However it is well known, see for example Adams and Razar [1], that each step of the continued fraction expansion corresponds to addition of the divisor at infinity on the relevant elliptic or hyperelliptic curve; that readily explains Zagier’s explanation. 1. Some Brief Reminders 1.1. The numerical case We need little more than the following. Suppose ω is a quadratic irrational integer ω , defined by ω 2 − tω + n = 0, and greater than the other root ω of its defining equation. Denote by a the integer part of ω and set P0 = a − t, Q0 = 1. Then 1 Centre

for Number Theory Research, 1 Bimbil Place, Killara, Sydney, NSW 2071, Australia, e-mail: [email protected] ∗ This survey was written at Brown University, Providence, Rhode Island where the author held the position of Mathematics Distinguished Visiting Professor, Spring semester, 2005. The author was also supported by his wife and by a grant from the Australian Research Council. AMS 2000 subject classifications: primary 11A55, 11G05; secondary 14H05, 14H52. Keywords and phrases: continued fraction expansion, function field of characteristic zero, hyperelliptic curve, Somos sequence.

212

Somos sequences

213

the continued fraction expansion of ω0 := (ω + P0 )/Q0 is a two-sided sequence of lines, h in Z, ω + Ph+1 ω + Ph = ah − ; Qh Qh

in brief ωh = ah − ρh ,

with (ω + Ph+1 )(ω + Ph+1 ) = −Qh Qh+1 defining the integer sequences (Ph ) and (Qh ). Obviously Q0 divides (ω + P0 )(ω + P0 ). This suffices to ensure that the integrality of the sequence (ah ) of partial quotients guarantees that always Qh divides the norm n + tPh + Ph2 = (ω + Ph )(ω + Ph ). Comment 1. Consider the Z-module ih =
Qh , Ph + ω. It is a pleasant exercise to confirm that ih is an ideal of the domain Z[ω] if and only if Qh does indeed divide the norm (ω + Ph )(ω + Ph ) of its numerator. One says that a real quadratic irrational ωh is reduced if ωh > 1 and its conjugate ω h asatisfies −1 < ω h < 0. If the partial quotient ah is always chosen as the integer part of ωh then ω0 reduced entails all the ωh and ρh are reduced; and, remarkably, ah — which starts life as the integer part of ωh — always also is the integer part of ρh . Then conjugation of the continued fraction tableau retrieves the negative half of the expansion of ω0 from the continued fraction expansion of ρ0 . Comment 2. What a continued fraction expansion does. Suppose α = [ a0 , a1 , a2 , . . . ] with α > 1, is our very favourite expansion, so much so that sometimes we go quite alpha — expanding arbitrary complex numbers β = β0 by the ‘alpha’ rule βh = ah + (βh − ah )

and (βh − ah )−1 = βh+1 = ah+1 + · · · etc.

What can one say about those ‘alpha d’ complete quotients βh ? Quite a while ago, in 1836, Vincent reports that either (i) all βh > 1, in which case β = α ; or (ii) for all sufficiently large h, |βh | < 1 and the real part βh of βh satisfies −1 < βh < 0; in other words, all those βh lie in the left hand half of the unit circle. Proof : Straightforward exercise; or see [4] or, better, the survey [2]. Of this result Uspensky [24] writes: ‘This remarkable theorem was published by Vincent in 1836 in the first issue of Liouville’s Journal, but later [was] so completely forgotten that no mention of it is found even in such a capital work as the Enzyclop¨ adie der mathematischen Wissenschaften. Yet Vincent’s theorem is the basis of the very efficient method for separating real roots . . . ’. The bottom line is this: When we expand a real quadratic irrational α then, willy-nilly, by conjugation we also expand α . By Vincent’s theorem, its complete quotients eventually arrive in the left hand-half of the unit circle and, once ‘reduced’, they stay that way. One readily confirms that the integers Ph and Qh are bounded by 0 < 2Ph + t < ω − ω

and

0 < Qh < ω − ω .

It follows by the box principle that the continued fraction expansion of ω is periodic. More, the adjustment whereby we replace ω by ω0 = ω + a − t arranges that ω0 is reduced. Yet more, by conjugating the tableau one sees immediately that (an observation credited to Galois) for any h the expansion of ωh is purely periodic. Comment 3. On conjugating the tableau, a putative preperiod becomes a ‘postperiod’ in the expansion of ρh ; which is absurd.

A. J. van der Poorten

214

1.2. The function field case Here, ‘polynomial’ replaces ‘integer’. Specifically, set Y 2 = D(X) where D is a monic polynomial of even degree deg D = 2g + 2 and defined over the base field, K say. Then we may write D(X) = A(X)2 + 4R(X)

(2)

where A is the polynomial part of the square root Y of D , so deg A = g + 1, and the remainder R satisfies deg R ≤ g . It is in fact appropriate to study the expansion of Z := 21 (Y + A). Plainly C : Z 2 − AZ − R = 0 with deg Z = g + 1 and deg Z < 0.

(3)

Comment 4. Note that Y is given by a Laurent series A + d−1 X −1 + d−2 X −2 + · · · , an element of K((X −1 )); in effect we expand around infinity. Felicitously, by restricting our attention to Z , and forgetting our opening remarks, the story we tell below makes sense over all base fields of arbitrary characteristic, including characteristic two. However, for convenience, below we mostly speak as if K = Q. In the present context we study the continued fraction expansion of an element Z0 of K(Z, X) leading to the expansion consisting of a tableau of lines, h ∈ Z, Z + Ph Z + Ph+1 = ah − , Qh Qh+1

in brief Zh = ah − Rh , say,

(4)

initiated by the conditions deg P0 < g , deg Q0 ≤ g and Q0 divides the norm (Z + P0 )(Z + P0 ) = −R + P0 (A + P0 ). Indeed, the story is mutatis mautandis precisely as in the numerical case, up to the fact that a function of K(Z, X) is reduced exactly when it has positive degree but its conjugate has negative degree. Here, analogously, we find that therefore all the Ph and Qh satisfy deg Ph < g

and

deg Qh ≤ g ,

(5)

the conditions equivalent to the Zh and Rh all being reduced. 1.2.1. Quasi-periodicity If the base field K is infinite then the box principle does not entail periodicity. In a detailed reminder exposition on continued fractions in quadratic function fields at Section 4 of [17], we are reminded that periodicity entails the existence of a non-trivial unit, of degree m say, in K[Z, X] . Conversely however, the exceptional existence of such a unit implies only ‘quasi-periodicity’ — in effect, periodicity ‘twisted’ by multiplication of the period by a nonzero element of K. The existence of an exceptional unit entails the divisor at infinity on the curve C being torsion of order dividing m. If quasi-periodic, the expansion of the reduced element Z0 = (Z + P0 )/Q0 is purely quasi-periodic. Comment 5. Consider the surprising integral
6x dx √ 4 3 x + 4x − 6x2 + 4x + 1  = log x6 + 12x5 + 45x4 + 44x3 − 33x2 + 43   + (x4 + 10x3 + 30x2 + 22x − 11) x4 + 4x3 − 6x2 + 4x + 1 ,

Somos sequences

a nice example of a class of pseudo-elliptic integrals
   f (x)dx  = log a(x) + b(x) D(x) . D(x)

215

(6)

Here we take D to be a monic polynomial defined over Q, of even degree 2g + 2, and not the square of a polynomial; f , a, and b denote appropriate polynomials. We suppose a to be nonzero, say of degree m at least g + 1. One sees readily that necessarily deg b = m − g − 1, that deg f = g , and that f has leading coefficient m. In the example, m = 6 and g = 1. 2 2 The  is a non-zero constant and √ trick is to recognise that obviously a − b D a + b D is a unit of degree m in the domain Q(x, D(x) and is not necessarily of norm ±1 — it is this that corresponds to quasi -periodicity; for details see [14].

1.2.2. Normaility of the expansion In the sequel, I suppose that Z0 has been so chosen that its continued fraction expansion is normal : namely, all its partial quotients are of degree 1. This is the generic case if K is infinite. Since I have the case K = Q in mind, I refer to elements of K as ‘rational numbers’. Comment 6. Na¨ıvely, it is not quite obvious that the case all partial quotients of degree one is generic, let alone that this generic situation can be freely arranged in our partcular situation. I comment on the latter matter immediately below but the former point is this: when one inverts d−1 X −1 + d−2 X −2 + · · · , obtaining a polynomial plus e−1 X −1 + e−2 X −2 + · · · it is highly improbable that e−1 vanishes because e−1 is a somewhat complicated rational function of several of the d−i . See my remarks in [15]. 1.2.3. Ideal classes If K is infinite, choosing P0 and Q0 is a matter of selecting one of infinitely many different ideal classes of Z-modules {
Qh , Z + Ph  : h ∈ Z}. Only a thin subset of such classes fails to give rise to a normal continued fraction expansion. Of course our choice of P0 and Q0 will certainly avoid the blatantly singular principal class: containing the ideal
1, Z. 1.2.4. Addition on the Jacobian Here’s what the continued fraction expansion does. The set of zeros {ωh,1 , . . . , ωh,g } of Qh defines a rational divisor on the hyperelliptic curve C . In plain language: any one of these zeros ωh is the X co-ordinate of a point on C , here viewing C as defined over the algebraic extension K(ωh ), generically of degree g over the base field K. In particular, in the case g = 1, when C is an elliptic curve, the unique zero wh of Qh provides a point on C defined over the base field; see §2.2 at page 217 below. Selecting P0 and Q0 is to choose a divisor class, say M , thus a point on the Jacobian Jac(C) of the curve C defining Z . Let S denote the divisor class defined by the divisor at infinity. Then each complete quotient Zh = (Z + Ph )/Qh has divisor in the class Mh+1 := M + hS . I show this in [17] for g = 1 making explicit

A. J. van der Poorten

216

remarks of Adams and Razar [1]. See further comment at §2.2. For higher genus cases, one wilkl find helpful the introduction to David Cantor’s paper [7] and the instructive discussion by Kristin Lauter in [13]. A central theme of the paper [3] is a generalisation of the phenomenon to Pad´e approximation in arbitrary algebraic function fields. My suggestion that partial quotients of degree one are generic in our examples is the same remark as that divisors on C/K typically are given by g points on C/F, where F is some algebraic extension of K. 2. The Continued Fraction Expansion Evidently, the polynomials Ph and Qh in Z + Ph Z + Ph+1 = ah − , Qh Qh+1

(7)

are given sequentially by the formulas Ph + Ph+1 + A = ah Qh

and

−Qh Qh+1 = (Z + Ph+1 )(Z + Ph+1 ) = −R + Ph+1 (A + Ph+1 ).

(8)

2.1. A na¨ıve approach At first glance one might well be tempted to use this data by spelling out the first recursion in terms of g equations linearly relating the coeficients of the Ph , and the second recursion in terms of 2g + 1 equations quadratically relating the coefficients of the Qh . Even for g = 1, doing that leads to a fairly complicated analysis; see [16], and the rather less clumsy [17]. For g = 2, I had to suffice myself with the special case deg R = 1 so that several miracles (I called it ‘a ridiculous computation’) could yield a satisfying result [18]. I realised that a quite different view of the problem would be needed to say anything useful in higher genus cases. 2.2. Less explicit use of the recursions Denote by ωh a zero of Qh (X). Then Ph (ωh ) + Ph+1 (ωh ) + A(ωh ) = 0   so − Qh Qh+1 = −R + Ph+1 A + Ph+1 becomes R(ωh ) = −Ph+1 (ωh )Ph (ωh ).

(9)

This, together with a cute trick, already suffices to tame the g = 1 case: that is, the case beginning as the square root of a monic quartic polynomial. Indeed, if g = 1 then deg Ph = g − 1; so the Ph are constants, say Ph (X) =: eh . Also, deg Qh = 1, say Qh (X) =: vh (X − wh ), so ωh = wh . Further, deg R ≤ 1, say R(X) =: v(X − w). First, (9) tells us directly that v(w − wh ) = eh eh+1 . Second, this is the ‘cute trick’, obviously −R(wh ) = v(w − wh ) = Qh (w) · v/vh . An intelligent glance at (8) reminds us that vh−1 vh = −eh . Hence, by (8) and R(w) = 0,   eh−1 e2h eh+1 = −Qh−1 (w)Qh (w) · v 2 /eh = v 2 eh + A(w) .

Somos sequences

217

The exciting thing about the recursion   eh−1 e2h eh+1 = v 2 eh + A(w)

(10)

E : V 2 − vV = monic cubic in U with zero constant coefficient,

(11)

is that, among the parameters varying with h, it involves the eh alone; moreover, its coefficients v 2 and v 2 A(w) depend only on the curve C and not on the initial conditions P0 = e0 and Q0 = v0 (X − w0 ). More, if X = wh then Z = −eh or Z = −eh+1 ; yielding pairs of rational points on C . Moreover, the transformation U = Z , V − v = XZ transforms the curve C to a familiar cubic model

essentially by moving one of the two points SC , say, at infinity on C to the origin SE = (0, 0) on E . As at §1.2.4 on page 215 above, denote the point (wh , −eh ) on C by Mh+1 . Recall that an elliptic curve is an abelian group with group operation denoted by +. Set M1 = M . One confirms, see [17] for details, that Mh+1 = M + hSC by seeing that this plainly holds on E where the addition law is that F + G + H = 0 if the three points F , G, H on E lie on a straight line. I assert at §1.2.4 that precisely this property holds also in higher genus. There, however, one is forced to use the gobbledegook language of ‘divisor classes on the Jacobian of the curve’ in place of the innocent ‘points on the curve’ allowed in the elliptic case. Comment 7. By the way, because each rational number −eh is the U co-ordinate of a rational point on E it follows that its denominator must be the square of an integer. 2.3. More surely useful formulas Set ah (X) = (X + νh )/uh ; so uh is the leading coefficient of Qh (its coefficient of X g ) and ah vanishes at −νh . Below, we presume that dh denotes the leading coefficient of Ph (its coefficient of X g−1 ). Then R(X) + Ph+1 (X)Ph (X) = Qh (X)Qh+1 (X) + Ph+1 (X)(X + νh )Qh (X)/uh = Qh−1 (X)Qh (X) + Ph (X)(X + νh )Qh (X)/uh . Plainly, we should divide by Qh (X)/uh and may set     Ch (X) := R(X) + Ph+1 (X)Ph (X) / Qh (X)/uh = uh Qh+1 (X) + Ph+1 (X)(X + νh ) = uh Qh−1 (X) + Ph (X)(X + νh ).

(12)

Since deg R ≤ g , and the P all have degree g − 1, and the Q degree g , it follows that the polynomial C has degree g − 2 if g ≥ 2, or is constant in the case g = 1. If g = 2 it of course also is constant and then, with R(X) = u(X 2 − vX + w), its leading coefficient is dh dh+1 + u so we have, identically, Ch (X) = dh dh+1 + u . If g ≥ 3 then Ch is a polynomial with leading coefficient dh dh+1 . 2.4. The case g = 2 If deg A = 3 then R = u(X 2 − vX + w) is the most general remainder. In the continued fraction expansion we may take Ph (X) = dh (X + eh ). As just above,

A. J. van der Poorten

218

denote by uh the leading coefficient of Qh (X) and note that uh−1 uh = −dh . In the case g = 1 we succeeded in finding an identity in just the parameters eh ; here we seek an identity just in the dh . First we note, using (12) and Ch (X) = dh dh+1 + u, that dh dh+1 + u = Ch (−eh ) = uh R(−eh )/Qh (−eh ) = uh Qh−1 (−eh ) . dh−1 dh + u = Ch−1 (−eh ) = uh−1 R(−eh )/Qh−1 (−eh ) = uh−1 Qh (−eh ) .

(13) (14)

Hence, cutely, (dh−1 dh + u)(dh dh+1 + u) = uh−1 uh R(−eh ) = −udh (e2h + veh + w) .

(15)

Set R(X) =: u(X − ω)(X − ω). Also from (12), we have Ch (ω)Qh (ω) = Ph+1 (ω)Ph (ω) and therefore  2 Ch−1 (ω)Ch (ω)Qh−1 (ω)Qh (ω) = Ph−1 (ω) Ph (ω) Ph+1 (ω) .   But −Qh−1 (ω)Qh (ω) = Ph (ω) A(ω) + Ph (ω) . Together with (15) we find that    u3 A(ω) + dh (ω + eh ) A(ω) + dh (ω + eh ) = dh−1 d3h dh+1 (dh−2 dh−1 + u)(dh+1 dh+2 + u).

(16)

In the special case u = 0, that is: R(X) = −v(X − w), a little less argument yields the more amenable   −v 3 A(ω) + dh (w + eh ) = dh−2 d2h−1 d3h d2h+1 dh+2 .

In this case, we have −vdh (w + eh ) = −dh R(−eh ) = dh−1 d2h dh+1 , finally providing dh−2 d2h−1 d3h d2h+1 dh+2 = v 2 dh−1 d2h dh+1 − v 3 A(w) .

(17)

Comment 8. This reasonably straightforward argument removes the ‘miraculous’ aspects from the corresponding discussion in [18]. Moreover, it gives a result for u = 0. However, I do not yet see how to remove the dependence on eh in (16) so as to obtain a polynomial relation in the ds. 3. Somos Sequences The complexity of the various parameters in the continued fraction expansions increases at frantic pace with h. For instance the logarithm of the denominators of eh of the elliptic case at §2.2 is readily proved to be O(h2 ) and the same must therefore hold for the logarithmic height of each of the parameters. Denote by A2h the denominator of eh (recall the remark at Comment 7 on page 217 that these denominators in fact are squares of integers). It is a remarkable fact holding for the co-ordinates of multiples of a point on an elliptic curve that in general (18) Ah−1 Ah+1 = eh A2h Moreover, it is a simple exercise to see that (18) entails Ah−2 Ah+2 = eh−1 e2h eh+1 A2h .  Thus, on multiplying the identity (10), namely eh−1 e2h eh+1 = v 2 eh + A(w) , by A2h we find that (19) Ah−2 Ah+2 = v 2 Ah−1 Ah+1 + v 2 A(w)A2h gives a quadratic recursion for the ‘denominators’ Ah .

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Comment 9. My remark concerning the co-ordinates of multiples of a point on an elliptic curve is made explicit in Rachel Shipsey’s thesis [21]. The fact this also holds after a translation is shown by Christine Swart [23] and is in effect again proved here by way of the recursion (10). My weaselling ‘in general’ is to avoid my having to chat on about exceptional primes — made evident by the equation defining the elliptic curve — at which the Ah may not be integral at all. In other words, in true generality, there may be a finite set S of primes so that the Ah actually are just S -integers: that is, they may have denominators but primes dividing such denominators must belong to S . 3.1. Michael Somos’s sequences Some fifteen years ago, Michael Somos noticed [11, 20], that the two-sided sequence Ch−2 Ch+2 = Ch−1 Ch+1 +Ch2 , which I refer to as 4-Somos in his honour, apparently takes only integer values if we start from Ch−1 = Ch = Ch+1 = Ch+2 = 1. Indeed Somos went on to investigate also the width 5 sequence, Bh−2 Bh+3 = Bh−1 Bh+2 + Bh Bh+1 , now with five initial 1s, the width 6 sequence Dh−3 Dh+3 = Dh−2 Dh+2 +Dh−1 Dh+1 +Dh2 , and so on, testing whether each when initiated by an appropriate number of 1s yields only integers. Naturally, he asks: “What is going on here?” By the way, while 4-Somos (A006720) 5-Somos (A006721), 6-Somos (A006722), and 7-Somos (A006723) all do yield only integers; 8-Somos does not. The codes in parentheses refer to Neil Sloane’s ‘On-line encyclopedia of integer sequences’ [22]. 3.2. Elliptic divisibility sequences Sequences generalising those considered by Somos were known in the literature. Morgan Ward had studied anti-symmetric sequences (Wh ) satisfying relations 2 (20) − Wm−n Wm+n Wh2 . Wh−m Wh+m Wn2 = Wh−n Wh+n Wm    He shows that if W1 = 1 and W2 W4 then ab implies that Wa Wb ; that is, the sequences become divisibility sequences (compare the Fibonacci numbers). For a brief introduction see Chapter 12 of [9]. There is a drama here. The recurrence relation

Wh−2 Wh+2 = W22 Wh−1 Wh+1 − W1 W3 Wh2 , and four nonzero initial values, already suffices to produce (Wh ). Thus (20) for all m and n is apparently entailed by its special case n = 1 and m = 2. The issue is whether the definition (20) is coherent. One has to go deep into Ward’s memoir [25] to find an uncompelling proof that there is in fact a solution sequence, namely one defined in terms of quotients of Weierstrass sigma functions. More to the point, given integers W1 = 1, W2 , W3 , and W4 , there always is an associated elliptic curve. In our terms, there is a curve C : Z 2 − AZ − R = 0 with deg A = 2, deg R = 1 and the sequence (Wh ) arises from the continued fraction expansion of Z1 = Z/(−R). I call (Wh ) the singular sequence because in that case e1 = 0 — so the partial quotient a0 (X) is not linear. My ‘translated’ sequences (Ah ) were extensively studied by Christine Swart in her thesis [23].

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3.3. Somos sequences It is natural to generalise Michael Somos’s questions and to study recurrences of the kind he considers but with coefficients and initial values not all necessarily 1. Then our recurrence (19) is a general instance of a Somos 4 sequence, an easy computation confirms that −v = W2 and v 2 A(w) = −W3 , and given the recursion and four consecutive Ah one can readily identify the curve C and the initial ‘translation’ M = (w0 , −e0 ). Comment 10. One might worry (I did worry) that Ah−2 Ah+2 = aAh−1 Ah+1 +bA2h does not give a rationally defined elliptic curve if a is not a square. No worries. One perfectly happily gets a quadratic twist by a of a rationally defined curve. For example, 4-Somos, the sequence (Ch ) = (. . . , 2, 1, 1, 1, 1, 2, 3, 7, . . . ) with Ch−2 Ch+2 = Ch−1 Ch+1 + Ch2 arises from C : Z 2 − (X 2 − 3)Z − (X − 2) = 0 with M = (1, −1); equivalently from E : V 2 − V = U 3 + 3U 2 + 2U with ME = (−1, 1). Christine Swart and I found a nice inductive proof [19] that if (Ah ) satisfies (19) then for all integers m and n, 2 Ah−n Ah+n − Wm−n Wm+n A2h . Ah−m Ah+m Wn2 = Wm

Our argument obviates any need for talk of transcendental functions and is purely algebraic. It also is plain that Ah−1 Ah+1 = eh A2h yields Ah−1 Ah+2 = eh eh+1 Ah Ah+1 and Ah−2 Ah+3 = eh−1 e2h e2h+1 eh+2 Ah Ah+1 . However, although (10) directly entails   eh−1 e2h e2h+1 eh+2 = −v 2 A(w)eh eh+1 + v 4 + 2wv 3 A(w) , it requires some effort to see this. Whatever, (19) eventually also gives W1 W2 Ah−m Ah+m+1 = Wm Wm+1 Ah−1 Ah+2 − Wm−1 Wm+2 Ah Ah+1 .

(21)

For details see [17] and [19]. The case m = 2 of (21) includes all Somos 5 sequences. The sequence 5-Somos, (Bh ) = ( . . . , 2, 1, 1, 1, 1, 1, 2, 3, 5, 11, . . . ) with Bh−2 Bh+3 = Bh−1 Bh+2 + Bh Bh+1 , arises from Z 2 − (X 2 − 29)Z + 48(X + 5) = 0 with M = (−3, −8); equivalently from E : V 2 + U V + 6V = U 3 + 7U 2 + 12U with ME = (−2, −2). Actually, a Somos 5 sequence (Ah ) may also be viewed as a pair (A2h ) and (A2h+1 ) of Somos 4 sequences coming from the same elliptic curve but with different translations (in fact differing by half its point S ); see the discussion in [17]. 3.4. Higher genus Somos sequences My purpose in studying the elliptic case was to be able to make impact on higher genus cases. That’s been only partly achieved, what with little more than (17) at page 218 to show for the effort. I tame (17) by defining a sequence (Th ), one hopes of integers, by way of Th−1 Th+1 = dh Th2 . We already know that then Th−2 Th+2 =

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dh−1 d2h dh+1 Th2 ; also Th−3 Th+3 = dh−2 d2h−1 d3h d2h+1 dh+2 Th2 follows with only small extra effort. Thus (17) yields Th−3 Th+3 = v 3 Th−2 Th+2 − v 2 A(w)Th2 .

(22)

Then the sequence (Th ) = (. . . , 2, 1, 1, 1, 1, 1, 1, 2, 3, 4, 8, 17, 50, . . .) satisfying Th−3 Th+3 = Th−2 Th+2 + Th2 is readily seen to derive from the genus 2 curve C : Z 2 − Z(X 3 − 4X + 1) − (X − 2) = 0 . (23) A relevant piece of the associated continued fraction expansion is Z + 2X − 1 Z +X = X − X2 − 1 X2 − 1 Z +X −1 Z +X = −X − 2 −(X − 2) −(X 2 − 2) Z0 :=

Z +X −1 Z +X −1 =X +1− 2 2 X −X −1 X −X −1 Z +X −1 Z +X = −X − 2 −(X − 2) −(X 2 − 2) Z +X Z + 2X − 1 =X− 2 X −1 X2 − 1 ···

illustrating that M is the divisor class defined by the pair of points (ϕ, ϕ) and (ϕ, ϕ) — here, ϕ is the golden ratio, a happenstance that I expect will please adherents to the cult of Fibonacci. Comment 11. There does remain an issue here. Although (10) is concocted on the presumption that eh is never zero, it continues to make sense if some eh should vanish — for instance in the singular case. However, (17) is flat out false if dh−1 dh dh+1 = 0. Indeed, a calm study of the argument yielding (17) sees us dividing by v(eh + w) at a critical point. These considerations together with Cantor’s results [8] suggest that (22) should always be reported as multiplied by Th and, more to the point, that the general genus 2 relation when u = 0 will be cubic rather than quadratic. 4. Other Viewpoints My emphasis here has been on continued fraction expansions producing sequences M + hS of divisors — in effect the polynomials Qh — obtained by repeatedly adding a divisor S to a starting ‘translation’ M . That viewpoint hints at arithmetic reasons for the integrality of the Somos sequences but does not do that altogether convincingly in genus greater than one. 4.1. The Laurent phenomenon As it happens, the integrality of the Somos sequences is largely a combinatorial phenomenon. In brief, as an application of their theory of cluster algebras, Fomin and

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Zelevinsky [10] prove results amply including the following. Suppose the sequence (yh ) is defined by a recursion yh+n yh = αyh+r yh+n−r + βyh+s yh+n−s + γyh+t yh+n−t , with 0 < t < s < r ≤ 21 n. Then the yh are Laurent polynomials∗ in the variables y0 , y1 , . . ., yn−1 and with coefficients in the ring Z[α, β, γ] . That deals with all four term and three term quadratic recursions and thus with the cases Somos 4 to Somos 7. Rather more is true than may be suggested by the given example. 4.2. Dynamic methods Suppose we start from a Somos 4 relation Ah−2 Ah+2 = αAh−1 Ah+1 + βAh2 and appropriate initial values A0 , A1 , A2 , A3 . Then one obtains rationals eh = Ah−1 Ah+1 /A2h satisfying the difference equation eh+1 =

1 eh−1 eh



α+

β . eh

The point is that this equation has a first integral given by  1 1 β J := J(eh−1 , eh ) = eh−1 eh + α + = J(eh , eh+1 ) + eh−1 eh eh−1 eh and one can now construct an underlying Weierstrass elliptic function ℘. Indeed, the readily checked assertion that given y ∈ C there are constants α and β so that   2     ℘(x + y) − ℘(y) ℘(x) − ℘(y) ℘(x − y) − ℘(y) = −α ℘(x) − ℘(y) + β reveals all; particularly that α = ℘ (y)2 , β = ℘ (y)2 (℘(2y) − ℘(y)). Specifically, after fixing x by J ≡ ℘ (x) one sees that −eh = ℘(x + ny) − ℘(y) . A program of this genre is elegantly carried out by Andy Hone in [12] for Somos 4 sequences. In [5], the ideas of [12] are shown to allow a Somos 8 recursion to be associated with adding a divisor on a genus 2 curve. Incidentally, that result coheres with the guess mooted at Comment 11 above that there always then is a cubic relation of width 6. References [1] Adams, William W. and Razar, Michael J. (1980). Multiples of points on elliptic curves and continued fractions. Proc. London Math. Soc. 41, 481–498. MR591651. [2] Alesina, Alberto and Galuzzi, Massimo (1998). A new proof of Vincent’s theorem. Enseign. Math. (2) 44.3–4, 219–256. MR1659208; Addendum (1999). Ibid. 45.3–4, 379–380. MR1742339. [3] Bombieri, Enrico and Cohen, Paula B. (1997). Siegel’s lemma, Pad´e approximations and Jacobians’ (with an appendix by Umberto Zannier, and dedicated to Enzio De Giorgi). Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25, 155–178. MR1655513. ∗A

Laurent polynomial in the variable x is a polynomial in x and x−1 .

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[4] Bombieri, Enrico and van der Poorten, Alfred J. (1995). Continued fractions of algebraic numbers. In Computational Algebra and Number Theory (Sydney, 1992) Math. Appl., 325, Kluwer Acad. Publ., Dordrecht, 137–152. MR1344927. [5] Braden, Harry W., Enolskii, Victor Z., and Hone, Andrew N. W. (2005). Bilinear recurrences and addition formulæ for hyperelliptic sigma functions’. 15pp: at http://www.arxiv.org/math.NT/0501162. [6] Cassels, J. W. S. and Flynn, E. V. (1966). Prolegomena to a Middlebrow Arithmetic of Curves of Genus 2, London Mathematical Society Lecture Note Series, 230. Cambridge University Press, Cambridge, 1996. xiv+219 pp. MR1406090. [7] Cantor, David G. (1987). Computing in the Jacobian of a hyperelliptic curve. Math. Comp. 48.177, 95–101. MR866101. [8] Cantor, David G. (1994). On the analogue of the division polynomials for hyperelliptic curves. J. f¨ ur Math. (Crelle), 447, 91–145. MR1263171. [9] Everest, Graham, van der Poorten, Alf, Shparlinski, Igor, and Ward, Thomas (2003). Recurrence Sequences. Mathematical Surveys and Monographs 104, American Mathematical Society, xiv+318pp. MR1990179. [10] Fomin, Sergey and Zelevinsky, Andrei (2002). The Laurent phenomenon. Adv. in Appl. Math., 28, 119–144. MR1888840. Also 21pp: at http://www.arxiv.org/math.CO/0104241. [11] Gale, David (1991). The strange and surprising saga of the Somos sequences. The Mathematical Intelligencer 13.1 (1991), 40–42; Somos sequence update. Ibid. 13.4, 49–50. [12] Hone, A. N. W. (2005). Elliptic curves and quadratic recurrence sequences. Bull. London Math. Soc. 37, 161-171. MRMR2119015 [13] Lauter, Kristin E. (2003). The equivalence of the geometric and algebraic group laws for Jacobians of genus 2 curves. Topics in Algebraic and Noncommutative Geometry (Luminy/Annapolis, MD, 2001), 165–171, Contemp. Math., 324, Amer. Math. Soc., Providence, RI. MR1986121. [14] Pappalardi, Francesco and van der Poorten, Alfred J. (2004). Pseudo-elliptic integrals, units, and torsion. 12pp: at http://www.arxiv.org/math.NT/0403228. [15] van der Poorten, Alfred J. (1998). Formal power series and their continued fraction expansion. In Algorithmic Number Theory (Proc. Third International Symposium, ANTS-III, Portland, Oregon, June 1998), Springer Lecture Notes in Computer Science 1423, 358–371. MR1726084. [16] van der Poorten, Alfred J. (2004). Periodic continued fractions and elliptic curves. In High Primes and Misdemeanours. Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, Fields Institute Communications 42, American Mathematical Society, 353–365. MR2076259. [17] van der Poorten, Alfred J. (2005). Elliptic curves and continued fractions. 12pp: at http://arxiv.org/math.NT/0403225. [18] van der Poorten, Alfred J. (2005). Curves of genus 2, continued fractions, and Somos sequences. 6pp: at http://arxiv.org/math.NT/0412372. [19] van der Poorten, Alfred J. and Swart, Christine S. (2005). Recurrence relations for elliptic sequences: every Somos 4 is a Somos k . 7pp: at http://arxiv.org/math.NT/0412293. [20] Propp, Jim. The Somos Sequence Site. http://www.math.wisc.edu/∼propp/somos.html.

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[21] Shipsey, Rachel (2000). Elliptic divisibility sequences, PhD Thesis, Goldsmiths College, University of London. http://homepages.gold.ac.uk/rachel/. [22] Sloane, Neil On-Line Encyclopedia of Integer Sequences. http://www.research.att.com/∼njas/sequences/. [23] Swart, Christine (2003). Elliptic curves and related sequences. PhD Thesis, Royal Holloway, University of London. [24] Uspensky, J. V. (1948). Theory of Equations, McGraw-Hill Book Company. [25] Ward, Morgan (1948). Memoir on elliptic divisibility sequences Amer. J. Math. 70, 31–74. MR0023275. [26] Zagier, Don (1966) Problems posed at the St Andrews Colloquium, Solutions, 5th day; see http://www-groups.dcs.st-and.ac.uk/∼john/Zagier/Problems.html.

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 225–236 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000248

Old and new results on normality Martine Queff´ elec1 Universit´ e Lille1 Abstract: We present a partial survey on normal numbers, including Keane’s contributions, and with recent developments in different directions.

1. Introduction A central tool in the papers of Mike Keane is the ingenious and recurrent use of measure theory (probability and ergodic theory) in the various problems he investigated, paving the road for successors. So did he with normality, one of his favorite topics. Let q ≥ 2 be an integer. A real number θ ∈ I := [0, 1) is said to be normal to base q, if for every m ≥ 1, every d1 d2 . . . dm ∈ {0, 1, . . . , q − 1}m occurs
finite word −n in the q-adic expansion of θ, n≥1 n (θ)q , with frequency q −m . This means that every pattern appears infinitely often with the “good frequency” in the expansion of x. A number θ ∈ I is said to be simply normal to base q if this property holds for m = 1. It is said to be q (absolutely) normal if it is normal to every base q, and it is simply normal if it is simply normal to every base q. Normal numbers have been introduced in 1909 by E. Borel in [7], where he proved that–with respect to Lebesgue measure m–almost every number is normal. In fact, the digits (j (θ))j≥1 of θ in base q behave as independent identically distributed random variables. Borel’s result is thus a consequence of the Strong Law of Large Numbers, applied for each q to disjoint blocks km+1 (θ) km+2 (θ) . . . (k+1)m (θ), k ≥ 0. Normality to base q can also be defined in terms of dynamics. The q-expansion of θ is obtained by iterating on θ the q-transformation Tq defined on I by Tq (x) = qx mod 1; if x0 = x and xn+1 = qxn mod 1 for n ≥ 1, then n+1 (x) = [qxn ]. It is well known that the Lebesgue measure m on I is the unique absolutely continuous Tq -invariant measure, and that the dynamical system (I, Tq , m) is ergodic. Borel’s result is thus a consequence of the Birkhoff Ergodic Theorem. So, normal numbers lie at the junction between two areas of expertise of Mike Keane, who gave illuminating proofs of both the Strong Law of Large Numbers and the Birkhoff Ergodic Theorem [19, 22], and wrote “Measure-preserving transformations are beautiful” with the evident corollary “Invariant measures are beautiful!” 1 Universit´ e

Lille1, UMR 8546, France, e-mail: [email protected] AMS 2000 subject classifications: primary 11K, 37A; secondary 11J. Keywords and phrases: normal number, uniform distribution, continued fractions, Hausdorff dimension, Fourier transform.

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2. Combinatorial point of view From the definition, explicit q-normal numbers and non-normal numbers can be exhibited; for example Champernowne’s number [11], whose expansion consists in concatenating all consecutive words to base q, is q-normal, for example c2 = 0.0100011011000001010011100101110111 . . . when q = 2. √ However, nobody knows whether classical arithmetical constants such as π, e, 2, ζ(n), . . . are normal numbers to a fixed base, say q = 2. Even weaker assertions are unresolved. For example,√it is not known whether there exist infinitely many 7s in the decimal expansion of 2; or whether√arbitrarily long blocks of zeros appear in its binary expansion, i.e., lim inf n→∞ {2n 2} = 0? Returning to Champernowne’s number c2 , the question arises whether c2 is normal to any other base. Even though almost every number is normal to every base q, no explicit example is known of a number which is both p and q-normal, with p = q. It is also not known how can one get the 3-expansion from the 2-expansion of some irrational number. A plausible conjecture (suggested by Borel’s work) is that every irrational algebraic number is normal. Already a partial answer to this conjecture such as every irrational algebraic number is simply normal, would be of great interest as we shall see.
∞ an An unsolved question from Mahler [26] is the following. Suppose that α = 1 2n
∞ and β = 1 a3nn are algebraic numbers, with an ∈ {0, 1}. Are α and β necessarily rational? If any irrational algebraic number would be simply normal, β must be rational, and consequently the sequence (an ) is ultimately periodic; This would imply that α is rational as well, thus answering Mahler’s question positively. In recent investigations on normality, the statistical property of “good frequency” is replaced by the combinatorial property of “good complexity,” asserting that p(n), the number of words of length n occurring in the q-expansion of x, is maximal, i.e., p(n) = q n , n ≥ 1, leading to the following conjecture. Conjecture. Every irrational algebraic number has maximal complexity function p(n). Using a p-adic version of Roth’s theorem, which appeared in previous work by Ridout [35], Ferenczi and Mauduit [12] obtained the following description of a class of transcendental numbers (in a revisited formulation by Adamczewski and Bugeaud). Theorem 2.1 (Ferenczi–Mauduit). Let x be an irrational number, whose qexpansion begins with 0.Un Vns for every n ≥ 1, with (i) s > 2; (ii) |Vn |, the length of the word Vn , is increasing; (iii) |Un |/|Vn | is bounded. Then x is a transcendental number. As a corollary they get the transcendence of Sturmian numbers: If there exists q such that the expansion of x to base q is a Sturmian sequence on q letters, then x is a transcendental number. Remembering that a Sturmian sequence on q letters satisfies p(n) = n + q − 1 for n ≥ 1, this can be viewed as the first result on complexity in this setting:

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Theorem 2.2. If x is an irrational algebraic number, then its complexity function to base q satisfies lim inf (p(n) − n) = +∞. n→∞

Roughly speaking Theorem 2.1 means that the sequence of digits begins with arbitrarily long prefixes of the form Wn Wn Wn
; but these conditions eliminate some natural numbers; for example, the Thue-Morse sequence on {0, 1} being without overlap of size 2 + ε, and the associated Thue-Morse number as well. Involving a suitable version of the Schmidt Subspace Theorem, which may be considered as a multi-dimensional extension of previous quoted results by Roth and Ridout, Adamczewski and Bugeaud [1] improved this theorem by replacing in (i) s > 2 by s > 1. This proves as a by-result that the Thue-Morse number is transcendental. As a corollary they get Theorem 2.3 (Adamcewski–Bugeaud). If x is an irrational algebraic number, then its complexity function to base q satisfies lim inf n→∞

p(n) = +∞. n

Thus, every non-periodic sequence with sub-linear complexity gives rise to a transcendental number; this is the case with generalized Morse sequences [20, 21]. 3. Metric point of view We identify throughout [0, 1) and the circle T = R/Z, and let M (T) be the set of bounded Borel measures on T. In this section we turn to the characterization of the sets Nq , of q-normal numbers, and of N = ∩q Nq as well as their negligible complements, raising many connected questions. 1. Under what conditions on p and q (if any) does there exist a number normal to base p but non-normal to base q? 2. In this case, what is the cardinality of Np ∩ Nqc ? 3. What is the Hausdorff dimension of such a set? Another approach to normality will be useful. Recall that a real sequence (un )n is said to be uniformly distributed mod 1 if ∀0 ≤ a < b ≤ 1,

1 |{n ≤ N ; {un } ∈ [a, b]}| → b − a, N

as N → ∞, where {x} is the fractional part of x. Equivalently, by the Weyl’s criterion, if and only if ∀k = 0,

1  ek (un ) → 0, N n≤N

where ek (x) := e2iπkx . This means that the sequence of probability measures on I: 1  δ{un } → m weak∗ N n≤N

where δx is the unit mass at x.

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It was proved by Wall in [39] that x ∈ I is normal to base q if and only the sequence (q n x)n is uniformly distributed mod 1. This point of view leads to the ergodic proof of Borel’s theorem since, m-almost everywhere,  1  n ek ◦ Tq → ek dm = 0. N I n
3.1. Schmidt’s results In 1960, W. Schmidt solved questions 1. and 2. We write p ∼ q if there exist positive p integers r, s with pr = q s (or log log q ∈ Q), and p ∼ q otherwise. Theorem 3.1 (W. Schmidt). 1. Assume p ∼ q; then any number normal to base p is normal to base q: Np = Nq . 2. If p ∼ q, then the set of p-normal numbers which are not even simply normal to base q has the power of the continuum. Proof. 1. The identity Npr = Np can be established by using the combinatorial definition. This gives the first assertion immediately. Np goes as follows: if x ∈ Npr so are An alternative proof of
the inclusion Npr ⊂
2 r−1 rn j px, p x, . . . , p x, since n
(s)

νN =



δ{qsn x} → m weak∗

n≤N −1 (s)

by proving that m is the unique weak∗ limit point of the sequence (νN ). If F ⊂ I is a closed set such that m(F
) = 0, then by hypothesis νsN (F ) ≤ ε for N ≥ NF . 1 The evident inequality sN n≤N −1 δ{qsn x} (F ) ≤ νsN (F ) yields that (s)

νN (F ) ≤ sε, N ≥ NF . (s)

Let σ be a weak∗ limit point of νN in the weak∗ compact set of probability measures on I. From the above, σ(F ) = 0 whenever F is closed with m(F ) = 0. For every Borel set A with m(A) = 0, one has σ(A) =

σ(K) = 0 sup compact K⊂A

K

since m(K) = 0, and σ m. But σ is Tqs -invariant and m is the unique absolutely continuous Tqs -invariant measure. Hence σ = m and x ∈ Nqs . 2. The proof of the second assertion, which is rather complicated in [37], has been simplified by many authors, first of them being Keane and Pierce [23]. The basic idea is that any q-invariant, ergodic and non absolutely continuous probability measure µ supports Nqc . By choosing k with µ ˆ(k) = 0, and using the ergodic theorem, one gets 1  ek (q n x) → µ ˆ(k) = 0 µ − a.e. N n
and µ-almost x is in Nqc .

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Keane and Pierce constructed a Cantor measure (depending on q but not on p) supported by Np ∩ Nqc . Later, generalizations have been obtained by Brown, Moran & Pierce [9] by employing Riesz products, by Feldman & Smorodinski [13] then Bisbas [6] with Bernoulli convolution measures, by Kamae [17] with specific singular measures; Host [14] improved the result in view of F¨ urstenberg’s conjecture. c Pollington [31] showed that the set Np ∩ Nq has full Hausdorff dimension. We give here a revisited proof by Kamae [17] where g-measures, introduced by Keane [21], play an important role. Let us recall the notion of g-measures. Let T be an m-to-1 covering transformation on
a compact metric space X, and g a positive continuous function on X such that g(y) = 1. We define the transfer operator L on C(X) by T y=x

Lf (x) =



f (y)g(y).

T y=x

A g-measure is a T -invariant Borel probability measure, which is an eigenmeasure to the maximal eigenvalue of the adjoint operator L∗ . Uniqueness holds whenever g is strictly positive with Lipschitz regularity, and in this case µ is strongly mixing [21]. Natural examples appear in computing the correlation measure of finitely-valued q-multiplicative sequences of modulus 1 (which are nothing else than generalized Morse sequences). Fix q ≥ 2; consider Sq (n) the sum of the q-digits of n and ρ := ρq the correlation measure of the sequence α(n) = e2iπαSq (n) , α ∈ R; ρ is the strongly mixing g 1  measure P (q n x), with P (x) = | α(k)e(kx)|2 and g = 1q P . It is proved in q 0≤k
[32] that ρ has the constant modulus property; we need only a weaker assertion: Lemma 1. Denote by Γ the dual group of T when acting multiplicatively. If χρ ∈ Γρ , the set of limit points for the topology σ(L∞ (ρ), L1 (ρ)) of sequences (γn ) ⊂ Γ, then |χρ | = C, C constant. As explained above, ρ(Nq ) = 0. Now, for every k = 0,   2 1   1  ρˆ(kpn − kpm ) ek (pn x) dρ(x) = 2  N N n
m
and the property ρ(Np ) = 1 for p ∼ q follows readily from the next one: Lemma 2. For every integer a = 0, lim n,m→∞ ρˆ(a(pn − pm )) = 0. n>m Proof. Fix a > 0 and denote by χρ ∈ Γρ a limit point in the topology σ(L∞ (ρ), L1 (ρ)) of the sequence (γapn ) ⊂ Γ, where γk = ek . By Lemma 1, |χρ | = C. Since |χρ |2 is a to prove that C = 0. limit point of the sequence (γapn −apm )n>m , it suffices  If C = 0, there exists c = 0 and γ = γb with | χρ γb dρ| = c; now, by extracting a subsequence if necessary,  n ρˆ(ap + b) → χρ γb dρ. Let 0 < ε 1, and ρ being continuous, let m > 0 be such that |ˆ ρ(m)| < εc. If

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q N −1 ≤ m < q N , from hypotheses on p, q, there exists nk , mk going to ∞ such that pnk m → N ∈ [0, 1[. m q k aq Thus, apnk = mq mk −N + rk where rk = o(q mk ), and for k big enough, apnk + b = mq mk −N + rk + b, mk > N, rk + b < q mk −N . By the strong mixing property, ρˆ(apnk + b) ∼ ρˆ(m)ˆ ρ(rk + b). But ρˆ(apnk + b) → c and |ˆ ρ(m)ˆ ρ(rk + b)| < εc, which yields a contradiction if such a c = 0 exists. This implies that χρ = 0 and the lemma is proved. This lemma has been proved in a different way by Kamae with help of a deep result from Strauss and Senge. 3.2. Sums of normal numbers Observe first that a Borel set E ⊂ T with m(E) > 1/2 satisfies E + E = T. This follows from the fact that for every x ∈ T, m (E ∩ (x − E)) > 0, hence there exist u, v ∈ E such that x = u + v. Now if m(E) = 0 but E supports a probability measure µ satisfying µ ∗ µ = m, then every number in T can be expressed as the sum of four numbers in E since m(E + E) = 1. But this is far from optimal. It is well-known that K3 , the triadic Cantor set, is such that K3 + K3 = T. However, if µ3 is the Cantor measure (the probability measure uniformly distributed over those points of T whose ternary expansions contain only 0’s and 2’s), µ3 ∈ M0 (T) = {µ ∈ M (T); limn→∞ µ ˆ(n) = 0} and µ3 ∗ µ3 is not even absolutely continuous with respect to the Lebesgue measure m on T. More striking: Erd¨ os gave an example showing that there exist arbitrarily thin sets, in the sense of Hausdorff measures, whose sum set contains an interval. Now, if E and F have positive Cantor measure, E + F has positive Lebesgue measure, since we have the following inequality m(E + F ) ≥ µ3 (E)α µ3 (F )α

(3.1)

where α = log 3/ log 4; this follows from the combinatorial one: If A, B ⊂ {0, 1}n , then |A+B| ≥ |A|α |B|α . (Note that 2α = dim m/ dim µ3 ). The extension of (3.1) to the Cantor measure µm on Km , the m-adic Cantor set, m ≥ 3, has been established in [8]: m  µm+1 (Ej )α(m) m(E1 + · · · + Em ) ≥ j=1

where α(m) = log(m + 1)/m log 2. Similar observations can be made about the negligible sets N (p) = ∩q∼p Nq ∩Npc . Keane and Pierce have shown in [23] that µp (N (p)) = 1. Combining this with all previous remarks yields that every number in T can be expressed as the sum of 2(p − 1) numbers in N (p). But various measures living on N (p) (as we saw) lead to different decomposition results. By using Hausdorff measures, Pollington proved in [31] the following result. Theorem 3.2. For each p ≥ 2, N (p) + N (p) = T.

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3.3. More about non-normal numbers So what more can be said about measures living on the negligible sets Nqc ? Probability measures µ on T whose Fourier transform vanishes at infinity rather fast are still annihilating the sets Nqc . More precisely, in [25] the following was shown. If
φ(n) |ˆ µ(n)| ≤ φ(|n|) whenever φ is a non-increasing function such that n log n < ∞, c then µ(Nq ) = 0. This is a consequence of the following Strong Law of Large Numbers for weakly correlated random variables, applied to Xn = ekqn . Theorem 3.3 (Davenport–Erd¨ os–LeVeque). Let µ be a probability measure
N on E, (Xn ) a sequence of bounded random variables and SN = N1 n=1 Xn . If  1  |SN |2 dµ < ∞, N N ≥1

then SN → 0 µ-a.e. On the other hand, Lyons constructed a probability measure concentrated on N2c
φ(n) with |ˆ µ(n)| ≤ φ(|n|), where φ is a non-increasing function such that n log n = ∞. It is a convolution type measure with respect to which the binary digits as random variables are independent by blocks. His sharpest result is the following theorem. Theorem 3.4 (Lyons). There exists a probability measure µ ∈ M0 (T) such that K 1  1[0,1/2] (2k x) = 1 lim sup K K→∞ 1

µ − a.e.

Then arised the following question: Does there exist x and I, closed arc of length 1/2, such that {2n x} ∈ I for all n ≥ 0? 4. Normality and continued fractions Every real number admits a regular continued fraction expansion, which is more handy than the q-expansions, and furnishes the best rational approximations. Although it is difficult to get one expansion from the other, there exist classes of examples for which both are known, but no example of normal number with bounded partial quotients has been exhibited yet. We recall some facts concerning the regular continued fraction algorithm (RCF). Given an irrational number x ∈ I, x is the limit of the infinite sequence of rational numbers 1 Pk (x) = , 1 Qk (x) a1 (x) + 1 a2 (x) + · · · 1 ak−1 (x) + ak (x) Pk = [0; a1 , . . . , ak ], x = where the integers ak (x) ≥ 1 if k ≥ 1; we write shortly Q k [0; a1 , a2 , . . .] where the partial quotients of x, a1 , a2 , . . . , are uniquely defined. The shift on the RCF expansion is conjugated to an expanding transformation, 1 the Gauss map, defined on X = [0, 1]\Q by T x = mod 1. The partial quotients x

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an = an (x), n ≥ 1, are then given by 1 a1 (x) = [ ], ak+1 (x) = ak (T x). x The measure µ on X (called the Gauss measure) defined by µ(A) =  probability 1 dx for every Borel-set A, is preserved by T and is T -ergodic. Furthermore, log 2 A 1+x µ is the unique absolutely continuous T -invariant probability measure. Applying Birkhoff’s ergodic theorem, we get that 1 (a1 (x) + · · · + an (x)) = +∞ µ − a.e. n→∞ n Since µ is equivalent to Lebesgue measure, the set BAD of numbers in [0, 1) with bounded partial quotients has zero Lebesgue measure. A significant difference between adic- and RCF expansions is the behavior of the digits as random variables; the an (x) are no more independent identically distributed variables on (X, µ), but quasi-independent in the following sense. lim

Theorem 4.1 (Philipp). Consider the σ-algebras Ak = σ(a1 , . . . , ak ) and Bk = σ(ak+1 , . . .). Then there exists θ ∈]0, 1[ such that |µ(A ∩ B) − µ(A)µ(B)| ≤ θn µ(A)µ(B) for every A ∈ Ak and B ∈ Bk+n , k ≥ 1, n ≥ 1. For complementary notations and results on the regular continued fraction expansion we refer to [16] and [5]. In [24], a class of normal numbers with explicit RCF expansion is exhibited; this combines two ideas.
1. Every number of the form k A1k , with Ak integers ≥ 2 such that A2k divides Ak+1 , has an explicit RCF expansion. This is a consequence of classical identities around symmetry, for example [28, P = [0; a1 , . . . , aN ], aN ≥ 2, x ≥ 1, 38], if Q P (−1)N + = [0; a1 , . . . , aN −1 , aN , x − 1, 1, aN − 1, aN −1 , . . . , a1 ]. 



  Q xQ2
∞ 1 In this way, Mahler proved that the transcendental number k=0 2k , u ≥ 2, is in u BAD.
−λk −µk 2. Every number α of the form , where (λk ), (µk ) are increasing q kp sequences of integers ≥ 1, p, q ≥ 2 relatively prime integers and µk ≥ pλk , is normal to base q. This is due to sharp estimates on the distribution of digits in periodic fractions and a consequence of the structure in base q of the numbers α. It can be seen that the q-expansion of an α looks like 0a W0s0 W0 W1s1 W1 . . . Wnsn Wn . . . , Wn being a prefix of the word Wn , |Wn | = tn the order of q mod pλn and |Wnsn Wn | = µn+1 − µn .
−2k −p2k Theorem 4.2 (Korobov). Numbers of the form α = q with p, q kp relatively prime integers, are normal to base q with explicit RCF expansion. Of course in these cases xk = Ak+1 /A2k → ∞ and such numbers α ∈ BAD. Montgomery [29] in his book posed the following problem: Does there exist normal numbers with bounded partial quotients ? In 1980, Kaufman [18], using the structure of the sets F (N ) = {x ∈ [0, 1); x = [0; a1 , a2 , . . . ] with ai ≤ N ∀i ≥ 1}, obtained the following deep result.

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Theorem 4.3 (Kaufman). For N ≥ 3 F (N ) carries a probability measure µ such that |ˆ µ(t)| ≤ c|t|−η for a certain η > 0 (Kaufman’s measure). When Kaufman’s paper appeared, R.C. Baker observed in [29] that combining the above result with a corollary of the Davenport–Erd¨ os–Leveque’s theorem (already used by Lyons for non-normal sets), yields the existence of normal numbers in BAD. More generally, if A is a finite alphabet of integers ≥ 1, |A| ≥ 2, and F (A) = {x ∈ [0, 1); x = [0; a1 , a2 , . . .] with ai ∈ A ∀i ≥ 1}, In [34] we improved Kaufman’s result as follows. Theorem 4.4. If dimH (F (A)) > 1/2, F (A) supports a Kaufman’s measure and contains infinitely many normal numbers. In particular there exist infinitely many normal numbers with partial quotients ∈ {1, 2}. Note that no explicit normal numbers in BAD have been constructed yet. Now, it is natural to ask about the existence of RCF-normal numbers. Such a number x is defined by the property that for every m ≥ 1, every finite word d1 d2 . . . dm ∈ N∗m occurs in the RCF expansion of x, with frequency µ([d1 d2 . . . dm ]), the Gauss measure of the cylinder [d1 d2 . . . dm ]. Once more from ergodicity (or quasi-independence), almost all x ∈ [0, 1) are normal for the continued fraction transformation and once more constructing one RCF-normal number raises difficulties. A successful proceedure has been carried out by Adler, Keane and Smorodinsky [2]. Theorem 4.5. Let Qn be the ordered set of all rationals in [0, 1) with denominator n. The RCF expansion obtained by concatenating the RCF expansions of the numbers in Q2 , Q3 , . . . leads to an RCF-normal number. Hence the following problems: How to construct a number normal with respect to both adic- and Gauss transformations? Does there exist an RCF-normal number with low complexity relative to an adic-expansion? 5. More investigation 5.1. Topological point of view The sets Nqc , N c are small from the metric point of view but big from the topological one. This has been known for a long time, for example, a result from Helson and Kahane [15] goes as follows. For each q ≥ 2, Nqc intersects every open interval in a uncountable set. Also, if q ≥ 2, 0 ≤ r < q, x ∈ [0, 1) and Nn (r, x) is the number of occurrences of r in the first n terms of the q-expansion of x, then the set of limit points of the sequence (Nn (r, x)/n) is [0, 1], for all x but a set of first Baire category; as a consequence the set N is in first Baire category [36]. Real numbers with non-dense orbit under Tq (“q-orbit”) are in fact very interesting. If I is an open arc of T of length > 1/2, the set of x ∈ T whose 2-orbit does not intersect I must be finite, but there exist s ∈ [0, 1/2] whose 2-orbit is infinite and whole contained into [s, s + 1/2]; this furnishes an answer to the question raised at the end of section 4; in fact we have a complete description of these numbers s (see [4, 33]).

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ε Theorem 5.1 (Bullett–Sentenac). Let s = j≥1 2jj be an irrational number in [0, 1/2]. Then following assertions are equivalent. 1. The closed 2-orbit of s lies into [s, s + 1/2]. 2. The sequence of digits (εj )j is a characteristic sturmian sequence on the alphabet {0, 1}. 5.2. Generalizations Generalization of normality to non-integer bases or to endomorphisms leads to very interesting questions (see [3]). But this would be another story . . . Acknowledgments Thanks to the organizators of this colloquium in honor of Mike, thanks to the referee, and my gratitude to Mike himself for transmitting his enthousiasm through various lectures he gave us many years ago. References [1] Adamczewski, B. and Bugeaud, Y. (2004). On the complexity of algebraic numbers. Preprint. [2] Adler, R., Keane, M.S., and Smorodinsky, M. (1981). A construction of a normal number for the continued fraction transformation. J. Number Th. 13, 95–105. MR0602450 [3] Bertrand-Mathis, A. (1996). Nombres normaux. J. Th. Nombres de Bordeaux 8, 397–412. MR1438478 [4] Bullett, S. and Sentenac, P. (1994). Ordered orbits of the shift, square roots, and the devil’s staircase. Math. Proc. Camb. Philos. Soc. 115, 451–481. MR1269932 [5] Billingsley, P. (1978). Ergodic Theory and Information. Reprint of the 1965 original. Robert E. Krieger Publishing Co., Huntington, N.Y.. MR0524567 [6] Bisbas, A. (2003). Normal numbers from infinite convolution measures. Ergodic Th. Dyn. Syst. 23, 389–393.MR2061299 [7] Borel, E. (1909). Les probabilit´es d´enombrables et leurs applications arithm´etiques. Rend. Circ. Mat. Palermo 27, 247–271. [8] Brown, G., Keane, M.S., Moran, W. and Pearce, C. (1988). An inequality, with applications to Cantor measures and normal numbers. Mathematika 35, 87–94. MR0962738 [9] Brown, G., Moran, W. and Pearce, C. (1985). Riesz products and normal numbers. J. London Math. Soc. 32, 12–18. MR0813380 [10] Bugeaud, Y. (2002). Nombres de Liouville et nombres normaux. C.R. Acad. Sci. Paris 335 , 117–120. MR1920005 [11] Champernowne, D.G. (1933). The construction of decimals normal in the scale of ten. J. London Math. Soc. 8, 254–260. [12] Ferenczi, S. and Mauduit, C. (1997). Transcendence of numbers with a low complexity expansion. J. of Number Th. 67, 146–161. MR1486494 [13] Feldman, J. and Smorodinsky, M. (1992). Normal numbers from independent processes. Ergod. Th. Dynam. Sys. 12, 707–712. MR1200338 [14] Host, B. (1995). Nombres normaux, entropie, translations. Israel J. Math. 91, 419–428. MR1348326

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[15] Helson, H. and Kahane, J.P. (1964). Distribution modulo 1 and sets of uniqueness. Bull. Am. Math. Soc. 70, 259–261. [16] Hardy, G.H. and Wright, E.M. (1979). An Introduction to the Theory of Numbers. Clarendon Press, Oxford Univ. Press. MR0568909 [17] Kamae, T. (1987). Cyclic extensions of odometer transformations and spectral disjointness. Israel J. Math. 59, 41–63. MR0923661 [18] Kaufman, R. (1980). Continued fractions and Fourier transforms. Mathematika 27, 262–267. MR0610711 [19] Keane, M.S. (1991). Ergodic theory and subshifts of finite type. In Ergodic theory, Symbolic Dynamics and Hyperbolic Spaces, 35–70. MR1130172 [20] Keane, M.S. (1968). Generalized Morse sequences. Zeits. Wahr. 10, 335–353. MR0239047 [21] Keane, M.S. (1972). Strongly mixing g-measures. Invent. Math. 16, 304–324. MR0310193 [22] Keane, M.S. (1995). The essence of the law of large numbers. In Algorithms, Fractals, and Dynamics. Ed. by Y. Takahashi, Plenum Press, New York, 125– 129. MR1402486 [23] Keane, M.S. and Pierce, C.E.M., (1982). On normal numbers. J. Aust. Math. Soc. 32, 79–87. MR0643432 [24] Korobov, R. (1990). Continued fractions of some normal numbers (Russian) Mat. Zametki 47, 28–33; translation in Math. Notes 47, 128–132. MR1048540 [25] Lyons, R. (1986). The measure of non-normal sets. Invent. Math. 83, 605–616. MR0827371 [26] Mahler, K. (1976). Lectures on Transcendental Numbers. Lecture Notes in Mathematics No 546. Springer Verlag, Berlin/New York. MR0491533 ´s France, M. (1967). Nombres normaux. Applications aux fonctions [27] Mende pseudo-alatoires. J. d’Analyse Math. 20, 1–56. MR0220683 `s France, M. (1973). Sur les fractions continues limit´ees. Acta Arith. [28] Mende 23, 207–215.MR0323727 [29] Montgomery, H. L. (1994). Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis. CBMS Regional Conf. Series in Math. 84, A.M.S. Providence, RI.MR1297543 [30] Moran, W. and Pollington, A. (1997). The discrimination theorem for normality to non-integer bases. Israel J. Math. 100, 339–347. MR1469117 [31] Pollington, A. (1981). The Hausdorff dimension of a set of normal numbers. Pacific J. Math. 95, 193–204. MR0631669 ´lec, M. (1979). Mesures spectrales associ´ees `a certaines suites [32] Queffe arithm´etiques. Bull. Soc. Math. France 107, 385–421. MR0557078 ´lec, M. (2002). Approximations diophantiennes des nombres stur[33] Queffe miens. J. Th. Nombres de Bordeaux 14, 613–628. MR2040697 ´lec, M. and Ramare ´, O. (2003). Analyse de Fourier des frac[34] Queffe tions continues a` quotients restreints. l’Enseignement Math. 49, 335–356. MR2028020 [35] Ridout, D. (1957). Rational approximations to algebraic numbers. Mathematika 4, 125–131. MR0093508 [36] Salat, T. (1966). A remark on normal numbers. Rev. Roum. Math. Pure Appl. 11, 53–56. MR0201386 [37] Schmidt, W. (1960). On normal numbers. Pacific J. Math. 10, 661–672. MR0117212

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[38] Van der Poorten, A.J. (1986). An introduction to continued fractions. London Math. Soc. Lecture Note Ser. 109, Cambridge University Press, 99–138. MR0874123 [39] Wall, D.D. (1949). Normal numbers. PhD Thesis, University of California, Berkeley. ¨ [40] Weyl, H. (1916). Uber die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77, 313–352. MR1511862

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 237–247 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000257

Differentiable equivalence of fractional linear maps Fritz Schweiger1 University of Salzburg Abstract: A Moebius system is an ergodic fibred system (B, T ) (see [5]) defined on an interval B = [a, b] with partition (Jk ), k ∈ I, #I ≥ 2 such that c +d x T x = ak +bk x , x ∈ Jk and T |Jk is a bijective map from Jk onto B. It is k k well known for #I = 2 the invariant density can be written in the form
that dy where B ∗ is a suitable interval. This result does not hold h(x) = ∗ B (1+xy)2 for #I ≥ 3. However, in this paper for #I = 3 two classes of interval maps are determined which allow the extension of the before mentioned result.

1. Introduction Definition 1. Let B be an interval and T : B → B be a map. We assume that there is a countable collection of intervals (Jk ), k ∈ I, #I ≥ 2 and an associated sequence of matrices   ak b k α(k) = c k dk where det α(k) = ak dk − bk ck = 0, with the properties:  • k∈I Jk = B, Jm ∩ Jn = ∅ if n = m. kx • T x = ackk+d +bk x , x ∈ Jk • T |Jk is a bijective map from Jk onto B. Then we call (B, T ) a Moebius system. Examples of Moebius systems are abundant. We mention the g-adic map T x = gx mod1, g ≥ 2, g ∈ N or the map related with regular continued fractions T x = x1 mod1. Another important example is the R´enyi map T : [0, 1] → [0, 1] Tx =

1 x ,0 ≤ x ≤ ; 1−x 2

Tx =

1−x 1 , ≤ x ≤ 1. x 2

A Moebius system is a special case of a fibred system [5]. Since T |Jk is bijective the inverse map Vk : B → Jk exists. The corresponding matrix will be denoted by   dk −bk β(k) = . −ck ak 1 Dept.

of Mathematics, University of Salzburg, Hellbrunnerstr. 34, A-5020 Salzburg, Austria, e-mail: [email protected] AMS 2000 subject classifications: primary 37A05, 37A05; secondary 11K55, 37E05. Keywords and phrases: measure preserving maps, interval maps.

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k −bk ck | We denote furthermore ω(k; x) := |Vk (x)| = |a(dkkd−b 2 . Then a nonnegative meak x) surable function h is the density of an invariant measure if and only if the Kuzmin equation  h(x) = h(Vk x)ω(k; x)

k∈I

is satisfied. Remark. It is easy to see that we can assume B = [a, b], B = [a, ∞[ or B =]−∞, b] but B = R is excluded: Since #I ≥ 2 there must exist a Moebius map from a subinterval onto B = R which is impossible. Definition 2. A Moebius system (B ∗ , T ∗ ) is called a natural dual of (B, T ) if there ∗ ky is a partition {Jk∗ }, k ∈ I, of B ∗ such that T ∗ y = abkk+d +ck y , y ∈ Jk i.e. the matrix α∗ (k) is the transposed matrix of α(k). Theorem 1. If (B ∗ , T ∗ ) is a natural dual of
(B, T ) then the density of the invariant dy measure for the map T is given as h(x) = B ∗ (1+xy) 2. Proof. Starting with the Kuzmin equation this follows from the relation (1 +

−bk + ak y 2 −ck + ak x 2 y) (dk − bk x)2 = (1 + x) (dk − ck y)2 . dk − b k x dk − c k y

Details are given in [5]. Definition 3. The Moebius system (B ∗ , T ∗ ) is differentiably isomorphic to (B, T ) if there is a bijective map ψ : B → B ∗ such that ψ  exists almost everywhere and the commutativity condition ψ ◦ T = T ∗ ◦ ψ holds. As an example we consider the R´enyi map. The system T ∗ : [0, ∞[→ [0, ∞[ T ∗y =

1−y , 0 < y ≤ 1; T ∗ y = −1 + y, 1 ≤ y y

is a natural dual which is differentiably isomorphic under ψ(t) = map.

1−t t

to the R´enyi

Theorem 2. If the natural dual system (B ∗ , T ∗ ) is differentiably isomorphic to b+dt (B, T ) then ψ(t) = a+bt . Proof. Assume for simplicity B = [0, 1]. Then h(x) =



B∗

dy = (1 + xy)2



0

1

ψ(1) ψ(0) |ψ  (z)| dz = | − |. 2 (1 + xψ(z)) 1 + xψ(1) 1 + xψ(0)

Since the invariant density for T ∗ is given as h∗ (y) = |ψ  (x)| 1+ψ(x) .

1 1+y

we also find h(x) =

Integration gives ψ(x) =

ψ(0) + x(ψ(1) + ψ(1)ψ(0) − ψ(0)) . 1 + xψ(0)

Remark. Note that if the natural dual system (B ∗ , T ∗ ) is differentiably isomorphic c+dt then it follows from the proof of Theorem 2 that to (B, T ) with a map ψ(t) = a+bt b = c is satisfied.

Differentiable equivalence

Lemma 1. Such a map ψ : B → B ∗ , ψ(t) =

b+dt a+bt

239

exists if and only if the conditions

abk + b(dk − ak ) − dck = 0, k ∈ I are satisfied. Proof. From ψ ◦ T = T ∗ ◦ ψ we get the equations       a b ak b k ak c k a b =ρ b d c k dk b k dk b d with a constant ρ = 0. If aak +bck = 0 or bbk +ddk = 0 then ρ = 1 and the equation abk + bdk = ak b + ck d remains. If aak + bck = 0 and bbk + ddk = 0 then we see that ρ2 = 1. Only the case ρ = −1 needs to be considered. We obtain the equations aak + bck = 0 bbk + ddk = 0 abk + b(ak + dk ) + dck = 0. A non-trivial solution exists if (ak + dk )(bk ck − ak dk ) = 0. Since det α(k) = 0 we obtain ak + dk = 0. But then α(k)2 = (a2k + bk ck )1. This means that T 2 x = x on Jk which is not possible. Therefore only ρ = 1 remains. Remark. Let I = {1, 2} then the system of two equations abk + b(dk − ak ) − dck = 0, k = 1, 2 is always soluble. This explains the result given in [4]. Note that the degenerate case that B ∗ reduces to a single point carrying Dirac measure is included. This happens for T x = 2x, 0 ≤ x < 12 ; T x = 2x − 1, 12 ≤ x < 1. Then formally we obtain 2y two branches T ∗ y = 2y and T ∗ y = 1−y . Here B ∗ = {0} and h(x) = 1. Remark. Haas [1] constructs invariant measures for a family of Moebius systems. It is easy to verify that in all cases (B, T ) has a natural dual (B ∗ , T ∗ ) which explains the invariant densities given under corollary 2 in [1]. Definition 4. A natural dual system (B ∗ , T ∗ ) which is not differentiably isomorphic to (B, T ) is called an exceptional dual. We will first give an example of an exceptional dual. We consider a special case of Nakada’s continued fractions [3]. Setting √ 1 −1 + 5 , k = [| | + 1 − g] g= 2 x 1 1 B = [g − 1, g]; T x = − k, x > 0; T x = − − k, x < 0 x x 1 2 1 1 1 2 1 B ∗ = [0, ]; T ∗ y = − k, < y < ; T ∗ y = − + k, < y < 2 y 2k + 1 k y k 2k − 1 then we consider the equation ∗ ∗ ψ ◦ T2 ◦ T−3 ◦ T−4 = T2∗ ◦ T−3 ◦ T−4 ◦ ψ.

However, the equation       a b 26 −7 18 5 a b , |ρ| = 1 =ρ b d 11 −3 11 3 b d leads to a = b = d = 0.

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In this paper we will consider in more detail the case I = {1, 2, 3}. To avoid complications with parameters we fix the partition as 0 < 12 < 32 < 1. Note that a different partition would give a Moebius system which is not isomorphic by a c+dt . Moebius map ψ(t) = a+bt 2. Differentiably isomorphic dual systems It is easy to see that on every subinterval [0, 21 ],[ 12 , 32 ],[ 32 , 1] we can choose a fractional linear map depending on one parameter λ, µ, ν say. Furthermore, T restricted to one of these subintervals can be increasing or decreasing. We call T of type (1 , 2 , 3 ) where j = 1 stands for ”increasing” and j = −1 for ”decreasing”. The parameters satisfy the equations 1 det α(1) = λ, 2 det α(2) = µ, 3 det α(3) = ν. By the choice of the parameters λ, µ, ν we have λ > 0, µ > 0, and ν > 0 but due to the fact that no attractive fixed point is allowed some additional restrictions hold e. g. 0 < λ ≤ 1 if 1 = 1 or 1 ≤ ν if 3 = 1. Now let us assume that a natural dual (B ∗ , T ∗ ) exists. Then the branches of T ∗ corresponding to the parameters λ, µ, ν could appear in six possible orders from left to right. More precisely, if Jλ , Jµ , Jν are the intervals such that T ∗ is defined piecewise by the matrices α1∗ , α2∗ , α3∗ then the three intervals Jλ , Jµ , Jν could be arranged in six possible orders, namely λµν, νµλ, λνµ, µνλ, µλν, νλµ. We give a list of the matrices α, β, and β ∗ . 1 = 1      λ 1 − 2λ 1 2λ − 1 1 0 0 1 0 λ 2λ − 1 λ 1 = −1       2 1 2 λ−2 −1 −λ + 2 λ − 2 −1 1 −1 −1 2 2 = 1      2 1 2 3µ − 2 2µ − 1 2 − 3µ 3µ − 2 2µ − 1 1 2µ − 1 −1 2 2 = −1      µ − 2 3 − 2µ 3 2µ − 3 3 2 −2 3 2 µ−2 2µ − 3 µ − 2 3 = 1      3 2 3 ν−3 ν − 2 −ν + 3 ν−3 ν−2 2 ν−2 −2 3 3 = −1      2ν − 1 1 − 3ν 1 3ν − 1 1 1 −1 1 1 2ν − 1 3ν − 1 2ν − 1 Lemma 2. The natural dual (B ∗ , T ∗ ) is differentiably isomorphic to (B, T ) if and only if the following condition C holds.    b 1 d1 − a1 c 1     b 2 d2 − a2 c 2  = 0    b 3 d3 − a3 c 3  Proof. This is clear from Lemma 1.

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Lemma 3. If (B ∗ , T ∗ ) is differentiably isomorphic to (B, T ), then only the orders λµν, νµλ can appear. b+dt a+bt

Proof. Since ψ(t) = is immediate.

is either order preserving or order reversing the assertion

Theorem 3. If the natural dual (B ∗ , T ∗ ) has the order λµν or νµλ, then (B ∗ , T ∗ ) is differentiably isomorphic to (B, T ). In other words: No exceptional dual exists with orders λµν or νµλ. Proof. The proof considers all types (1 , 2 , 3 ). It first lists the form of condition C and then the ’boundary conditions’ which means that the three maps of T ∗ fit together to form a Moebius system. Type (1, 1, 1)

2λµ + 2µ = λν + λ Vλ∗ ξ = ξ, Vµ∗ ξ = Vλ∗ σ, Vν∗ ξ = Vµ∗ σ, σ = Vν∗ σ. Then we find 2λ − 1 + λξ = ξ which gives 2λ − 1 = ξ(1 − λ). Furthermore σ=

ν − 3 + (ν − 2)σ 3 + 2σ

2σ 2 + σ(−ν + 5) − ν + 3 = 0. If σ = −1 then Vµ∗ σ = µ − 1, Vλ∗ σ = λ − 1, Vµ∗ ξ = λµ + µ − 1, Vν∗ ξ = λν−1−λ λ+1 . Hence we obtain the equations λµ + µ = λ, λµ + µ = λν which show ν = 1 and 2λµ + 2µ = λν + λ. Therefore we concentrate on σ = ν−3 2 . ∗ Then we calculate Vµ ξ = µ + λµ − 1,Vλ∗ σ = λ+λν−2 from which we again arrive at 2 2λµ + 2µ = λν + λ. Type (1, 1, −1) λµ + µ = λν + λ + ν Vλ∗ ξ = ξ, Vµ∗ ξ = Vλ∗ β, Vν∗ β = Vµ∗ β, Vν∗ ξ = β. We find again 2λ − 1 = ξ(1 − λ). Then we calculate Vµ∗ ξ = µ + λµ − 1, Vν∗ ξ = λν+ν−λ = β,Vλ∗ β = λν + λ + ν − 1 and we see that µλ + µ − 1 = νλ + λ + ν − 1 λ which is condition C. Type (1, −1, −1) 2λν + λ + 2ν = µ Vλ∗ ξ = ξ, Vµ∗ ξ = Vν∗ β, Vν∗ ξ = β, Vλ∗ β = Vµ∗ β. We find again 2λ − 1 = ξ(1 − λ). Further we calculate Vν∗ ξ = 2λν+2ν−1 ,Vµ∗ ξ = µ−λ−1 λ+1 λ+1 . This shows µ − λ = 2λν + 2ν.

λν+ν−λ λ

= β,Vν∗ β =

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Type (1, −1, 1) λν = µ Vλ∗ ξ = ξ, Vµ∗ ξ = Vν∗ ξ, Vλ∗ σ = Vµ∗ σ, σ = Vν∗ σ. As before 2λ − 1 = ξ(1 − λ) and further calculations show Vµ∗ ξ = −λ+µ−1 λ+1 , −λ+λν−1 ∗ ∗ ∗ Vν ξ = hence λν = µ. It is easy to see that Vλ σ = Vµ σ gives the same λ+1 condition. Note that σ = −1 corresponds to ν = 1. Type (−1, 1, −1) 2λµ + µ = 2λν + λ Vλ∗ β = α, Vµ∗ α = Vλ∗ α, Vµ∗ β = Vν∗ β, β = Vν∗ α. Then α = Vλ∗ Vν∗ α and we obtain α2 (1 + 2ν) + α(5ν + 2 − λ) + 3ν − λ + 1 = 0. −3ν+λ−1 then Vλ∗ α = −1+2λν−ν and Therefore α = −1 or α = −3ν+λ−1 1+2ν . If α = 1+2ν 1+λ+ν −1−ν−λ+µ+2µλ ∗ which gives immediately condition C. If α = −1 then β = ∞ Vµ α = 1+λ+ν and an easy calculation gives Vλ∗ α = λ−1, Vµ∗ α = µ−1,Vν∗ β = 2ν −1, Vµ∗ β = 2µ−1 which shows λ = µ = ν. Type (−1, 1, 1)

4µλ + µν + µ = λν + ν Vλ∗ σ = α, Vµ∗ α = Vλ∗ α, Vµ∗ σ = Vν∗ α, Vν∗ σ = σ. −1 ∗ ∗ As before σ = −1 or σ = ν−3 2 . If σ = −1 then α = λ − 1, Vλ α = 1+λ ,Vµ α = µ−1−λ+2µλ ∗ , Vν∗ α = λν−2λ−1 1+λ 1+2λ , Vµ σ = µ − 1. Hence µ + 2λµ = λ, µ + 2λµ = λν which implies ν = 1 and C is satisfied. If σ = ν−3 2 then calculation gives

α=

4λµ + µν + µ λν + λ 2λ − ν − 1 ∗ , Vµ α = − 1, Vλ∗ α = − 1, ν+1 2λ + ν + 1 2λ + ν + 1

2µν 2λν − 1, Vµ∗ σ = −1 4λ + ν + 1 ν+1 hence λν + λ = µν + 4λµ + µ. Vν∗ α =

Type (−1, −1, 1)

2λν = 2λµ + µν + µ Vλ∗ σ = α, Vµ∗ σ = Vλ∗ α, Vµ∗ α = Vν∗ α, Vν∗ σ = σ. −1 , Vµ∗ σ = µ − 1, Vµ∗ α = If σ = −1 then α = λ − 1. Calculation shows Vλ∗ α = 1+λ λµ−2λ+µ−1 , Vν∗ α = λν−2λ−1 2λ+1 2λ+1 . This gives λ = µ + λµ, λν = µ + λµ and ν = 1 as expected. Condition C is satisfied. the a similar calculation shows again that condition C is Now suppose σ = ν−3 2

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satisfied. Type (−1, −1, −1)

4λν + λ + ν = µν + λµ + µ Vλ∗ β

= α, Vµ∗ β = Vλ∗ α, Vµ∗ α = Vν∗ β, Vν∗ α = β.

Since α = Vλ∗ Vν∗ α, we obtain α2 (1 + 2ν) + α(5ν − λ + 2) + 3ν − λ + 1 = 0. ∗ ∗ If α = −1 then β = ∞,Vλ∗ α = λ − 1,Vµ∗ β = µ−2 2 , Vµ α = µ − 1,Vν β = 2ν − 1. therefore ν = λ and µ = 2λ. Condition C is satisfied. Now let α = λ−3ν−1 2ν+1 . then (λ − ν)β = 2ν + 2νλ − λ. Furthermore we obtain

Vλ∗ α = −1 +

λµ + 2λµν 2λν + λ , Vµ∗ β = −1 + 1+λ+ν λ + ν + 4λν

ν + 4λν − λ − 1 −1 + µ + µν + λµ − 2λ ∗ , Vν β = 2λ + 1 2λ + 1

Vµ∗ α =

µ 1 = λ+ν+4λν and from Vµ∗ α = Vν∗ β we get From Vµ∗ β = Vλ∗ α we obtain 1+λ+ν −1 + µ + µν + λµ − 2λ = ν + 4λν − λ − 1. Both equations lead to condition C.

2. Exceptional dual systems Exceptional dual systems exist for some orders. A full discussion of all possible cases is not only lengthy but also of limited value since the method is similar in all cases. Therefore we give some examples for existence and non-existence of such systems. Type (1, 1, −1) Orders λνµ and µνλ ξ = Vλ∗ ξ, Vλ∗ γ = Vν∗ γ, Vν∗ ξ = Vµ∗ ξ, γ = Vµ∗ γ We find ξ(1 − λ) = 2λ − 1 as before. Then Vν∗ ξ = ν+λν−λ , Vµ∗ ξ = µλ + µ − 1. This λ gives the equation ν + νλ = µλ2 + µλ and eventually ν = µλ. From Vµ∗ γ = γ we find that γ 2 + γ(3 − 2µ) − 3µ + 2 = 0. We use Vλ∗ γ = 2λ − 1 + λγ and Vλ∗ =

3ν−1+(2ν−1)γ 1+γ

and find the equation

γ 2 λ + γ(3λ − 2ν) + 2λ − 3ν = 0. Comparing both equations we find again ν = λµ. Type (1, 1, −1) Orders µλν and νλµ γ = Vµ∗ γ, Vλ∗ γ = Vµ∗ β, Vλ∗ β = Vν∗ β, Vν∗ γ = β

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Again we obtain

γ 2 + γ(3 − 2µ) − 3µ + 2 = 0.

Therefore γ = −1. From Vν∗ γ = β we obtain β = −1 + 3ν+2νγ 1+γ . The equation Vλ∗ γ = Vµ∗ β gives β(2µ−2λ−λγ) = −3µ+4λ+2λγ. If 2µ−2λ−λγ = 0 then −3µ + 4λ + 2λγ = 0 which gives µ = 0, a contradiction. Therefore (3ν − 1 + γ(2ν − 1))(2µ − 2λ − λγ) = (1 + γ)(−3µ + 4λ + 2λγ) which gives the equation γ 2 (λ + 2λν) + γ(3λ − µ + 7λν − 4µν) − µ + 2λ − 6µν + 6λν = 0. Therefore 3 − 2µ =

3λ − µ + 7λν − 4µν −µ + 2λ − 6µν + 6λν , −3µ + 2 = . λ + 2λν λ + 2λν

From this we get 4λµν + λν − 4µν + 2λµ − µ = 0, 6λµν + 2λν − 6µν + 3λµ − µ = 0. Eliminating λµν we obtain λν + µ = 0 which is a contradiction since λ, µ, ν > 0. Therefore no exceptional system exists with this configuration. If one uses Vλ∗ β = Vν∗ β one gets the equation β 2 λ + β(3λ − 2ν) + 2λ − 3ν = 0. ν If we insert γ = −1 + β−2ν+1 in the quadratic equation for γ we obtain a second equation for β which leads again to µ + νλ = 0.

Let us give another example of an exceptional dual system with an unforeseen complicated condition. Type (1, 1, 1) Orders λνµ and µνλ ξ = Vλ∗ ξ, Vν∗ ξ = Vλ∗ γ, Vν∗ γ = Vµ∗ ξ, γ = Vµ∗ γ We find again ξ(1 − λ) = 2λ − 1 and γ 2 + γ(3 − 2µ) − 3µ + 2 = 0. Note that γ = Calculation gives λν − λ − 1 ∗ Vν∗ ξ = , Vλ γ = 2λ − 1 + λγ λ+1 ν − 3 + γ(ν − 2) . Vµ∗ ξ = λµ + µ − 1, Vν∗ γ = 3 + 2γ Equating Vν∗ ξ = Vλ∗ γ gives γ = −1 + λµ+µ 2µλ+2µ−ν . From this the condition

ν−λ−1 λ+1

−3 2 .

and Vν∗ γ = Vµ∗ ξ gives γ = −1 −

2λµν + 2µν + λν + ν = ν 2 + 2µλ + µλ2 + µ follows. If we use (γ + 1)2 + (γ + 1)(1 − 2µ) − µ = 0 and substitute γ + 1 = ν−λ−1 λ+1 we 1 4 get the same condition. As the example λ = 2 , µ = 15 , ν = 2 shows this condition

Differentiable equivalence

245

can be satisfied easily. Type (1, 1, 1) Orders νλµ and µλν σ = Vν∗ σ, Vλ∗ σ = Vν∗ γ, Vµ∗ σ = Vλ∗ γ, γ = Vµ∗ γ We get γ 2 + γ(3 − 2µ) − 3µ + 2 = 0. From Vλ∗ σ = Vν∗ γ and Vµ∗ σ = Vλ∗ γ we calculate σ = −2 +

ν + νγ µ , σ = −2 + 3λ + 2λγ 2µ − 2λ − λγ

and eventually we get γ 2 λν + γ(3λν + 2λµ − 2µν) + 3λµ − 2µν + 2λν = 0. Therefore comparing the coefficients of the quadratic equations for γ we get 3−2µ = , 2 − 3µ = 2 + 3λµ−2µν which shows µν = 0, a contradiction. 3 + 2λµ−2µν λν λν Type (1, −1, −1) Orders νλµ and µλν δ = Vµ∗ β, Vλ∗ δ = Vµ∗ δ, Vν∗ β = Vλ∗ β, β = Vν∗ δ Therefore Vν∗ Vµ∗ β = β which gives the quadratic equation β 2 µ + β(3µ − 2νµ − 2ν) − 4νµ − 3ν + 2µ = 0. The equation Vν∗ β = Vλ∗ β gives β 2 λ + β(3λ − 2ν) + 2λ − 3ν = 0. Therefore 3µ − 2νµ − 2ν 3λ − 2ν −4νµ − 3ν + 2µ 2λ − 3ν = , = µ λ µ λ and then νµλ + νλ = µν, 4νµλ + 3νλ = 3µν. Hence νµλ = 0, a contradiction. Type (1, −1, −1) Orders λνµ and µνλ ξ = Vλ∗ ξ, Vλ∗ δ = Vν∗ δ, Vν∗ ξ = Vµ∗ δ, δ = Vµ∗ ξ ∗ We note ξ(1 − λ) = 2λ − 1, Vν∗ ξ = −1 + ν+λν λ , Vµ ξ = δ = −1 + Vλ∗ δ = Vν∗ δ gives δ 2 λ + δ(3λ − 2ν) + 2λ − 3ν = 0.

If one inserts δ = −1 +

µ λ+1

µ λ+1 .

we get

µ2 λ + µλ2 + µλ = 2νµλ + νλ2 + 2νµ + 2λν + ν.

The equation

F. Schweiger

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2

+µ ν+νλ ∗ ∗ We further calculate Vµ∗ δ = −1 + λµ+µ λ+2µ+1 . The equation Vν ξ = Vµ δ = −1 + λ gives the same condition. Therefore exceptional systems exist. An example is given by λ = 1, µ = 4, ν = 65 .

Type (1, −1, 1) Orders λνµ and µνλ ξ = Vλ∗ ξ, Vλ∗ δ = Vν∗ ξ, Vν∗ δ = Vµ∗ δ, δ = Vµ∗ ξ µ . The condition Vν∗ δ = Vµ∗ δ gives Calculation gives ξ(1 − λ) = 2λ − 1, δ = −1 + λ+1 ∗ ∗ ν = λ+µ+1 but the condition Vλ δ = Vν ξalso leads to λ2 +λµ+λ = λν. Since λ > 0 this is an equivalent condition. Examples are easy to find (λ = 1, µ = 1, ν = 3).

Type (1, −1, 1) Orders µλν and νλµ Vµ∗ δ = Vλ∗ δ, Vµ∗ σ = δ, Vν∗ δ = Vλ∗ σ, σ = Vν∗ σ or σ = −1 (which is formally included for ν = 1) and δ = We obtain σ = ν−3 2 µν+µ −1 + 2ν . The equation Vν∗ δ = Vλ∗ σ gives λ(ν + 1)(ν + µ + µν) = µν(ν + 1). Since ν + 1 > 0 we obtain λ(ν + µ + µν) = µν. The equation Vµ∗ δ = Vλ∗ δ leads to the same condition. An example is given by λ = 41 , µ = 12 , ν = 1. 3. Closing remarks The question remains what can be said about a suitable dual system (B ∗ , T ∗ ) (see [4] for the general notion of a dual system) if no natural dual system exists. The following conjecture seems reasonable. There is a closed set B ∗ such that (B ∗ , T ∗ ) is a fibred system where T ∗ is piecewise defined by the transposed matrices. Generally, the set B ∗ supports a 1-conformal measure (see e.g. [2] and [6]). This property is based on the fact that the equation 

k1 ,...,ks

1 ω ∗ (k1 , . . . , ks ; y) = ∗ 1 + V (k1 , . . . , ks )y 1+y

holds. In the case of a dual Moebius system B ∗ is an interval and this measure clearly is Lebesgue measure. However, there exist examples where B ∗ is a union of infinitely many intervals. We give one such example. B = [0, 1] and

B∗ =

∞

k=0 ]2k, 2k

 1 x   1−2x , if 0 ≤ x < 3 , 1 1 T x = 1−2x x , if 3 ≤ x < 2 ,   1−x if 21 ≤ x < 1, x ,

+ 1] and  ∞  −y + 2, if y ∈ k=1 ]2k, 2k + 1], ∞ 1 1 if y ∈ k=0 ] 2k+2 , 2k+1 ], T y = 1−y y ,  ∞  1−2y 1 1 if y ∈ k=0 ] 2k+3 , 2k+2 ]. y ,

Differentiable equivalence

247

Therefore the invariant density for T is given as h(x) =

∞ 

k=0

(

2k + 1 2k − ). 1 + (2k + 1)x 1 + 2kx

I want to express my sincere thanks to the referee whose advice was very helpful. References [1] Haas, A. (2002). Invariant measures and natural extensions. Canad. Math. Bull. 45, 1, 97–108. MR1884139 ´ski, M. (1996). Dimensions and measures in [2] Mauldin, R. D. and Urban infinite iterated function systems. Proc. London Math. Soc. (3) 73, 1, 105–154. MR1387085 [3] Nakada, H. (1981). Metrical theory for a class of continued fraction transformations and their natural extensions. Tokyo J. Math. 4, 2, 399–426. MR646050 [4] Schweiger, F. (1983). Invariant measures for piecewise linear fractional maps. J. Austral. Math. Soc. Ser. A 34, 1, 55–59. MR683178 [5] Schweiger, F. (2000) Multidimensional Continued Fractions. Oxford University Press, Oxford. MR2121855 ´ski, M. (2003). Measures and dimensions in conformal dynamics. Bull. [6] Urban Amer. Math. Soc. (N.S.) 40, 3, 281–321 (electronic). MR1978566

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 248–251 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000266

Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem Michael Keane1 and Karl Petersen2 Wesleyan University and University of North Carolina Abstract: We give a short proof of a strengthening of the Maximal Ergodic Theorem which also immediately yields the Pointwise Ergodic Theorem.

Let (X, B, µ) be a probability space, T : X → X a (possibly noninvertible) measurepreserving transformation, and f ∈ L1 (X, B, µ). Let k−1 1
j Ak f = fT , k j=0

∗ fN = sup Ak f, 1≤k≤N

∗ f ∗ = sup fN , N

and A = lim sup Ak f. k→∞

When λ is a constant, the following result is the Maximal Ergodic Theorem. Choosing λ = A −  covers most of the proof of the Ergodic Theorem. Theorem. Let λ be an invariant (λ ◦ T = λ a.e.) function on X with λ+ ∈ L1 . Then  (f − λ) ≥ 0. {f ∗ >λ}

Proof. We may assume that λ ∈ L1 {f ∗ > λ}, since otherwise  (f − λ) = ∞ ≥ 0. {f ∗ >λ}

But then actually λ ∈ L1 (X), since on {f ∗ ≤ λ} we have f ≤ λ, so that on this set λ− ≤ −f + λ+ , which is integrable. Assume first that f ∈ L∞ . Fix N = 1, 2, . . . , and let ∗ EN = {fN > λ}.

Notice that (f − λ)χEN ≥ (f − λ), since x ∈ / EN implies (f − λ)(x) ≤ 0. Thus for a very large m  N , we can break up m−1
(f − λ)χEN (T k x) k=0

1 Department

of Mathematics, Wesleyan University, Middletown, CT 06457, USA, e-mail: [email protected] 2 Department of Mathematics, CB 3250, Phillips Hall, University of North Carolina, Chapel Hill, NC 27599 USA, e-mail: [email protected] AMS 2000 subject classifications: primary 37A30; secondary 37A05. Keywords and phrases: maximal ergodic theorem, pointwise ergodic theorem. 248

Proofs of the Ergodic Theorem and Maximal Ergodic Theorem

249

into convenient strings of terms as follows. There is maybe an initial string of 0’s during which T k x ∈ / EN . Then there is a first time k when T k x ∈ EN , which initiates a string of no more than N terms, the sum of which is positive (using on each of these terms the fact that (f − λ)χEN ≥ (f − λ)). Beginning after the last term in this string, we repeat the previous analysis, finding maybe some 0’s until again some T k x ∈ EN initiates another string of no more than N terms and with positive sum. The full sum of m terms may end in the middle of either of these two kinds of strings (0’s, or having positive sum). Thus we can find j = m−N +1, . . . , m such that m−1


(f − λ)χEN (T k x) ≥

m−1


(f − λ)χEN (T k x) ≥ −N (f ∞ + λ+ (x)).

k=j

k=0

Integrating both sides, dividing by m, and letting m → ∞ gives  m (f − λ) ≥ −N (f ∞ + λ+ 1 ), E  N −N (f ∞ + λ+ 1 ), (f − λ) ≥ m EN  (f − λ) ≥ 0. EN

Letting N → ∞ and using the Dominated Convergence Theorem concludes the proof for the case f ∈ L∞ . To extend to the case f ∈ L1 , for s = 1, 2, . . . let φs = f · χ{|f |≤s} , so that φs ∈ L∞ and φs → f a.e. and in L1 . Then for fixed N ∗ (φs )∗N → fN

a.e. and in L1

Therefore 0≤



∗ µ ({(φs )∗N > λ} {fN > λ}) → 0.

and

(φs − λ) → {(φs )∗ >λ} N



(f − λ), ∗ >λ} {fN

again by the Dominated Convergence Theorem. The full result follows by letting N → ∞. Corollary (Ergodic Theorem). The sequence (Ak f ) converges a.e. Proof. It is enough to show that 

A≤



f.

For then, letting A = lim inf Ak f , applying this to −f gives   − A ≤ − f, so that

and hence

 

A≤



f≤

(A − A) = 0,



A≤



A,

so that A = A a.e.

250

M. Keane and K. Petersen

Consider first f + and its associated A, denoted by A(f + ). For any invariant function λ < A(f + ) such that λ+ ∈ L1 , for example λ = A(f + ) ∧ n − 1/n, we have {(f + )∗ > λ} = X, so the Theorem gives    + f ≥ λ  A(f + ). Thus (A)+ ≤ A(f + ) is integrable (and, by a similar argument, so is (A)− ≤ A(f − ).) Now let  > 0 be arbitrary and apply the Theorem to λ = A −  to conclude that    f ≥ λ  A. Remark. This proof may be regarded as a further development of one given in a paper by Keane [10], which has been extended to deal also with the Hopf Ratio Ergodic Theorem [8] and with the case of higher-dimensional actions [11], and which was itself a development of the Katznelson-Weiss proof [9] based on Kamae’s nonstandard-analysis proof [7]. (It is presented also in the Bedford-Keane-Series collection [1].) Our proof yields both the Pointwise and Maximal Ergodic Theorems essentially simultaneously without adding any real complications. Roughly contemporaneously with this formulation, Roland Zweim¨ uller prepared some preprints [21, 22] also giving short proofs based on the Kamae-Katznelson-Weiss approach, and recently he has also produced a simple proof of the Hopf theorem [23]. Without going too deep into the complicated history of the Ergodic Theorem and Maximal Ergodic Theorem, it is interesting to note some recurrences as the use of maximal theorems arose and waned repeatedly. After the original proofs by von Neumann [18], Birkhoff [2], and Khinchine [12], the role and importance of the Maximal Lemma and Maximal Theorem were brought out by Wiener [19] and Yosida-Kakutani [20], making possible the exploration of connections with harmonic functions and martingales. Proofs by upcrossings followed an analogous pattern. It also became of interest, for instance to allow extension to new areas or new kinds of averages, again to prove the Ergodic Theorem without resort to maximal lemmas or theorems, as in the proof by Shields [16] inspired by the Ornstein-Weiss proof of the ShannonMcMillan-Breiman Theorem for actions of amenable groups [14], or in Bourgain’s proofs by means of variational inequalities [3]. Sometimes it was pointed out, for example in the note by R. Jones [6], that these approaches could also with very slight modification prove the Maximal Ergodic Theorem. Of course there are the theorems of Stein [17] and Sawyer [15] that make the connection explicit, just as the transference techniques of Wiener [19] and Calder´ on [4] connect ergodic theorems with their analogues in analysis like the Hardy-Littlewood Maximal Lemma [5]. In many of the improvements over the years, ideas and tricks already in the papers of Birkhoff, Kolmogorov [13], Wiener, and Yosida-Kakutani have continued to play an essential role. Acknowledgment This note arose out of a conversation between the authors in 1997 at the Erwin Schr¨ odinger International Institute for Mathematical Physics in Vienna, and we thank that institution for its hospitality. Thanks also to E. Lesigne and X. M´ela for inducing several clarifications.

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References [1] Bedford, T., Keane, M., and Series, C. (1991). Ergodic Theory, Symbolic Fynamics, and Hyperbolic Spaces. Oxford Science Publications. The Clarendon Press Oxford University Press, New York. MR1130170 [2] Birkhoff, G. D. (1931). Proof of the ergodic theorem, Proc. Nat. Acad. Sci. U.S.A. 17, 656–660. [3] Bourgain, J. (1988). On the maximal ergodic theorem for certain subsets of the integers. Israel J. Math. 61, 1, 39–72. MR937581 ´ n, A.-P. (1968). Ergodic theory and translation-invariant operators. [4] Caldero Proc. Nat. Acad. Sci. U.S.A. 59, 349–353. MR227354 [5] Hardy, G.H., and Littlewood, J.E., (1930). A maximal theorem with function-theoretic applications, Acta Math. 54, 81–116. [6] Jones, R. L. (1983). New proofs for the maximal ergodic theorem and the Hardy-Littlewood maximal theorem. Proc. Amer. Math. Soc. 87, 4, 681–684. MR687641 [7] Kamae, T. (1982). A simple proof of the ergodic theorem using nonstandard analysis. Israel J. Math. 42, 4, 284–290. MR682311 [8] Kamae, T. and Keane, M. (1997). A simple proof of the ratio ergodic theorem. Osaka J. Math. 34, 3, 653–657. MR1613108 [9] Katznelson, Y. and Weiss, B. (1982). A simple proof of some ergodic theorems. Israel J. Math. 42, 4, 291–296. MR682312 [10] Keane, M. (1995). The essence of the law of large numbers. In Algorithms, Fractals, and Dynamics (Okayama/Kyoto, 1992). Plenum, New York, 125–129. MR1402486 [11] Keane, M., unpublished. [12] Khinchine, A.I., (1933). Zu Birkhoffs L¨ osung des Ergodenproblems, Math. Ann. 107, 485–488. ¨ [13] Kolmogorov, A. (1928). Uber die Summen durch den Zufall bestimmter unabh¨ angiger Gr¨ ossen, Math. Ann. 99, 309–319. [14] Ornstein, D. and Weiss, B. (1983). The Shannon–McMillan–Breiman theorem for a class of amenable groups. Israel J. Math. 44, 1, 53–60. MR693654 [15] Sawyer, S. (1966). Maximal inequalities of weak type. Ann. of Math. (2) 84, 157–174. MR209867 [16] Shields, P. C. (1987). The ergodic and entropy theorems revisited. IEEE Trans. Inform. Theory 33, 2, 263–266. MR880168 [17] Stein, E. M. (1961). On limits of seqences of operators. Ann. of Math. (2) 74, 140–170. MR125392 [18] von Neumann, J., (1932). Proof of the quasi-ergodic hypothesis, Proc. Nat. Acad. Sci. U.S.A. 18, 70–82. [19] Wiener, N. (1939). The ergodic theorem, Duke Math. J. 5, 1–18. [20] Yosida, K. and Kakutani, S. (1939). Birkhoff’s ergodic theorem and the maximal ergodic theorem. Proc. Imp. Acad., Tokyo 15, 165–168. MR355 ¨ller, R. (1997). A simple proof of the strong law of large numbers, [21] Zweimu Preprint, see http://www.mat.sbg.ac.at/staff/zweimueller/. ¨ller, [22] Zweimu R. (1997). A unified elementary approach to some pointwise ergodic theorems, Preprint, May 1997, see http://www.mat.sbg.ac.at/staff/zweimueller/. ¨ller, R. (2004). Hopf’s ratio ergodic theorem by inducing. Colloq. [23] Zweimu Math. 101, 2, 289–292. MR2110730

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 252–261 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000275

Purification of quantum trajectories Hans Maassen1 and Burkhard K¨ ummerer2 Radboud University Nijmegen and Technische Universit¨ at Darmstadt Abstract: We prove that the quantum trajectory of repeated perfect measurement on a finite quantum system either asymptotically purifies, or hits upon a family of ‘dark’ subspaces, where the time evolution is unitary.

1. Introduction A key concept in the modern theory of open quantum systems is the notion of indirect measurement as introduced by Kraus [5]. An indirect measurement on a quantum system is a (direct) measurement of some quantity in its environment, made after some interaction with the system has taken place. When we make such a measurement, our description of the quantum system changes in two ways: we account for the flow of time by a unitary transformation (following Schr¨ odinger), and we update our knowledge of the system by conditioning on the measurement outcome (following von Neumann). If we then repeat the indirect measurement indefinitely, we obtain a chain of random outcomes. In the course of time we may keep record of the updated density matrix Θt , which at time t reflects our best estimate of all observable quantities of the quantum system, given the observations made up to that time. This information can in its turn be used to predict later measurement outcomes. The stochastic process Θt of updated states, is the quantum trajectory associated to the repeated measurement process. By taking the limit of continuous time, we arrive at the modern models of continuous observation: quantum trajectories in continuous time satisfying stochastic Schr¨ odinger equations [3], [4], [2], [1]. These models are employed with great success for calculations and computer simulations of laboratory experiments such as photon counting and homodyne field detection. In this paper we consider the question, what happens to the quantum trajectory at large times. We do so only for the case of discrete time, not a serious restriction indeed, since asymptotic behaviour remains basically unaltered in the continuous time limit. We focus on the case of perfect measurement, i.e. the situation where no information flows into the system, and all information which leaks out is indeed observed. In classical probability such repeated perfect measurement would lead to a further and further narrowing of the distribution of the system, until it either becomes pure, i.e. an atomic measure, or it remains spread out over some area, thus leaving a certain amount of information ‘in the dark’ forever. Using a fundamental inequality of Nielsen [7] we prove that in quantum mechanics the situation is quite comparable: the density matrix tends to purify, until it hits upon some family of 1 Radboud University Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands, e-mail: [email protected] 2 Technische Universitt Darmstadt, Fachbereich Mathematik, Schloßgartenstr. 7, 64289 Darmstadt, Germany, e-mail: [email protected] AMS 2000 subject classifications: primary 82C10; secondary 60H20, 47N50. Keywords and phrases: quantum trajectory, stochastic Schr¨ odinger equation.

252

Quantum trajectories

253

‘dark’ subspaces, if such exist, i.e. spaces from which no information can leak out. A crucial difference with the classical case is, however, that even after all available information has been extracted by observation, the state continues to move about in a random fashion between the ‘dark’ subspaces, thus continuing to produce ‘quantum noise’. The structure of this paper is as follows. In Section 2 we introduce quantum measurement on a finite system, in particular Kraus measurement. In Section 3 repeated measurement and the quantum trajectory are introduced, and in Section 4 we prove our main result. Some typical examples of dark subspaces are given in Section 5. 2. A single measurement Let A be the algebra of all complex d×d matrices. By S we denote the space of d×d density matrices, i.e. positive matrices of trace 1. We think of A as the observable algebra of some finite quantum system, and of S as the associated state space. A measurement on this quantum system is an operation which results in the extraction of information while possibly changing its state. Before the measurement the system is described by a prior state θ ∈ S, and afterwards we obtain a piece of information, say an outcome i ∈ {1, 2, . . . , k}, and the system reaches some new (or posterior) state θi
: θ −→ (i, θi
). Now, a probabilistic theory, rather than predicting the outcome i, gives a probability distribution (π1 , π2 , . . . , πk ) on the possible outcomes. Let Ti : θ → πi θi
,

(i = 1, . . . , k).

(1)

Then the operations Ti , which must be completely positive, code for the probabilities πi = tr(Ti θ) of the possible outcomes, as well as for the posterior states θi
= Ti θ/tr(Ti θ), conditioned on these outcomes. The k-tuple (T1 , . . . , Tk ) describes the quantum measurement completely. Its mean effect on the system, averaged over all possible outcomes, is given by the trace-preserving map T : θ →

k
i=1

πi θi
=

k


Ti θ.

i=1

Example 1 (von Neumann measurement). Let p1 , p2 , . . . pk be mutually orthogonal projections in A adding up to 1, and let a ∈ A be a self-adjoint matrix whose eigenspaces are the ranges of the pi . Then according to von Neumann’s projection postulate a measurement of a is obtained by choosing for Ti the operation Ti (θ) = pi θpi . Example 2 (Kraus measurement). The following indirect measurement procedure was introduced by Karl Kraus [5]. It contains von Neumann’s measurement as an ingredient, but is considerably more flexible and realistic. Our quantum system A in the state θ is brought into contact with a second system, called the ‘ancilla’, which is described by a matrix algebra B in the state β. The two systems interact for a while under Schr¨ odinger’s evolution, which results in a rotation over a unitary u ∈ B ⊗ A. Then the ancilla is decoupled again, and

H. Maassen and B. K¨ ummerer

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is subjected to a von Neumann measurement given by the orthogonal projections p1 , . . . , pk ∈ B. The outcome of this measurent contains information about the system, since system and ancilla have become correlated during their interaction. In order to assess this information, let us consider an event in our quantum system, described by a projection q ∈ A. Since each of the projections pi ⊗1 commutes with 1 ⊗ q, the events of seeing outcome i and then the occurrence of q are compatible, so according to von Neumann we may express the probability for both of them to happen as:    ∗ P[outcome i and then event q] = tr ⊗ tr u(β ⊗ θ)u (pi ⊗ q) . Therefore the following conditional probability makes physical sense.    tr ⊗ tr u(β ⊗ θ)u∗ (pi ⊗ q)  . P[event q|outcome i] =  tr ⊗ tr u(β ⊗ θ)u∗ (pi ⊗ 1) This expression, which describes the posterior probability of any event q ∈ A, can be considered as the posterior state of our quantum system, conditioned on the measurement of an outcome i on the ancilla, even when no event q is subsequently measured. As above, let us therefore call this state θi
. We then have   tr (Ti θ)q
, tr(θi q) = tr(Ti θ) where Ti θ takes the form    Ti θ = tr ⊗ id u(β ⊗ θ)u∗ (pi ⊗ 1) . Here, id denotes the identity map S → S. The expression for Ti takes a simple form in the case which will interest us here, namely when the following three conditions are satisfied: (i) B consists of all k × k-matrices for some k; (ii) the orthogonal projections pi ∈ B are one-dimensional (say pi is the matrix with i-th diagonal entry 1, and all other entries 0); (iii) β is a pure state (say with state vector (β1 , . . . , βk ) ∈ Ck ). These conditions have the following physical interpretations. (i) The ancilla is purely quantummechanical; (ii) the measurement discriminates maximally; (iii) no new information is fed into the system. If these conditions are satisfied, u can be written as a k × k matrix (uij ) of d × d matrices, and Ti may be written Ti θ = ai θa∗i , where ai =

k
j=1

βj uij .

(2)

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We note that, by construction, k


a∗i ai =

k k
k



βj u∗ji uij
βj
= β2 · 1.

i=1 j=1 j
=1

i=1

This basic rule expresses the preservation of the trace by T . Definition 1. By a perfect measurement on A we shall mean a k-tuple (T1 , . . . , Tk ) k of operations on S, where Ti θ is of the form ai θa∗i with i=1 a∗i ai = 1.

Mathematically speaking, the measurement (T1 , . . . , Tk ) is perfect iff the Stinespring decomposition of each Ti consists of a single term. We note that every perfect Kraus measurement is a perfect measurement in the above sense, and that every perfect measurement can be obtained as the result of a perfect Kraus measurement. 3. Repeated measurement By repeating a measurement on the quantum system A indefinitely, we obtain a Markov chain with values in the state space S. This is the quantum trajectory which we study in this paper. Let Ω be the space of infinite outcome sequences ω = {ω1 , ω2 , ω3 , . . .}, with ωj ∈ {1, . . . , k}, and let for m ∈ N and i1 , . . . , im ∈ {1, . . . , k} the cylinder set Λi1 ,...,im ⊂ Ω be given by Λi1 ,...,im := {ω ∈ Ω | ω1 = i1 , . . . , ωm = im }. Denote by Σm the Boolean algebra generated by these cylinder sets, and by Σ the σ-algebra generated by all these Σm . Let T1 , . . . , Tk be as in Section 2. Then for every initial state θ0 on A there exists a unique probability measure Pθ0 on (Ω, Σ) satisfying Pθ0 (Λi1 ,...,im ) = tr(Tim ◦ · · · ◦ Ti1 (θ0 )). Indeed, according to the Kolmogorov–Daniell reconstruction theorem we only k need to check consistency: since T = i=1 Ti preserves the trace, k
i=1

k
  tr Ti ◦ Tim ◦ · · · ◦ Ti1 (θ0 ) = tr(T ◦ Tim ◦ · · · ◦ Ti1 (θ0 )) Pθ0 (Λi1 ,...,im ,i ) = i=1

    = tr Tim ◦ · · · ◦ Ti1 (θ0 ) = Pθ0 Λi1 ,...,im .

On the probability space (Ω, Σ, Pθ0 ) we now define the quantum trajectory (Θn )n∈N as the sequence of random variables given by Θn : Ω → S : ω →

T ◦ · · · ◦ Tω1 (θ0 ) .  ωm tr Tωm ◦ · · · ◦ Tω1 (θ0 )

We note that Θn is Σn -measurable. The density matrix Θn (ω) describes the state of the system at time n under the condition that the outcomes ω1 , . . . , ωn have been seen. The quantum trajectory (Θn )n∈N is a Markov chain with transitions θ −→ θi
=

Ti θ tr(Ti θ)

with probability tr(Ti θ).

(3)

H. Maassen and B. K¨ ummerer

256

4. Purification In a perfect measurement, when Ti is of the form θ → ai θa∗i , a pure prior state θ = |ψ ψ| leads to a pure posterior state: θi
=

ai |ψ ψ|a∗i = |ψi  ψi |, ψ, a∗i ai ψ

where ψi =

ai ψ . ai ψ

Hence in the above Markov chain the pure states form a closed set. Experience with quantum trajectories leads one to believe that in many cases even more is true: along a typical trajectory the density matrix tends to purify: its spectrum approaches the set {0, 1}. In Markov chain jargon: the pure states form an asymptotically stable set. There is, however, an obvious counterexample √ to this statement in general. If every ai is proportional to a unitary, say ai = λi ui with u∗i ui = 1, then θi
=

ai θa∗i = ui θu∗i ∼ θ, ∗ tr(ai θai )

where ∼ denotes unitary equivalence. So in this case the eigenvalues of the density matrix remain unchanged along the trajectory: pure states remain pure and mixed states remain mixed with unchanging weights. In this section we shall show that in dimension 2 this is actually the only exception. (Cf. Corollary 2.) In higher dimensions the situation is more complicated: if the state does not purify, the ai must be proportional to unitaries on a certain collection of ‘dark’ subspaces, which they must map into each other. (Cf. Corollary 8.) In order to study purification we shall consider the moments of Θn . By the m-th moment of a density matrix θ ∈ S we mean tr (θm ). We note that two states θ and ρ are unitarily equivalent iff all their moments are equal. In dimension d equality of the moments m = 1, . . . , d suffices.   Definition 2. We say that the quantum trajectory Θn (ω) n∈N purifies when   ∀m∈N : lim tr Θn (ω)m = 1. n→∞ Note that the only density matrices ρ satisfying tr(ρm ) = 1 are one-dimensional projections, the density matrices of pure states. In fact, it suffices that the second moment be equal to 1. We now state our main result concerning repeated perfect measurement.   Theorem 1. Let Θn n∈N be the Markov chain with initial state θ0 and transition probabilities (3). Then at least one of the following alternatives holds. (i) The paths of (Θn )n∈N (the quantum trajectories) purify with probability 1, or: (ii) there exists a projection p ∈ A of dimension at least two such that ∀i∈{1,...,k} ∃λi ≥0 :

pa∗i ai p = λi p.

Condition (ii) says that ai is proportional to an isometry in restriction to the range of p. Note that this condition trivially holds if p is one-dimensional. Corollary 2. In dimension d = 2 the quantum trajectory of a repeated perfect measurement either purifies with probability 1, or all the ai ’s are proportional to unitaries.

Quantum trajectories

257

If the ai are all proportional to unitaries, the coupling to the environment is essentially commutative in the sense of [6]. Our proof starts from an inequality of Michael Nielsen [7] to the effect that for all m ∈ N and all states θ: k
i=1

  πi tr (θi
)m ≥ tr(θm ),

where

ai θa∗i . tr(ai θa∗i ) Nielsen’s inequality says that the expected m-th moment of the posterior state is as least as large as the m-th moment of the prior state. In terms of the associated Markov chain we may express this inequality as    m  ∀m,n∈N : E tr Θn+1 Σn ≥ tr (Θm n ), πi := tr(ai θa∗i ) and

θi
:=

(m)

i.e. the moments Mn := tr (Θm n )n∈N are submartingales. Clearly all moments take values in [0, 1]. Therefore, by the martingale convergence theorem they must converge almost surely to some random variables M (m) . This suggests the following line of proof for our theorem: Since the moments converge, the eigenvalues of (Θn )n∈N must converge. Hence along a single trajectory the states eventually become unitarily equivalent, i.e. eventually ∀i :

Θn (ω) ∼

ai Θn (ω)a∗i . tr(ai Θn (ω)a∗i )

But this seems to imply that either Θn purifies almost surely, or the ai ’s are unitary on the support of Θn . In the following proof of Theorem 1 we shall make this suggestion mathematically precise. Lemma 3. In the situation of Theorem 1 at least one of the following alternatives holds. (i) For all m ∈ N: lim tr (Θm n ) = 1 almost surely; n→∞

(ii) there exists a mixed state ρ ∈ S such that ∀i=1,...,k ∃λi ≥0 :

ai ρa∗i ∼ λi ρ.

Proof. For each m ∈ N we consider the continuous function m 2   k
ai θa∗i m ∗ − tr(θ ) . tr(ai θai ) tr δm : S → [0, ∞) : θ → tr(ai θa∗i ) i=1 Then, using (2) and (3),

(m)

 2  (m) (m)  Σn . δm (Θn ) = E Mn+1 − Mn 

Since (Mn )n∈N is a positive submartingale bounded by 1, its increments must be square summable: ∞
  E δm (Θn ) ≤ 1. ∀m∈N : n=0

H. Maassen and B. K¨ ummerer

258

In particular lim

n→∞

d
  E δm (Θn ) = 0.

(4)

m=1

Now let us assume that (i) is not the case, i.e. for some (and hence for all) m ≥ 2 the expectation E(M (m) ) =: µm is strictly less than 1. For any n ∈ N consider the event  

 (2) µ + 1 2 An := ω ∈ Ω  Mn ≤ . 2 (2)

Then, since E(Mn ) is increasing in n, we have for all n ∈ N: E(Mn(2) ) ≤ E(M (2) ) = µ2 < 1. Therefore for all n ∈ N,   (2) µ2 ≥ E Mn · 1[M (2) > µ2 +1 ] n 2

µ µ2 + 1 2+1 (2) P Mn > ≥ 2 2 µ2 + 1 = (1 − P(An )) , 2 so that

1 − µ2 . (5) 1 + µ2 On the other hand, An is Σn -measureable and therefore it is a union of sets of the from Λi1 ,...,in . Since Θn is Σn -measureable, Θn is constant on such sets; let us call the constant Θn (i1 , . . . , in ). We have the following inequality:   d

  1 P (Λi1 ,...,in ) δm Θn (i1 , . . . , in ) P(An ) m=1 P(An ) ≥

Λi1 ,...,in ⊂An

d
  1 ≤ E δm (Θn ) . P(An ) m=1

On the dleft hand side we have an average of numbers which are each of the form m=1 δm (Θn (i1 , . . . , in )), hence we can choose (i1 , . . . , in ) such that ρn := Θn (i1 , . . . , in ) satisfies, by (5), d  µ2 + 1
 E δm (Θn ) . δm (ρn ) ≤ µ2 − 1 m=1 m=1 d


Since Λi1 ,...,in ⊂ An , the sequence (ρn )n∈N lies entirely in the compact set  

 µ2 + 1 2  θ ∈ S  tr(θ ) ≤ . 2

Let ρ be a cluster point of this sequence. Then, since E(δm (Θn )) tends to 0 as n → ∞, and δm is continuous, we may conclude that for m = 1, . . . , d: δm (ρ) = 0,

and

tr(ρ2 ) ≤

µ2 + 1 < 1. 2

Quantum trajectories

259

So ρ is a mixed state, and, by the definition of δm , tr(ai ρa∗i )

  m 2 ai ρa∗i m tr − tr(ρ ) = 0 tr(ai ρa∗i )

for all m = 1, 2, 3, . . . , d and all i = 1, . . . , k. Therefore either tr(ai ρa∗i ) = 0, i.e. ai ρa∗i = 0, proving our statement (ii) with λi = 0; or tr(ai ρa∗i ) > 0, in which case ρ
i := ai ρa∗i /tr(ai ρa∗i ) and ρ itself have the same moments of orders m = 1, 2, . . . , d, so that they are unitarily equivalent. This proves (ii). From Lemma 3 to Theorem 1 is an exercise in linear algebra: k Lemma 4. Let a1 , . . . , ak ∈ Md be such that i=1 a∗i ai = 1. Suppose that there exists a density matrix ρ ∈ Md such that for i = 1, . . . , k ai ρa∗i ∼ λi ρ. Let p denote the support of ρ. Then for all i = 1, . . . , k: pa∗i ai p = λi p.

Proof. Let us define, for a nonnegative matrix x, the positive determinant detpos (x) to be the product of all its strictly positive eigenvalues (counted with their multiplicities). Then, if p denotes the support projection of x, we have the implication detpos (x) = detpos (λp)

=⇒

tr(xp) ≥ tr(λp)

(6)

with equality iff x = λp. (This follows from the fact that the sum of a set of positive numbers with given product is minimal iff these numbersare equal.) Now let p be the support of ρ as in the Lemma. Let vi pa∗i ai p denote the polar decomposition of ai p. Then we have by assumption, detpos (λi ρ) = detpos (ai ρa∗i ) = detpos (ai pρpa∗i )   = detpos (vi pa∗i ai pρ pa∗i ai pvi∗ )   = detpos ( pa∗i ai pρ pa∗i ai p) = detpos (pa∗i ai p)detpos (ρ). Now, since detpos (λi ρ) = detpos (λi p) · detpos (ρ) and detpos (ρ) > 0, it follows that detpos (λi p) = detpos (pa∗i ai p). By the implication (6) we may conclude that tr(pa∗i ai p) ≥ trλi p).

(7)

On the other hand, k
i=1

λi =

k
i=1

tr(λi ρ) =

k
i=1

  k 
tr(ai ρa∗i ) = tr ρ a∗i ai = trρ = 1, i=1

H. Maassen and B. K¨ ummerer

260

where in the second equality sign the assumption was used again. Then, by (7),   k k k


λi trp = tr p. tr(λi p) = tr(pa∗i ai p) ≥ tr p = i=1

i=1

i=1

So apparently, in this chain, we have equality. But then, since equality is reached in (6), we find that pa∗i ai p = λi p. 5. Dark subspaces By considering more than one step at a time the following stronger conclusion can be drawn. Corollary 5. In the situation of Theorem 1, either the quantum trajectory purifies with probability 1 or there exists a projection p of dimension at least 2 such that for all l ∈ N and all i1 , . . . , il there is λi1 ,...,il ≥ 0 with pa∗i1 · · · a∗il ail · · · ai1 p = λi1 ,...,il p.

(8)

We shall call a projection p satisfying (8) a dark projection, and its range a dark subspace.  √ Let p be a dark projection, and let vi pa∗i ai p = λi vi p be the polar decomposition of ai p. Then the projection p
i := vi pvi∗ satisfies: λi p
i a∗i1 · · · a∗im aim · · · ai1 p
i = λi (vi pvi∗ )a∗i1 · · · a∗im aim · · · ai1 (vi pvi∗ ) = vi pa∗i a∗i1 · · · a∗im aim · · · ai1 ai pvi∗ = λi,i1 ,...,im · p
i . Hence if p is dark, and λi = 0 then also p
i is dark with constants λ
i1 ,...,im = λi,i1 ,...,im /λi . We conclude that asymptotically the quantum trajectory performs a random walk between dark subspaces of the same dimension, with transition probabilities p −→ p
i equal to λi , the scalar value in pa∗i ai p = λi p. In the trivial case that the dimension of p is 1, purification has occurred. Inspection of the ai should reveal the existence of nontrivial dark subspaces. If none exist, then purification is certain. We end this Section with two examples where nontrivial dark subspaces occur. Example 1. Let d = l · e and let H1 , . . . , Hl be mutually orthogonal e-dimensional subspaces of H = Cd . Let (πij ) be an l × l matrix of transition probabilities. Define aij ∈ A by √ aij := πij vij , where the maps vij : Hi → Hj are isometric. Then the matrices aij , i, j = 1, . . . , l define a perfect measurement whose dark subspaces are H1 , . . . , Hl . Example 2. The following example makes clear that nontrivial dark subspaces need not be orthogonal.

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Let H = C2 ⊗ D, where D is some finite dimensional Hilbert space, and for i = 1, . . . , k let ai := bi ⊗ ui , where the 2 × 2-matrices bi satisfy the usual equality k


b∗i bi = 1,

i=1

and the ui are unitaries D → D. Suppose that the bi are not all proportional to unitaries. Then the quantum trajectory defined by the ai has dark subspaces ψ ⊗D, with ψ running through the unit vectors in C2 . Physically this example describes a pair of systems without any interaction between them, one of which is coupled to the environment in an essentially commutative way, whereas the other purifies. References ˘, M. and Maassen, H. (2004). Stochastic Schr¨ [1] Bouten, L., Gut ¸a odinger equations. J. Phys. A 37, 9, 3189–3209. MR2042615 [2] Carmichael, H.J. (1993). An Open systems Approach to Quantum Optics. Springer-Verlag, Berlin–Heidelberg, New York. [3] Davies, E. B. (1976). Quantum Theory of Open Systems. Academic Press [Harcourt Brace Jovanovich Publishers], London. MR489429 [4] Gisin, N. (1984). Quantum measurements and stochastic processes. Phys. Rev. Lett. 52, 19, 1657–1660. MR741987 [5] Kraus, K. (1983). States, Effects, and Operations. Lecture Notes in Physics, Vol. 190. Springer-Verlag, Berlin. MR725167 ¨mmerer, B. and Maassen, H. (1987). The essentially commutative di[6] Ku lations of dynamical semigroups on Mn . Comm. Math. Phys. 109, 1, 1–22. MR879030 [7] Nielsen, M. A. (2000). Continuity bounds for entanglement. Phys. Rev. A (3) 61, 6, 064301, 4. MR1767484

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 262–273 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000284

Finitary Codes, a short survey Jacek Serafin1 Wroc
law University of Technology Abstract: In this note we recall the importance of the notion of a finitary isomorphism in the classification problem of dynamical systems.

1. Introduction The notion of isomorphism of dynamical systems has been an object of an extensive research from the very beginning of the modern study of dynamics. Various invariants of isomorphism have been introduced, like ergodicity, mixing of different kinds, or entropy. Still, the task of deciding whether two dynamical systems are isomorphic remains a notoriously difficult one. Independent processes, referred here to as Bernoulli schemes (defined below), are most easily described examples of dynamical systems. However, even in this simplest situation, the isomorphism problem remained a mystery from around 1930, when it was stated, until late 1950’s, when Kolmogorov implemented Shannon’s ideas and introduced mean entropy into ergodic theory; later Sinai was able to compute entropy for Bernoulli schemes and show that Bernoulli schemes of different entropy were not isomorphic. In his ground-breaking work Ornstein (see [20]) produced metric isomorphism between Bernoulli schemes of the same entropy, and later he and his co-workers discovered a number of conditions which turned out to be very useful in studying the isomorphism problem. Among those conditions are notions of Weak Bernoulli, Very Weak Bernoulli and Finitely Determined. As those will not be of our interest in this note, we shall not go into the details, and the interested reader should for example consult [20] or [24]. What is relevant for us here is the fact that the mappings which Ornstein constructed had the following disadvantageous property: in order to determine a target sequence one should examine the entire input sequence (the whole infinite past and future). On the other hand, it seemed necessary for applications that the codes should enjoy some kind of continuity property. 2. Finitary Codes Numerous attempts had been done, after Kolmogorov introduced entropy into the study of dynamical systems, to construct effective codes between independent processes and between Markov processes. Meshalkin ([17]), Blum and Hanson [3], Monroy and Russo [18], among others, have successfully constructed isomorphisms between specific independent or Markov processes, and all of those codes enjoyed the following property: to determine a value of any given coordinate in an output sequence, one should only examine finitely many coordinates of the source sequence, 1 Institute of Mathematics and Computer Science, Wroc
law University of Technology, Wybrze˙ ze Wyspia´ nskiego 27, 50-370 Wroc
law, Poland, e-mail: [email protected] AMS 2000 subject classifications: 37A35, 28D20. Keywords and phrases: finitary coding, Bernoulli scheme, entropy, code length.

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this finite number of coordinates depending upon the input sequence under consideration. A fundamental object of our study is a dynamical system: a Lebesgue probability space (X, A, µ) together with a measure-preserving, ergodic automorphism S of X. However, as it is well-known ([14]) that every ergodic dynamical system of finite entropy can be represented as a shift transformation on a sequence space with a finite underlying alphabet, from now on we shall focus our attention to the case of sequence spaces. Consequently, we assume that X is a sequence space, A is a product σ-algebra generated by the coordinate mappings, S is the left shift transformation and µ is a shift-invariant measure. We are now ready to define the principal objects of our interest. Definition 1. A homomorphism (factor map) φ from a dynamical system (X, A, µ, S) to a dynamical system (Y, B, ν, T ) is a measurable map φ from a subset of measure one of X to Y such that ν = µφ−1 and φS = T φ. The homomorphism φ is called finitary if it becomes a continuous map after discarding subsets of measure zero from both spaces X and Y . If φ is invertible and φ−1 is continuous outside subsets of measure zero, then we call φ a finitary isomorphism. For shift spaces, an equivalent definition is the following: Definition 2. A homomorphism φ between X and Y is finitary if for almost every x ∈ X there exist positive integers q = q(x), r = r(x), q ≤ r, such that if y ∈ X and [x−q , . . . , xr ] = [y−q , . . . , yr ] and if φ(y) is defined, then (φ(x))0 = (φ(y))0 . Let us note in passing that definitions of almost-continuity can be given in a more general setting, and for this we refer the interested reader to [5]. The following random variable, called code length function, will be of special interest to us: Definition 3 (code length). For x ∈ X, let q = q(x) and r = r(x) be minimal positive integers, as in the previous definition. The code length for x ∈ X is C(x) = q(x) + r(x). Random variables q and r are sometimes called memory and anticipation of the coding, respectively. Definition 4. We say that dynamical systems are finitarily isomorphic with finite expected code times (fect) if both the finitary isomorphism φ and its inverse φ−1 have finite expected code lengths. A systematic study of finitary coding had begun with the works of Keane and Smorodinsky ([10], [11], [12]), and Denker and Keane ([5], [6]). In their 1977 paper [10], and in subsequent articles [11], [12], Keane and Smorodinsky developed a theory and methodology of finitary coding, creating a new area of research, which has been (and is still being) extended in a multitude of ways. In this note we want to recall the marker methods of [10], and later discuss the existence of finitary homomorphism with finite expected code length or a finitary isomorphism with fect. It is perhaps worth noticing here that, despite major developments in the field, some of the most fundamental questions regarding classification remain open for more than 20 years now.

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2.1. Case of different entropies In this section we recall the main ideas behind the Keane-Smorodinsky construction [10]. The basic object of our study is a space of doubly-infinite sequences drawn from a finite alphabet consisting of a symbols (a ≥ 2), X = {1, . . . , a}Z , equipped with the product σ-algebra A, product measure µ = pZ and the left shift transformation S. Here p = (p1 , . . . , pa ) is a strictly positive probability vector assigning probabilities to symbols 1, . . . , a. A quadruple (X, A, µ, S) is commonly referred to as a Bernoulli scheme based on a probability vector p, and will be denoted BS(p). It is well-known that the entropy of BS(p) equals
h = h(p) = − i pi log pi . We will need another Bernoulli scheme, BS(q), based ¯ = on a probability vector q = (q1 , . . . , qb ) on b symbols, here the shift space is X {1, . . . , b}Z , the product σ-algebra is B, product measure is ν = qZ and the left shift ¯ = h(q). In 1969 Ornstein (see [20]) proved transformation is T , entropy of T is h that Bernoulli shifts of the same entropies were isomorphic and also showed the following: ¯ then Bernoulli scheme BS(q) is a homomorphic image of Theorem 1. If h > h BS(p). Later ([10]) Keane and Smorodinsky strengthened both the unequal and equal entropies statements to the case of finitary coding. In this section we want to focus on the following statement: ¯ then there exists a finitary homomorphism from Theorem 2 ([10]). If h > h, BS(p) to BS(q). Before we continue, let us mention that the isomorphism result [11] relies upon a beautiful refinement and improvement of the methods developed for unequal entropies case. An excellent exposition of the finitary isomorphism result (which has become standard in ergodic theory) appeared in Petersen’s book [24] (see also [4] or [23]). We now proceed to describe the Keane–Smorodinsky construction (see also [1]) in the case of unequal entropies, and then recall some subsequent results in which the new techniques were applied. Basic reduction allows to assume that there are two blocks, called markers, one in each scheme, of the same length and the same probability of appearance; it is also quite natural to demand that the coding procedure should map markers to markers, the main difficulty is in inventing a code for blocks, called fillers, occurring between markers. We define a marker, for either scheme, to be a block M = 1k−1 2 =

. . 1 2, 1 .

(k−1)−times

consisting of k − 1 consecutive 1’s followed by a 2. Let us note that a marker has the following non-overlapping property: none of its initial subblocks is equal to its terminal subblocks of the same length; this property guarantees the shift-invariance of the coding. By ergodicity, almost every source sequence x in X splits into runs of markers labeled in a natural manner by ±1, ±2, . . . and separating blocks labeled 0, ±1, ±2, . . . We assume that the 0-th coordinate of x is covered by either the run of markers

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labeled −1 or the subsequent 0-th separating block. By uj we denote the number of markers in the j-th run while lj stands for the length of the j-th separating block. For every r = 1, 2, . . . we denote by sr = sr (x) the skeleton of rank r. This is defined as the truncation of x to a finite segment around 0 such that the separating blocks in x are replaced by gaps of the same length, and with the property that the extreme left and right runs of markers contain each at least r markers while the internal runs, if any, contain each less than r markers. Moreover, neither the immediately preceding nor the immediately following k-block of x is a marker block. We denote by −mr < 0 and nr > 0 the label of the first and the last run of markers in sr , respectively. We may draw the following picture of the skeleton sr (x) of rank r at x, as a sequence of markers and spaces between the markers: M u−m

l−(m−1)

M u−(m−1) · · ·

l−1

M u−1

l0

M u1

l1

· · · M un−1

ln−1

M un

where m = m(r), n = n(r) ≥ 1, li ≥ 1, for i = −(m − 1), . . . , n − 1 and u−m , un ≥ r > u−(m−1) , . . . , un−1 . Clearly the rank one skeletons consist of two marker runs separated by one filler block. In order to avoid ambiguity we assume that the zero coordinate of x appears in the ‘interior’ of s1 (x), i.e. it corresponds to the blank part of s1 . The skeleton of rank r is obtained by looking to the left and to the right for the first appearance of M r . For a skeleton s we denote by l(s) the length of s minus the last run of markers, l(sr ) = ku−mr + l−mr +1 + · · · + ku−1 + l0 + ku1 + · · · + lnr −1 , as the final block of markers is only needed to determine the occurrence of sr (x) but is not considered to be part of that occurrence. A block in x occurring along a single run of markers followed by a separating block will be called an order one filler. The concatenation of all the order one fillers in sr will be referred to as the filler of sr . Clearly the length of the filler is equal to l(sr ). For a fixed non-indexed skeleton s the filler measure µs is defined on the l(s)blocks as the projection of the conditional measure µ(·|S) where S is the event that s occurs at [0, l(s) − 1] in x. According to [10], the filler measure µs is the product of the filler measures corresponding to order one subskeletons of s. Regardless of the skeleton rank, filler measures will be denoted by µs . ˆ defined by x Recall that the marker process is a stationary 0-1 process X ˆi = 0 iff xi . . . xi+k−1 is a marker block. If the marker length k is sufficiently large then the entropy of the marker process can be made as small as needed. It is clear that both Bernoulli schemes have the marker process as a common factor, and that the skeleton structure at x only depends upon the marker process x ˆ. We define the filler ˆ entropy f = h(X) − h(X). ¯ Let us fix  < (f − h)/3. A filler F in the skeleton sr (x) of the source sequence x ∈ X, is called good if µs (F ) ≤ e−l(sr )(f −
) . On the other hand, a corresponding ¯ ¯ will be called a good filler if µ l(sr )-block F¯ in x ¯∈X ¯(F¯ ) ≥ e−l(sr )(h+
) . According to [10], only good fillers will be encoded to good fillers. If a filler is bad, it will be encoded as a part of a longer good filler at a later stage. The coding is carried out for a given source sequence x by looking at the ascending skeletons sr (x), r = 1, 2, . . . By means of an ”assignment” defined in [10], the filled skeleton sr (x) will be encoded in a consistent way if the filler F is good, except for a small set of exceptional cases. Let us now be more specific about the coding procedure. ¯ r ) for the low entropy scheme is divided into The set of all possible fillers F(s ¯ r ) is equivalent ¯ r ) is good then no other filler in F(s equivalence classes. If F¯ ∈ F(s

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to it. Let us suppose that F¯ is bad; it is then possible that the restriction of F¯ to a subskeleton s
of sr is a good filler for that subskeleton s
. The equivalence ¯ with the property that G ¯ has class of a bad filler F¯ consists of all bad fillers G the same collection (as F¯ ) of subskeletons on which its restrictions are good, and those good restrictions agree with the corresponding (good) restrictions of F¯ . Let ¯ if F¯ and G ¯ are equivalent. A simple combinatorial argument in [10] us write F¯ ∼ G ¯ guarantees that there are no more than 2mr +nr −1+(h+
)l(sr ) equivalence classes in ¯ r ); we shall need that estimate later. An important step toward defining the F(s desired code is a notion of partial assignments. For a fixed skeleton s, a partial ¯ assignment Ps assigns to each element F¯ of F(s) a subset Ps (F¯ ) of F(s), in such a way that νs (F¯ ) ≤ µs (Ps (F¯ )). Partial assignment is an example of a more general notion of a society (for a detailed discussion of societies see [10] or [24]). The partial assignment Ps is good, if it respects the equivalence classes of F¯ (s), i.e. if ¯ ⇒ Ps (F¯ ) = Ps (G). ¯ F¯ ∼ G ¯ then the above condition guarantees that each filler F in If we suppose that F¯ ∼ G, ¯ and all other elements of the equivalence class of Ps (F¯ ) will be assigned to F¯ and G ¯ F . The Shannon–McMillan–Breiman theorem implies, however, that at some finite ¯ ∈ F(s ¯ r ) for a skeleton sr which stage F¯ becomes a part of a longer good filler H ¯ ¯ r ) is equivalent to has s as a subskeleton. Since H is good, no other element of F(s ¯ is assigned to exactly one filler in F(s ¯ r ), that filler being it, and each F ∈ Psr (H) ¯ of course H. Keane and Smorodinsky show in [10], using a version of the marriage lemma, that partial assignments can be consistently extended to so-called global assignments, in such a way that if at a finite stage of the above procedure a filler ¯ F ∈ F(s) is assigned to the unique F¯ ∈ F(s), then the assignments which take place for skeletons for which s is a subskeleton, respect the F → F¯ assignment. Finally, F¯ is defined as a homomorphic image of F . A natural question which one might ask, having constructed a finitary coding, is whether the average code length is finite? Even though it was believed that the expected code length should be finite in the case h(p) > h(q), the proof of this statement, quite nontrivial, was given much later in [30]. Shortly after [10], Akcoglu, del Junco and Rahe extended the result of Keane and Smorodinsky. They constructed a finitary coding between an ergodic Markov shift X and a mixing Markov shift Y of smaller entropy. Their construction is quite similar to that of [10], the essential role is being played by the low entropy marker process. Informally speaking, the presence of markers makes it possible to represent almost every source sequence x ∈ X as an ascending nested family of words, which fill longer and longer skeletons determined by the marker process. Fillers of sufficiently large rank are encoded to corresponding fillers in Y thus eventually defining the required finitary coding φ : X → Y . Akcoglu, del Junco, and Rahe also claim without proof that the code length should have a finite expectation. We will now very shortly sketch the proof of the main result in [8], which is a significant improvement of the result in [30], that the expected code length between Markov processes of unequal entropies is finite; we also indicate a number of differences between [1] and [10], as we proceed. One of the differences between [10] and [1] lies in the choice of markers. Unlike in [10], where the authors used 1k−1 2 as a marker, in the Markov case [1] a marker M is a collection of blocks of the same length k such that each word in M begins with the same symbol a1 , no word in M overlaps a word in M and arbitrarily long

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concatenations of words from M occur with positive probabilities. Also, the length k can be chosen arbitrarily large and the probability that a marker occurs at a given position decays exponentially with k. The coding between two mixing Markov processes is achieved in two steps, as a composition of two codes, using an intermediate Bernoulli scheme. In the first step, referred to as Markov-to-Bernoulli coding, we study a mixing Markov process ¯ < h = h(X). A marker ¯ µ ¯ =h (X, µ, T ) and a Bernoulli process (X, ¯, T¯) with h(X) in X can be selected as a single word a1 . . . ak in such a way that the filler entropy ¯ no marker is needed in the Bernoulli process X. ¯ f still exceeds the entropy h; Good and bad fillers are defined similarly as in the Bernoulli case, and the coding procedure follows to a large extent that of [10]. As far as the code length is concerned, the main results of [8] are the following: ¯ be mixing Markov and Bernoulli (or Bernoulli Theorem 3. Let the processes X, X ¯ then there exists a finitary coding from and Markov), respectively. If h(X) > h(X) ¯ such that for every p < 2 the code length is an Lp random variable. X to X Lemma 1. Let X, Y, Z be arbitrary stationary processes and let φ : X → Y and ψ : Y → Z be finitary codes. Assume that for some p1 , p2 > 1 with p1 ≤ p2 + 1 the code length of φ is in Lp for all p < p1 and the code length of ψ is in Lp for all p < p2 . Then the composed code ψ ◦φ has code length in Lp for all p < p1 p2 /(p2 +1). Composing the Markov-to-Bernoulli with Bernoulli-to-Markov codes, we obtain: Theorem 4. Let X1 and X2 be mixing Markov processes such that h(X1 ) > h(X2 ). Then there exists a finitary coding from X1 to X2 such that the code length is in Lp for all p < 4/3. Let us recall a basic fact about the nature of the Keane-Smorodinsky coding, that markers are mapped onto markers and fillers onto fillers, so the contents of good skeletons in one scheme define the contents of appropriate skeletons in the other scheme. Consequently, the code length function C only takes values cr+1 = k + l(sr+1 ) + k(r + 1) (depending upon the marker process), which are skeleton lengths plus the length of r + 1 markers in the terminal marker occurrence plus k which stands for a number of entries which have to be examined to make sure that there is no marker preceding the initial run of markers. From a combinatorial ¯ r ) it follows that the conditional bound on the number of equivalence classes in F(s probability, given a marker structure of x, that C(x) ≥ cr+1 , is bounded by (see [10], lemma 14) 2mr +nr −cl(sr ) + µs (F is bad) + µ ¯(F¯ is bad), ¯ ¯ respectively. ¯ where c = (f − h−2)/ log 2 > 0 and F, the sr -fillers in X, X,
F denote p p It is easy to see that EC is finite if Ecr+1 P (C ≥ cr+1 ) < ∞, so it suffices to show that the three following series converge ∞  r=1

p mr +nr −cl(sr )

E(cr+1 2

),

∞  r=1

p

E(cr+1 µs (F is bad)),

∞ 

E(cr+1 p µ ¯(F¯ is bad)).

r=1

Convergence is obtained by a careful study of analytic properties of the generating function of the filler length, the use of the Bernstein inequality to give exponential bounds for appropriate large deviation events, and repeated use of H¨ older inequality. ¯ is The second step is a Bernoulli-to-Markov coding. Now X is Bernoulli and X ¯ < h = h(X). Moreover, as in [1], by extending X ¯ =h ¯ mixing Markov with h(X)

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to another mixing Markov process with a slightly larger entropy (the extension is a ¯ in X ¯ and a singlecoding of length one) we may assume that there exist a marker M word marker M in X such that the corresponding marker processes have the same distribution. Therefore the two marker processes can be identified as a common ˆ of X and X. ¯ The bad fillers in X and X ¯ are defined as in the Markovfactor X p to-Bernoulli case and the finiteness of EC is concluded similarly by studying the three series (with some additional difficulties caused be the fact that the image is a Markov and not independent process). Remark 1. The above methods allowed to compute moments of order p with p < 2 for Markov-to-Bernoulli and Bernoulli-to-Markov coding, and left an open question whether the variance of the coding is finite. In a yet unpublished manuscript [7], the authors claim that there exists a universal finitary coding between Bernoulli schemes of unequal entropies which has exponential tails, i.e. Prob(C > n) decays exponentially as n → ∞, a condition which clearly implies that moments of all orders are finite. A code is universal in the following sense: if A and B are two ¯ and in addition an alphabets, and a Bernoulli scheme on B is given, of entropy h,  > 0 is given, then there exists a measurable subset of AZ and a mapping φ from that set to B Z such that if any Bernoulli scheme on A is given, of entropy larger ¯ + , then φ is finitary with exponential tails. than h Let us note, however, that this result does not imply that the Keane-Smorodinsky code has finite variance. 2.2. Case of equal entropies In [11] Keane and Smorodinsky improved upon the result of Ornstein and showed: Theorem 5. If h(p) = h(q), then BS(p) and BS(q) are finitarily isomorphic. A question was immediately posed as to under what additional assumptions the expected code length could be finite. It was known already that in the case of Meshalkin’s code, the average code length was infinite, as for that particular code the probability Prob(C > n) is equal to the probability that a simple random walk remains positive after n steps. First general statement in this direction was made by Parry ([21]), who provided a class of isomorphisms which had infinite expected code lengths, and therefore showed that entropy alone was not an invariant for finitary isomorphism with fect. An obstruction which was discovered by Parry involved the so-called information cocycle. Let us suppose that α = {A1 , . . . , Ak } is a time-zero partition (also called − state partition) of the shift space (X, S, µ), ∞that−iis, Ai = {x : x0 = i}. Let α denote the smallest σ-algebra containing i=1 S α, and define the information cocycle of S to be  IS = I(α| α− ) = − χAi log µ(Ai | α− ). i

In [21] Parry proved: Theorem 6. If S and T are finite state processes and if φ is a finitary isomorphism between S and T such that φ and φ−1 have finite expected code lengths, then the information cocycles IS and IT ◦ φ are cohomologous, with a finite valued and measurable transfer function, i.e. IS = IT ◦ φ + g ◦ S − g.

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Note that here the dynamical systems were not assumed to be Markov. It was then shown by Parry that particular dynamical systems which were known to be finitarily isomorphic by the results of Keane and Smorodinsky, had noncohomologous information cocycles and therefore could not be finitarily isomorphic with fect. Among the above was Meshalkin’s example of Bernoulli schemes BS( 12 , 18 , 18 , 81 , 18 ) and BS( 14 , 41 , 14 , 41 ) and some equal entropies Markov processes. First attempt in forming a set of invariants for isomorphism with fect between Markov shifts, was undertaken by Krieger. He (in [15]) defined, for a Markov shift with transition matrix P , a multiplicative subgroup ∆P of positive reals in the following way: P (i0 , i1 ) · · · P (in−1 , i0 ) }, ∆P = { P (i0 , j1 ) · · · P (jn−1 , i0 ) which are ratios of weights of cycles of equal lengths starting and ending in the same state. Krieger was then able to prove that if Markov shifts P and Q were finitarily isomorphic with fect then their respective delta groups ∆P and ∆Q were equal. He also gave examples of shifts with equal delta groups which could not be finitarily isomorphic with fect. In 1981 Tuncel ([33]) introduced the so-called β-function, as the spectral radius of the matrix P t (where P t (i, j) = (P (i, j))t ), and showed that this was an invariant of regular isomorphism. We shall not go into a detailed study of the classification up to a regular isomorphism; let us only note that this is a weaker notion than finitary isomorphism; an isomorphism φ is regular if both φ and its inverse φ−1 have bounded anticipation but can have infinite memory. Later Schmidt ([28], see also [29])improved on some of the results of Tuncel, showing in particular that β-function was an invariant for finitary isomorphism with fect. In 1984 Parry and Schmidt ([22]) extended the notion of ∆P -group to that the ΓP -group, generated by all weights P (i0 , i1 ) · · · P (in−1 , i0 ) of cycles. They showed that for an aperiodic transition matrix P , the quotient group ΓP /∆P is cyclic with a distinguished generator cP ∆P . The main statement of [22] is that ΓP , ∆P and cP ∆P are invariants for finitary isomorphism with fect, which together with the previous result of Schmidt allowed to establish the following conjecture. Conjecture 1. The quadruple (ΓP , ∆P , cP ∆P , βP ) forms a complete set of invariants for finitary isomorphism with fect. The above has been open for twenty years now, and the most general statement seems to be a recent result of Mouat and Tuncel: Theorem 7 ([19]). Let P and Q be primitive, stochastic matrices of the underlying Markov shifts, with the same Γ, ∆, c∆ and β invariants. If there exist states I0 , J0 of the P -shift and the Q-shift, respectively, and there exist a nontrivial column vector vr and a nontrivial row vector vl such that (P n vr )(I0 ) = (vl Qn )(J0 ) for all n ≥ 1, then the two Markov shifts are finitarily isomorphic with fect. 3. Applications In this section we discuss a number of results which are closely related to finitary coding.

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3.1. m-dependent processes The marker method of [11] and [12] was extended by Smorodinsky ([31]) to prove that m-dependent processes of equal entropy were finitarily isomorphic. Let us recall here that a stationary process is called m-dependent if its past and future become independent, if separated by m units of time. It is an easy observation that processes which are finite factors of independent processes are m-dependent, so Smorodinsky’s result implies that equal entropy, finite factors of independent processes are finitarily isomorphic. It is natural to ask about finitary instead of finite factors, and this is stated in [31] as a conjecture: Conjecture 2 (Finitary factors conjecture). Equal entropy, finitary factors of independent processes are finitarily isomorphic. Let us keep in mind a well-known fact from Ornstein’s theory that measurable factors of independent processes are (measurably) isomorphic, if they have the same entropy. 3.2. Zd -actions Another direction in which the results of [10] have been extended is the action of Zd , d ≥ 2, rather than the action of a single shift on Z. In [9] del Junco considered two random fields on Z2 : an ergodic Markov field X and an independent process Y such that h(X) > h(Y ). He then proved the existence of a finitary homomorphism from X to Y . It was left as an open question whether a mixing Markov field was always a finitary factor of a Bernoulli process of higher entropy. Later ([2]) an example was given by van den Berg and Steif, of a Markov field which was not a finitary factor of any independent process, so the finitary factors conjecture fails in d dimensions, d ≥ 2. Among the difficulties which arise when one considers Z2 actions instead of a Z action is the choice of markers. Let us recall that markers were nonoverlapping blocks; a marker in Z2 should therefore be a block which does not overlap itself under translation in any given direction, and that is a hard condition to fulfill. del Junco considers a multidimensional version of the Rokhlin tower lemma in order to define configurations in Z2 which have a number of disjoint shifts and which, together with a specific number of its shifts, almost cover the whole space; those configurations depend upon the multiple occurrences of blocks called markers. Skeletons of all ranks are defined in a highly nontrivial way, subsequently del Junco adapts and modifies when necessary the main ideas of [10], including the marriage lemma, to build the desired finitary coding between the random fields. We complete this section by mentioning that it is the subject of the current research of the author of this note to show that the average code volume for the finitary coding from a Markov random field into a Bernoulli field of strictly smaller entropy, is finite. The generalization of the notion of code length to that of code volume in Z2 is straightforward. We do, however, propose an alternative approach to that of del Junco. A marker is a fixed block of a low probability of occurrence; when a marker occurs at a fixed coordinate, we consider a skeleton at this coordinate as a set of coordinates for which this marker occurrence is the closest, in the L1 distance. This procedure gives rise to the partition of Z2 , into the so-called Voronoi regions; the details will appear elsewhere. A result of different flavor was obtained by Steif in [32], where he considered the so-called T, T −1 process (also known as Random Walk in Random Scenery)

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in d-dimensional integer lattice (for definitions see [32] and references therein). A classical result of Kalikow is that the T, T −1 process on Z associated to a simple random walk is not Bernoulli; Steif proves that in Zd the second coordinate of this process is not a finitary factor of an independent process, he also applies this to study some properties of the Ising model in statistical mechanics. 3.3. Countable state processes It turns out that the assumption that a Markov shift be a finite state process, is an essential one, as far as classification up to a finitary isomorphism is concerned. Smorodinsky proved (unpublished, see e.g. [16]) that an ergodic automorphism of compact abelian group can only be finitarily Bernoulli (that means: finitarily isomorphic to a Bernoulli shift) if it is exponentially recurrent (i.e. if U is an open set and rU is the first-return time function, then Prob(rU > n) decays exponentially fast with n). Smorodinsky used this result to construct a countable state Markov shift which is measurably isomorphic to a Bernoulli shift but has polynomially decaying return times, hence cannot be finitarily isomorphic to a Bernoulli scheme. Lind ([16]) went on in that direction to prove that ergodic automorphisms of compact abelian groups are exponentially recurrent, and that was a step forward in trying to resolve a question as to whether ergodic group automorphisms are finitarily isomorphic to Bernoulli shifts. Let us recall here that some special classes, like hyperbolic toral automorphisms, are known to be finitarily Bernoulli. Rudolph ([27]) completed Smorodinsky’s work in showing that a countable state, mixing Markov shift of finite entropy is finitarily Bernoulli if and only if the chain has exponentially decaying return times; in particular, it follows that the so-called βautomorphisms are finitarily isomorphic to independent processes of the same finite entropy. We wish to note that this result depends heavily upon the characterization of the processes finitarily isomorphic to Bernoulli shifts ([26]), a construction which is a very involved generalization of [11]. More recently, Keane and Steif ([13]) proved that T, T −1 process associated to a 1-dimensional random walk with positive drift is finitarily isomorphic to an independent process, using an intermediate countable state Markov shift, and results of [27]. Finally, we remark that Petit ([25]) extended the methods of Keane and Smorodinsky ([11]) to show that two infinite entropy Bernoulli schemes on countable state space are finitarily isomorphic. Let us close this paragraph by mentioning that despite all developments, the classification of countable state Markov processes up to a finitary isomorphism remains an intricate task, and there seems to be a need for methods which would both be more elementary and more easily understood than the present ones. References [1] Akcoglu, M. A., del Junco, A., and Rahe, M. (1979). Finitary codes between Markov processes. Z. Wahrsch. Verw. Gebiete 47, 3, 305–314. MR525312 [2] van den Berg, J. and Steif, J. E. (1999). On the existence and nonexistence of finitary codings for a class of random fields. Ann. Probab. 27, 3, 1501–1522. MR1733157 [3] Blum, J. R. and Hanson, D. L. (1963). On the isomorphism problem for Bernoulli schemes. Bull. Amer. Math. Soc. 69, 221–223. MR143862

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[4] Cornfeld, I. P., Fomin, S. V. and Sina˘ı, Ya. G. (1982). Ergodic Theory. Grundlehren der Mathematischen Wissenschaften, vol. 245. Springer-Verlag, New York, MR832433 [5] Denker, M. and Keane, M. (1979). Almost topological dynamical systems. Israel J. Math. 34, 1–2, 139–160 (1980). MR571401 [6] Denker, M. and Keane, M. (1980). Finitary codes and the law of the iterated logarithm. Z. Wahrsch. Verw. Gebiete 52, 3, 321–331. MR576892 [7] Harvey, N., Holroyd, A., Peres Y. and Romik, D. (2005) Universal finitary codes with exponential tails. Preprint. [8] Iwanik, A. and Serafin, J. (1999). Code length between Markov processes. Israel J. Math. 111, 29–51. MR1710730 [9] del Junco, A. (1980). Finitary coding of Markov random fields. Z. Wahrsch. Verw. Gebiete 52, 2, 193–202. MR568267 [10] Keane, M. and Smorodinsky, M. (1977). A class of finitary codes. Israel J. Math. 26, 3–4, 352–371. MR450514 [11] Keane, M. and Smorodinsky, M. (1979). Bernoulli schemes of the same entropy are finitarily isomorphic. Ann. of Math. (2) 109, 2, 397–406. MR528969 [12] Keane, M. and Smorodinsky, M. (1979). Finitary isomorphisms of irreducible Markov shifts. Israel J. Math. 34, 4, 281–286 (1980). MR570887 [13] Keane, M. and and Steif, J. (2003). Finitary coding for the onedimensional T, T −1 process with drift. Ann. Probab. 31, no. 4, 1979–1985. MR2016608 [14] Krieger, W. (1970). On entropy and generators of measure-preserving transformations. Trans. Amer. Math. Soc. 149, 453–464. MR259068 [15] Krieger, W. (1983). On the finitary isomorphisms of Markov shifts that have finite expected coding time. Z. Wahrsch. Verw. Gebiete 65, 2, 323–328. MR722135 [16] Lind, D. A. (1982). Ergodic group automorphisms are exponentially recurrent. Israel J. Math. 41, 4, 313–320. MR657863 [17] Meˇ salkin, L. D. (1959). A case of isomorphism of Bernoulli schemes. Dokl. Akad. Nauk SSSR 128, 41–44. MR110782 [18] Monroy, G. and Russo, L. (1975). A family of codes between some Markov and Bernoulli schemes. Comm. Math. Phys. 43, 2, 155–159. MR377018 [19] Mouat, R. and Tuncel, S. (2002). Constructing finitary isomorphisms with finite expected coding times. Israel J. Math. 132, 359–372. MR1952630 [20] Ornstein, D. S. (1974). Ergodic Theory, Randomness and Dynamical Systems, Yale University Press, New Haven, 1974. MR447525 [21] Parry, W. (1979). Finitary isomorphisms with finite expected code lengths. Bull. London Math. Soc. 11, 2, 170–176. MR541971 [22] Parry, W. and Schmidt, K. (1984). Natural coefficients and invariants for Markov-shifts. Invent. Math. 76, 1, 15–32. MR739621 [23] Parry, W. and Tuncel, S. (1982). Classification Problems in Ergodic Theory. London Mathematical Society Lecture Note Series, 67, Cambridge University Press, Cambridge, NY. MR666871 [24] Petersen, K. (1983) Ergodic Theory. Cambridge University Press, Cambridge. MR833286 [25] Petit, B. (1982). Deux sch´emas de Bernoulli d’alphabet d´enombrable et d’entropie infinie sont finitairement isomorphes. Z. Wahrsch. Verw. Gebiete 59, 2, 161–168. MR650608 [26] Rudolph, D. J. (1981). A characterization of those processes finitarily isomorphic to a Bernoulli shift. In Ergodic Theory and Dynamical systems, I (College

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IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 274–285 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000293

Entropy of a bit-shift channel Stan Baggen1 , Vladimir Balakirsky2 , Dee Denteneer1 , Sebastian Egner1 , Henk Hollmann1 , Ludo Tolhuizen1 and Evgeny Verbitskiy1 Philips Research Laboratories Eindhoven and Eindhoven University of Technology Abstract: We consider a simple transformation (coding) of an iid source called a bit-shift channel. This simple transformation occurs naturally in magnetic or optical data storage. The resulting process is not Markov of any order. We discuss methods of computing the entropy of the transformed process, and study some of its properties.

Results presented in this paper originate from the discussions we had at the “Coding Club” – the weekly seminar on coding theory at the Philips Research Laboratories in Eindhoven. Mike Keane, when his active travelling schedule permits, is also attending this seminar. We would like to use this opportunity to thank Mike for his active participation, pleasant and fruitful discussions, his inspiration which we had a pleasure to share. 1. Bit-shift channel In this paper we consider a simplified model for errors occurring in the readout of digital information stored on an optical recording medium like the Compact Disk (CD) or the Digital Versatile Disk (DVD). For more detailed information on optical storage see [9] or [16]. On optical disks the information is stored in a reflectivity pattern. For technical reasons, it is advantageous to use only two states, i.e. “low” and “high” reflectivity. Figure 1 shows the disk surfaces for two types of the DVD’s. While the presence of only 2 states greatly simplifies the detection of the state, it reduces the maximum spatial frequency, and hence storage capacity. In this situation it is better not to encode the information in the reflectivity state itself but rather in the location of the transitions: The reflectivity pattern consists of an alternating sequence of “high” and ”low” marks of varying length (an integer multiple of some small length unit), while each mark exceeds a minimal length, say d + 1 units. Hence, this “run-length limited” (RLL) encoding makes sure no mark is too short for the disk while the information density is only limited by the accuracy of determining the length of the marks, or equivalently the location of the transitions. For technical reasons (to recover the length unit from the signal itself) another constraint is imposed: No mark must exceed k + 1 units, k > d. (For the CD, (d, k) = (2, 10).) 1 Philips Research Laboratories, Prof. Holstlaan 4, 5656 AA, Eindhoven, The Netherlands, e-mail: [email protected] e-mail: [email protected] email: [email protected] e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] 2 TU Eindhoven, 5600 MB, Eindhoven, The Netherlands, e-mail: [email protected] AMS 2000 subject classifications: primary 94A17, 28D20; secondary 58F11. Keywords and phrases: Markov and hidden Markov processes, Entropy, channel capacity, Gibbs vs. non-Gibbs.

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Fig 1. Images of DVD disks. The left image shows a DVD-ROM. The track is formed by pressing the disk mechanically onto a master disk. On the right is an image of a rewritable disk. The resolution has been increased to demonstrate the irregularities in the track produced by the laser. These irregularities lead to higher probabilities of jitter errors.

It is customary to describe RLL sequences by their transitions: A (d, k)-RLL sequence has at least d and at most k ’0’s between ’1’s. So a “high” mark of 4 units, followed by a “low” of 3 units, followed by a “high” of 4 units correspond to the RLL-sequence 100010010001 written to the disk. At the time the RLL-sequence is read from disk, the transitions (the ‘1’s) might be detected at different positions due to noise, inter-symbol interference, clock jitter, and other distortions. In the simplest version of this “bit-shift channel model” each ‘1’ may be detected one unit early, on time, or one unit late with the probabilities (ε, 1 − 2ε, ε), 0 ≤ ε ≤ 1/2, and the shifts are independent. More formally, suppose X is the length of a continuous interval of low or high marks on the disk. Then, after reading, the detected length is Y = X + ωlef t − ωright ,

(1)

where ωlef t , ωright take values {−1, 0, 1}. And ω = 1, 0, −1 means that the transition between the ”low”-”high” or ”high”-”low” runs was detected one time unit too early, correctly, or one unit too late, respectively. Note that for two consecutive intervals ωright of of the first interval is ωlef t of the second. The simplest model for the distribution of time shifts ωlef t is to assume that they are independent for different intervals (runs), and P(ωlef t = −1) = P(ωlef t = 1) = ε,

P(ωlef t = 0) = 1 − 2ε,

for some ε ∈ [0, 1/2]. An important question then is: Given (d, k), ε, and some distribution for the input sequences (e.g. run-lengths uniformly distributed in {d, . . . , k}), what is the mutual information between input and output sequences? In other words, how much can be learned about the input from observing the output, on average. The problem of computing the mutual information is equivalent to computation of the entropy of the output sequence, see [2]. The supremum of this mutual information over all possible measures on space of input sequences is called “channel capacity”.

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1.1. Model Let us describe the bit-shift channel as a continuous transformation (factor) of a certain subshift of finite type. Let A = {d, . . . , k}, where d, k ∈ N, d < k and d ≥ 2. The input space then is A Z = {x = (xi ) : xi ∈ A }. Consider also a finite alphabet Ω with 9 symbols Ω = {(−1, −1), (−1, 0), (−1, 1), (0, −1), (0, 0), (0, 1), (1, −1), (1, 0), (1, 1)} . Finally, consider a subshift of finite type ΩJ ⊂ ΩZ defined as
 ΩJ = (ωn ) ∈ ΩN : ωn,2 = ωn+1,1 for all n ∈ Z , where ωn = (ωn,1 , ωn,2 ). The factor map φ is defined on A Z × ΩJ as follows: y = φ(x, ω) with yn = xn + ωn,1 − ωn,2 for all n. (2) Note that the output space O = φ(A Z × ΩJ ) is a subshift of B Z , with B = {d − 2, . . . , k + 2}. Clearly, O = B Z . For example, (d − 2, d − 2) cannot occur in any output sequence. Indeed, if yn = d − 2, then xn = d, ωn,1 = −1 and ωn,2 = 1. But then, yn+1 = xn+1 + ωn+1,1 − ωn+1,2 = xn+1 + ωn,2 − ωn+1,2 ≥ d + 1 − 1 = d. With a similar argument, one concludes that for any L ≥ 1 [d − 2, d, . . . , d, d − 2] or [d − 2, d, . . . , d, d − 1]       L times

L times

do not occur in any output sequence y ∈ O. Therefore, there is an infinite number of minimal forbidden words, i.e., forbidden words all of whose subwords are allowed. Hence, O is not a subshift of finite type. However, since A Z × ΩJ is a subshift of finite type, O is sofic [13]. 1.2. Capacity of a bit-shift channel Suppose J is a measure on ΩJ . For example, in this paper we will be mainly interested in Markov measures Jε on ΩJ , obtained in a natural way from Bernoulli measures on {−1, 0, 1} with probabilities ε, 1 − 2ε, and ε, respectively. If P is a translation invariant measure on A Z , then we obtain a measure Q on O, which is the push forward of P × J. We use a standard notation Q = (P × J) ◦ φ−1 . From the information-theoretical point of view, an important quantity is the capacity of the channel. The capacity of a bit-shift channel specified by J is defined as   −1 − h(J), (3) Cbitshift (J) = sup h (P × J) ◦ φ P∈P(A Z )

where the supremum is taken over P(A Z ) – the set of all translation invariant probability measures on A Z , and h(·) is the entropy. Even for ’Bernoulli’ measures Jε , the capacity Cbitshift (Jε ) is not known. It is relatively easy to see that the supremum in (3) is achieved. However, the properties of maximizing measures are not known. It is expected that maximizing measures are not Markov of any order. Finally, if one is interested in topological entropy of O: htop (O) = sup h(Q), Q∈P(O)

then htop (O) is easily computable using standard methods [13] or using the efficient numerical approach of [8].

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2. Entropy of a bit-shift channel Suppose that {Xn } are independent identically distributed random variables taking values in A = {d, . . . , k}, and let P be the corresponding distribution. What is the entropy of Q = (P × Jε ) ◦ φ−1 ? Note that (Xn , ωn ) is a Markov chain, and, hence, Yn , given by (2), is a function of a Markov chain. Let us start by recalling some methods of computing the entropy of processes which are functions of Markov chains. Suppose X = {Xn }, n ∈ Z, is a stationary ergodic Markov chain taking values in a finite alphabet A . Let φ : A → B be some map, and consider a process Y = {Yn }, defined by Yn = φ(Xn )

for all n ∈ Z.

The following result [3, 4], see also [5, Theorem 4.4.1], provides sharp estimates on the entropy of Y . Theorem 2.1. If X is a Markov chain and Y = φ(X ), then for every n ≥ 1 one has H(Y0 |Y1 , . . . , Yn−1 , Xn ) ≤ h(Y ) ≤ H(Y0 |Y1 , . . . , Yn−1 , Yn ). Moreover, as n  ∞ H(Y0 |Y1 , . . . , Yn−1 , Xn )  h(Y ),

H(Y0 |Y1 , . . . , Yn−1 , Yn ) h(Y ).

Birch [3, 4] has shown that under some additional conditions,the convergence is in fact exponential: |h(Y ) − H(Y0 |Y1 , . . . , Yn−1 , Xn )| ≤ Cρn , |h(Y ) − H(Y0 |Y1 , . . . , Yn−1 , Yn )| ≤ Cρn , where ρ ∈ (0, 1) is independent of the factor map φ. Let us give a proof of Theorem 2.1, since it is very short and provides us with some useful intuition. Proof of Theorem 2.1. An upper estimate of h(Y ) in terms of H(Y0 |Y1 , . . . , Yn ) and the monotonic convergence H(Y0 |Y1 , . . . , Yn ) to h(Y ) are standard facts. For the lower estimate we proceed as follows: for any m ∈ N one has H(Y0 |Y1 , . . . , Yn−1 , Xn ) = H(Y0 |Y1 , . . . , Yn−1 , Xn , . . . , Xn+m ) = H(Y0 |Y1 , . . . , Yn−1 , Yn , . . . , Yn+m , Xn , . . . , Xn+m ) ≤ H(Y0 |Y1 , . . . , Yn−1 , Yn , . . . , Yn+m ),

(4) (5) (6)

where in (4) we used the Markov property of X , and (5), (6) follow from the standard properties of conditional entropies. Since h(Y ) = lim H(Y0 |Y1 , . . . , Yn−1 , Yn , . . . , Yn+m ), k→∞

we obtain the lower estimate of h(Y ). Moreover, using standard properties of conditional entropies, we immediately conclude that H(Y0 |Y1 , . . . , Yn−1 , Xn ) is monotonically increasing with n.

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To prove that the lower bound actually converges to h(Y ), we proceed as follows. Note that H(Xn+1 ) ≥ H(Xn+1 |Yn ) − H(Xn+1 |Yn , . . . , Y0 ) =

n−1

H(Xn+1 |Yn , . . . , Yi+1 ) − H(Xn+1 |Yn , . . . , Yi )

n−1

H(Xn−i+1 |Yn−i , . . . , Y1 ) − H(Xn−i+1 |Yn−i , . . . , Y0 )

i=0

= =

i=0 n

H(Xj+1 |Yj , . . . , Y1 ) − H(Xj+1 |Yj , . . . , Y0 ) =

j=1

n

cj ,

j=1

where cj = H(Xj+1 |Yj , . . . , Y1 ) − H(Xj+1 |Yj , . . . , Y0 ), j = 1, . . . , n.

n Since cj ≥ 0 and j=1 cj < H(X1 ) < ∞ for all n, we conclude that cn → 0 as n → ∞. Moreover, cn = H(Xn+1 |Yn , . . . , Y1 ) − H(Xn+1 |Yn , . . . , Y0 ) = H(Y1 , . . . , Yn , Xn+1 ) − H(Y1 , . . . , Yn ) − H(Y0 , Y1 , . . . , Yn , Xn+1 ) + H(Y0 , Y1 , . . . , Yn ) = H(Y0 |Y1 , . . . , Yn ) − H(Y0 |Y1 , . . . , Yn , Xn+1 ). Finally, since H(Y0 |Y1 , . . . , Yn ) converges to h(Y ), so does H(Y0 |Y1 , . . . , Yn , Xn+1 ). Let us conclude this section with one general remark. Suppose Y is a factor of X , i.e. Y = φ(X ), where X is some ergodic process. For n, m ∈ N, let dn,m = H(Y0 |Y1 , . . . , Yn , Xn+1 , . . . , Xn+m ). Note that dn,m ≥ dn,m+1 ≥ 0, and hence limm→∞ dn,m =: Dn exists. Since for any n, m ∈ N dn,m = H(Y0 |Y1 , . . . , Yn , Xn+1 , . . . , Xn+m ) ≤ H(Y0 |Y1 , . . . , Yn , Yn+1 , . . . , Yn+m ), we conclude that Dn ≤ h(Y ). Note also that since dn,m ≤ dn+1,m−1 , one has Dn+1 ≥ Dn . The natural question is under which conditions does Dn converge to h(Y ) as n → ∞. For this we need a certain regularity of the conditional probabilities of the X -process. For example, if conditional probabilities are continuous, i.e., if    , . . .) , Xn+2 rn = sup sup P(X0 |X1 , . . . , Xn , Xn+1
X0 ,...,Xn X ,X”    − P(X0 |X1 , . . . , Xn , Xn+1 , Xn+2 , . . .) → 0, n → ∞,

then Dn → h(Y ). Gibbs measures and g-measures (see Section 4) have continuous conditional probabilities.

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2.1. Entropy via a prefix code In this section we recall the approach to efficient computation of entropies of factor processes Y = φ(X ), where X is Markov, which was originally proposed in [2, 7]. The inequalities of Theorem 2.1 can be rewritten as follows P(y1n )HP(·|y1n ) (Y0 ), (7) P(y1n )HP(·|y1n ) (Y0 |Xn+1 ) ≤ h(Y ) ≤ y1n ∈Bn

y1n ∈Bn

where we use the following notation y1n = (y1 , y2 , . . . , yn ) ∈ B n , P(y1n ) = P(Y1 = y1 , . . . , Yn = yn ), P(·|y1n ) = P(·|Y1 = y1 , . . . , Yn = yn ). The subindex P(·|y1n ) in (7) stresses that the entropy of Y0 and the conditional entropy of Y0 and Xn+1 is computed using P(·|y1n ). Note that the sum in (7) is taken over elements of a partition of B N into cylinders of length n:  
 y ∈ B Z : y˜1 = y1 , . . . , y˜n = yn }. Un = [y1n ] y1n ∈ B n , [y1n ] = {˜ In fact, an estimate similar to (7) holds for any partition of B Z into cylindric sets, see [7, Theorem 1]. Theorem 2.2. Let W be a finite partition of B Z into cylindric sets: M
. W = [wi ], wi = (wi,1 , . . . , wi,li ) i=1

Then



h1 (w) ≤ h(Y ) ≤

w∈W

where



h(w),

(8)

w∈W

h1 (w) = P(Y1 . . . Y|w| = w)H(Y0 |Y1 . . . Y|w| = w, X|w|+1 ), h(w) = P(Y1 . . . Y|w| = w)H(Y0 |Y1 . . . Y|w| = w).

Theorem (2.2) leads to the following algorithm. Suppose W is some partition into cylinders. We can refine the partition W by removing a certain word w from W and adding all words of the form wb, where b ∈ B, i.e., W  = W \ {w} ∪ {wb| b ∈ B}.

(9)

Suppose {Wk }k≥1 is a sequence of partitions such that for each k, Wk+1 is a refinement of Wk as in (9), and at each step a word w ∈ Wk is selected such that   h(w) − h1 (w) = max h(u) − h1 (u) . (10) u∈Wk

The greedy strategy (10), as well as some other strategies (e.g, uniform, |w| = minu∈W |u|), guarantees the convergence of the upper and lower estimates in (8), i.e.,

h(w) − h1 (w) = 0. lim k→∞

w∈Wk

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2.2. Entropy via renewal times. As before, suppose that {Xn } are independent and identically distributed in {d, . . . , k} with P(Xi = ) = p ,  = d, . . . , k. Assume also that pd > 0. Another method for estimating the entropy is based on the following observation. Suppose Yn = d − 2 for some n. This implies that Xn = d, ωn = (−1, 1). Since the sequence {ωk } forms a Markov chain, (. . . , ω1 , . . . , ωn−1 ) and (ωn+1 , ωn+2 , . . .), are independent given ωn . Therefore, since Yn = d−2 implies ωn = (−1, 1), we conclude that (. . . , ω1 , . . . , ωn−1 ) and (ωn+1 , ωn+2 , . . .) are independent given Yn = d − 2. Moreover, since Xn form an iid sequence, (. . . , Yn−2 , Yn−1 ) and (Yn+1 , Yn+2 , . . .) are also independent given Yn = d − 2. Consider our subshift O, and a set C = [d − 2] = {y ∈ O : y0 = d − 2}. Let S : O → O be a left shift, and consider an induced map SC on C: SC (y) = S RC (y) (y), where RC (y) = min{k ≥ 1 : yk = d − 2}. On C, the induced map SC has a natural Bernoulli partition
 [d − 2, y1 , . . . , yr , d − 2] : yj ∈ B, yj = d − 2, j = 1, . . . , r, r ∈ N . Finally, by the Abramov formula [1] h(Q) = −

∞



Q([d − 2, y1 , . . . , yr , d − 2]) log Q [d − 2, y1 , . . . , yr , d − 2])

r=1 y1 ,...,yr =d−2

+ Q([d − 2]) log Q([d − 2]).

(11)

Computation of entropy of images of Markov measures using the renewal times and induced map was used in the past, see e.g. [15]. However, in the case of bit-shift channel, the method based on (11) is extremely inefficient. 3. Numerics For illustration we present a numerical computation of the entropy using the prefix code method described in Section 2.1. The algorithm constructs a sequence of refined partitions Wk as described above. A particularly useful strategy is given by (10). This “greedy” heuristics selects the cylinder most responsible for the difference in upper and lower bound, in the hope that refining this cylinder will tighten the bounds quickly. This strategy is not optimal (as can be shown by example) but it has three advantages. Firstly, the bounds converge (eventually). Secondly, if in a particular word w ∈ W , the last symbol is the ”renewal” symbol d−2 (similarly k+2), this word will never be refined again. Thirdly, the next cylinder to expand can be found quickly by representing W as a “priority queue” data structure. For illustration, we run the algorithm for the model of the jitter channel described in Section 1.1. The parameters are inspired by the Compact Disc: The errorcorrection and modulation system of the CD essentially produces an RLL-sequence with parameters (d, k) = (2, 10). We model the run-lengths as independent identically distributed random variables with probabilities p = p2 γ −2 ,  ∈ {2, . . . , 10},

where γ = 0.658 and p2 is chosen such that p = 1. This truncated geometric

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2.6

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I(X;Y) rate EFM/CD

2.4

bits/run

2.2 2 1.8 1.6 1.4 0.001

0.01

0.1

epsilon Fig 2. Mutual information I(Y ; X ) = h(Y ) − h(Jε ) as a function of ε, for (d, k) = (2, 10) and the truncated geometric distribution for X .

model with γ = 0.658 is a very good approximation of the (marginal) run-length distribution observed on the CD. Figure 2 shows the mutual information I(Y ; X ) = h(Y ) − h(Jε ) = h(Y ) + 2ε log ε + (1 − 2ε) log(1 − 2ε) as a function of ε. The horizontal line represents the rate designed for the last stage of the encoding used in the CD (the so-called EFM code). If the jitter is so strong that the mutual information drops below this rate, reliable decoding is impossible. In practice, similar plots are used to evaluate the performance of particular encoding schemes with respect to various distortions introduced by the physical channel. Figure 3 compares the greedy and uniform heuristics. The standard estimate H(Y0 |Y1 , . . . , Yn ) in fact corresponds to the uniform refinement. Observe a superior rate of convergence for the greedy refinement strategy. 4. Thermodynamics of jittered measures Bernoulli and Markov measures belong to a wider class of the so-called Gibbs measures. Bernoulli and Markov measures are also examples of g-measures. In the seminal paper [10] M. Keane introduced a class of g-measures. These are the measures whose conditional probabilities are given by a continuous and strictly positive function g. For subshifts of finite type, the theory of g-measures is extensive. For sofic subshifts, the problem of defining g-measures is much more complicated. For the first results see the paper by W. Krieger [11] in this volume. The thermodynamic formalism allows to look at Gibbs measures from two different sides. First of all, locally, through the conditional probabilities; and secondly, globally, through the variational principles. Contrary to the class of g-measures, the class of Gibbs measures for a sofic subshift is well defined. The natural question is whether a ”jittered” measure Q = (P × Jε ) ◦ φ−1 is Gibbs. If the measure is Gibbs and the potential is identified, then,

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(upper - lower) bound [bit/run]

0.1

greedy uniform

0.01 0.001 0.0001 1e-05 1e-06 1e-07 1e-08 10

100

1000

10000

100000

# cylinders

[t]

Fig 3. Difference between upper and lower bounds on entropy as a function of |W |, the number of cylinders, in partitions built by the greedy and uniform refinement strategies.

using the variational principle, we obtain another method of computing the entropy of Q. The subshift O ⊂ B Z satisfies a specification property (as a factor of a subshift of finite type A Z × ΩJ which has a specification property [6]). Hence the results of [17] on existence of Gibbs measures for expansive dynamical systems with the specification property are applicable. If Q would be a Gibbs measure for potential f from the Bowen class V(O), then there would exist positive constants c, C such that for any n > 0 and every y ∈ O c≤

Q([y0 , . . . , yn ])

n ≤ C, exp( k=0 f (S i (y)) − (n + 1)P (f ))

(12)

where S : O → O is a left shift and P (f ) is the topological pressure of f . As a corollary of (12) one easily concludes that log Q(y0 |y1 , . . . , yn ) = log

Q([y0 , . . . , yn ]) Q([y1 , . . . , yn ])

should be bounded, which is not the case, see [18]. Hence, Q is not Gibbs for any potential from the large class of potentials V(O). Examples of measures Q such that estimates similar to (12) hold for some continuous f and subexponential bounds cn and Cn (limn n−1 log cn = limn n−1 log Cn = 0) have been considered [14, 20], and were shown to be weakly Gibbs. It is not known whether Q for the bit-shift channel is weakly Gibbs for some continuous potential f . Nevertheless, the thermodynamic formalism could be useful in estimating the capacity of the bit-shift channel. We recall the notion of a compensation function and some results summarized in [19]. First of all, we define the topological pressure of real-valued continuous functions defined on A Z × ΩJ and O. If f ∈ C(A Z × ΩJ ), g ∈ C(O), the topological pressures of f and g are defined as       Z f dS , P (g|O) = sup h(Q) + g dQ , P (f |A × ΩJ ) = sup h(S) + Q

S

A Z ×ΩJ

O

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where the suprema are taken over all translation invariant measures on A Z × ΩJ and O, respectively. A measure S on A Z × ΩJ is called an equilibrium state for f ∈ C(A Z × ΩJ ) if  Z (13) P (f |A × ΩJ ) = h(S) + f dS. We define equilibrium states on O in a similar way. It is well known that every measure is an equilibrium state: for every translation invariant measure S on A Z × ΩJ one can find a continuous function f : A Z × ΩJ → R such that (13) holds. Moreover, for any S = P × Jε , such an f is of a special form f (x, ω) = f˜(x) + jε (ω), where f˜ : A Z → R and jε : ΩJ → R are continuous functions. (In fact, jε can be found explicitly.) A continuous function F : A Z × ΩJ → R is a compensation function if P (F + g ◦ φ|A Z × ΩJ ) = P (g|O) for all g ∈ C(O). Compensation functions exist for factor maps defined on shifts of finite type [19]. An important result is the so-called relative variational principle [12, 19], which in our notation states that F is a compensation function if and only if for any invariant measure Q on O one has      h(Q) = sup h(S) + F dS  S ◦ φ−1 = Q . S

Suppose F is a compensation function, then for Q = (P × Jε ) ◦ φ−1 we obtain   h(Q) ≥ h(P × Jε ) + F d(P × Jε ) = h(P) + h(Jε ) + Fε dP, (14)

 where Fε (x) = ΩJ F (x, ω)Jε (dω). For the capacity of the bit-shift channel we obtain the following lower estimate Cbitshift (Jε ) =

h(Q) − h(Jε )

sup Q=(P×Jε )◦φ−1



≥ sup h(P) + P



 Fε dP = P (Fε |A Z ).

(15)

An interesting question is whether the inequalities in (14) and (15) are strict. The inequality (14) is most probably strict in the generic situation. Indeed, by Corollary 3.4 [19], if Q is an equilibrium state for g on O, and S is such that S ◦ φ−1 = Q and  h(Q) = h(S) + F d S, then S is an equilibrium state for F + g ◦ φ, and conversely. On the other hand if S = P × Jε , then S is an equilibrium state for f (x, ω) = f˜(x) + jε (ω). Therefore, for the equality in (14), it is necessary that F (x, ω) + (g ◦ φ)(x, ω) and f˜(x) + jε (ω) are physically equivalent, i.e., have the same set of equilibrium states. In fact, it is quite difficult to imagine how for a given compensation function F of the bit-shift

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channel and a generic g one could find f˜ to satisfy the requirement of physical equivalence. On the other hand, it is not very difficult to see that in fact Cbitshift (Jε ) = sup P (Fε |A Z ), F

(16)

where the supremum is taken over all compensation functions F . Indeed, suppose Q∗ = (P∗ × Jε ) ◦ φ−1 is a ‘maximal’ ergodic measure, i.e., Cbitshift (Jε ) = h(Q∗ ) − h(Jε ). Then there exist continuous functions g ∗ ∈ C(O) and f˜∗ ∈ C(A Z ) such that Q∗ and P∗ are equilibrium states for g ∗ and f˜∗ , respectively. But then F (x, ω) = f˜∗ (x) + jε (ω) − (g ∗ ◦ φ)(x, ω) is the compensation function for which the maximum in (16) is attained. Thus methods for dealing with factor systems developed in dynamical systems, could be applied to estimate channel capacities. The practicality of such estimates depends strongly on whether one is able to understand the structure of a class of compensation function for a given channel. Probably, in many concrete cases, a relatively large family of compensation functions will suffice as well. References [1] Abramov, L. M. (1959). The entropy of a derived automorphism. Dokl. Akad. Nauk SSSR 128, 647–650. MR0113984 [2] Baggen, S., and Balakirsky, V. (2003). An efficient algorithm for computing the entropy of output sequences for bitshift channels. Proc. 24th Int. Symposium on Information Theory in Benelux , 157–164. [3] Birch, J. J. (1962) Approximation for the entropy for functions of Markov chains. Ann. Math. Statist. 33, 930–938. MR0141162 [4] Birch, J. J. (1963) On information rates for finite-state channels. Information and Control 6, 372–380. MR0162651 [5] Cover, T. M., and Thomas, J. A. (1991). Elements of Information Theory. Wiley Series in Telecommunications. John Wiley & Sons Inc., New York. MR1122806 [6] Denker, M., Grillenberger, C., and Sigmund, K. (1976). Ergodic Theory on Compact Spaces. Springer-Verlag, Berlin. Lecture Notes in Mathematics 527. MR0457675 [7] Egner, S., Balakirsky, V., Tolhuizen, L., Baggen, S., and Hollmann, H. (2004) On the entropy rate of a hidden Markov model. Proceedings International Symposium Information Theory, ISIT 2004 . [8] Froyland, G., Junge, O., and Ochs, G. (2001). Rigorous computation of topological entropy with respect to a finite partition. Phys. D 154, 1–2, 68–84. MR1840806 [9] Immink, K. (1999). Codes for Mass Data Storage Systems. Shannon Foundation EEE Publishers, The Netherlands. [10] Keane, M. (1972). Strongly mixing g-measures. Invent. Math. 16, 309–324. MR0310193 [11] Krieger, W. (2006) On g-functions for subshifts. In Dynamics and Stochastics, IMS Lecture Notes-Monograph Series, Vol. 48, 306–316.

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[12] Ledrappier, F., and Walters, P. (1977). A relativised variational principle for continuous transformations. J. London Math. Soc. (2) 16, 3, 568–576. MR0476995 [13] Lind, D., and Marcus, B. (1995). An Introduction to Symbolic Dynamics and Coding. Cambridge University Press. MR1369092 [14] Maes, C., Redig, F., Takens, F., van Moffaert, A., and Verbitski, E. (2000). Intermittency and weak Gibbs states. Nonlinearity 13, 5, 1681–1698. MR1781814 [15] Marcus, B., Petersen, K., and Williams, S. ( 1984). Transmission rates and factors of Markov chains. In Conference in Modern Analysis and Probability (New Haven, Conn., 1982), vol. 26 of Contemp. Math. Amer. Math. Soc., Providence, RI, pp. 279–293. MR1369092 [16] Marcus, B. H., Roth, R. M., and Siegel, P. H. (2001). An Introduction to Coding of Constrained Systems. Lecture Notes, fifth edition. [17] Ruelle, D. (1992). Thermodynamic formalism for maps satisfying positive expansiveness and specification. Nonlinearity 5, 6, 1223–1236. MR1192516 [18] van Enter, A. C. D., and Verbitskiy, E. A. (2004). On the variational principle for generalized Gibbs measures. Markov Process. Related Fields 10, 3, 411–434. MR2097865 [19] Walters, P.(1986). Relative pressure, relative equilibrium states, compensation functions and many-to-one codes between subshifts. Trans. Amer. Math. Soc. 296, 1, 1–31. MR837796 [20] Yuri, M. (1999). Thermodynamic formalism for certain nonhyperbolic maps. Ergodic Theory Dynam. Systems 19, 5, 1365–1378. MR1721626

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 286–303 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000301

Nearly-integrable perturbations of the Lagrange top: applications of KAM-theory∗ H. W. Broer1 , H. Hanßmann2,3 , J. Hoo1 and V. Naudot1 Groningen University, RWTH Aachen and Groningen University Abstract: Motivated by the Lagrange top coupled to an oscillator, we consider the quasi-periodic Hamiltonian Hopf bifurcation. To this end, we develop the normal linear stability theory of an invariant torus with a generic (i.e., nonsemisimple) normal 1 : −1 resonance. This theory guarantees the persistence of the invariant torus in the Diophantine case and makes possible a further quasiperiodic normal form, necessary for investigation of the non-linear dynamics. As a consequence, we find Cantor families of invariant isotropic tori of all dimensions suggested by the integrable approximation.

1. The Lagrange top The Lagrange top is an axially symmetric rigid body in a three dimensional space, subject to a constant gravitational field such that the base point of the bodysymmetry (or figure) axis is fixed in space, see Figure 1. Mathematically speaking, this is a Hamiltonian system on the tangent bundle T SO(3) of the rotation group SO(3) with the symplectic 2-form σ. This σ is the pull-back of the canonical 2-form on the co-tangent bundle T ∗ SO(3) by the bundle isomorphism κ ˜ : T SO(3) → T ∗ SO(3) induced by a non-degenerate left-invariant metric κ on SO(3), where κ ˜ (v) = κ(v, ·). The Hamiltonian function H of the Lagrange top is obtained as the sum of potential and kinetic energy. In the following, we identify the tangent bundle T SO(3) with the product M = SO(3) × so(3) via the map vQ ∈ T SO(3) → (Q, TId L−1 Q v) ∈ SO(3) × so(3), where so(3) = TId SO(3) and LQ denotes left-translation by Q ∈ SO(3). We assume that the gravitational force points vertically downwards. Then, the Lagrange top has two rotational symmetries: rotations about the figure axis and the vertical axis e3 . We let S ⊂ SO(3) denote the subgroup of rotations preserving the vertical axis e3 . Then, for a suitable choice of the space coordinate system (e1 , e2 , e3 ), the two symmetries correspond to a symplectic right action Φr and to a symplectic left action Φl of the Lie subgroup S on M . By the Noether Theorem [1, 4], these Hamiltonian symmetries give rise to * Work

supported by grant MB-G-b of the Dutch FOM program Mathematical Physics and by grant HPRN-CT-2000-00113 of the European Community funding for the Research and Training Network MASIE. 1 Department of Mathematics, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands, e-mail: [email protected] e-mail: [email protected] e-mail: [email protected] 2 Institut f¨ ur Reine & Angewandte Mathematik, RWTH Aachen, 52056 Aachen,Germany. 3 Present address: Mathematisch Instituut, Universiteit Utrecht, Postbus 80010, 3508 TA Utrecht, The Netherlands, e-mail: [email protected] AMS 2000 subject classifications: primary 37J40; secondary 70H08. Keywords and phrases: KAM theory, quasi-periodic Hamiltonian Hopf bifurcation, singular foliation, the Lagrange top, gyroscopic stabilization. 286

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e3

Center of mass M e2

O

Gravity e1

Fig 1. The Lagrange top in the vertical gravitational force field.

integrals Mr and Ml of the Hamiltonian H: the angular momenta along the figure axis and along the vertical axis, respectively. These integrals induce the so-called energy-momentum map EM := (Mr , Ml , H) : M → R3 . The inverse images of the map EM divide the phase space M into invariant sets of the Hamiltonian system XH associated with H: for a regular value m = (a, b, h) ∈ R3 , the set EM−1 (m) ⊂ M is an XH -invariant 3-torus; for a critical value m, this set is a ‘pinched’ 3-torus, a 2-torus or a circle. This division is a singular foliation of the phase space by XH -invariant tori. This foliation gives rise to a stratification of the parameter space: the (a, b, h)-space is split into different regions according to the dimension of the tori EM−1 (a, b, h). To describe the stratification, one applies a regular reduction [1, 28] by the right symmetry Φr to the three-degrees-of-freedom Hamiltonian H as follows. For a fixed value a, we deduce from H a two-degrees-offreedom Hamiltonian Ha on the four-dimensional orbit space Ma = (Mr )−1 (a)/S under the Φr -action. The reduced
phase space Ma can be identified with  the four3 3 dimensional submanifold Ra = (u, v) ∈ R × R : u · u = 1, u · v = a , compare with [13, 16]. This manifold Ra inherits the symplectic 2-form ωa given by   ωa (u, v) (x, y), (p, q) = x ˆ · qˆ − yˆ · pˆ + v · (ˆ v × pˆ), (1.1) where (x, y) = (ˆ x × u, x ˆ × v + yˆ) ∈ T(u,v) Ra , (p, q) = (ˆ p × u, pˆ × v + qˆ) ∈ T(u,v) Ra . Here · and × denote the standard inner and cross product of R3 , respectively. The reduced Hamiltonian Ha : Ra → R obtains the form Ha (u, v) =

1 v · v + cu3 + ρa2 , 2

where c > 0 and ρ ∈ R. Observe that Pa = (0, 0, 1, 0, 0, a) ∈ Ra is an isolated equilibrium of Ha . For each a, the point Pa — a relative equilibrium of the full system H — corresponds to a periodic solution of H, namely a rotation about the vertical axis. It is known [13, 17] that a stability transition of the rotations Pa occurs√as the angular momentum value a passes through the critical value a = a0 = 2 c. Physically, this transition is referred to as gyroscopic stabilization of the Lagrange top. A mathematical explanation for the stabilization is that Floquet matrices Ωa of the periodic orbits Pa changes from hyperbolic into elliptic as a passes through a0 , see Figure 2. In fact, the Lagrange top undergoes a (non-linear) Hamiltonian Hopf bifurcation [13, 17]. A brief discussion of this bifurcation is given in Section 2.1

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i a 2 0

− 2i a0

Hyperbolic: a < a0

Resonant: a = a0

Elliptic: a > a0

Fig 2. Eigenvalue configuration of Floquet matrix Ωa as a passes through a0 .

above surface: Lagrangian 3-tori elliptic 2-tori

linearly unstable periodic solutions

linearly stable periodic solutions

µ2

Hamiltonian Hopf bifurcation point (resonant periodic orbit)

Fig 3. Sketch of the local stratification by invariant tori near the Hamiltonian Hopf bifurcation point in the (a, b, h)-space. After a suitable reparametrization, the surface is a piece of the swal2 lowtail catastrophe set [17, 29, 37]. The parameter µ2 is given by a4 − c.

below; for an extensive treatment see [29]. Following [13, 17, 29, 30], the local stratification of the parameter space — associated with the singular foliation by invariant tori — near the bifurcation point is described by a piece of the swallowtail catastrophe set from singularity theory [37] as follows. The one-dimensional singular part of this surface is the stratum associated with the periodic solutions, the regular part forms the stratum of 2-tori and the open region above the surface is the stratum of Lagrangian 3-tori, compare with Figure 3. We are interested in a perturbation problem where the Lagrange top is weakly coupled to an oscillator with multiple frequencies, e.g., the base point of the top is coupled to a vertically vibrating table-surface by a massless spring, see Figure 4. In this example, the spring constant, say ε, plays the role of the perturbation parameter. More generally, we consider the Hamiltonian perturbation Hε of the form Hε = H + G + εF,

(1.2)

where H and G are the Hamiltonians of the Lagrange top and of the quasi-periodic oscillator, respectively. Here the function F depends on the coupling between the top and the oscillator. We assume that the quasi-periodic oscillator has n ≥ 1 frequencies and is Liouville-integrable [1, 4]. Then, the unperturbed integrable Hamiltonian H0 = H + G contains invariant (n + 1)-, (n + 2)- and (n + 3)-tori. Near gyroscopic stabilization of the Lagrange top, this Hamiltonian H0 gives a similar

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Fig 4. The Lagrange top coupled with a vibrating table-surface by a spring.

(local) stratification by tori as sketched in Figure 3 in a parameter space, but with invariant (n + 1)-, (n + 2)- and (n + 3)-tori, for details see [7]. Our concern is with the fate of these invariant tori for small but non-zero ε. By KAM theory [2, 10, 12, 27, 33, 35, 39], the ‘majority’ of the invariant tori from the local stratification survives small perturbations. They form Whitney-smooth Cantor families, parametrized over domains with positive measure. The purpose of the present paper is to describe the persisting families of tori in terms of a quasi-periodic Hamiltonian Hopf bifurcation [7, 10]. From [13, 17] it is known that the Lagrangian torus bundle in the unperturbed Lagrangian top contains nontrivial monodromy. More precisely, we consider the local stratification by tori as sketched in Figure 3. Let D be a punctured disk in the stratum of Lagrangian 3-tori which transversally intersects the ‘thread’ (the 1-dimensional curve associated with unstable periodic solutions). Then, the Lagrangian torus bundle with the boundary ∂D ∼ = S1 as the base space is non-trivial. From this we conclude that the Lagrangian (n+2)-torus bundle of the unperturbed Hamiltonian H0 has non-trivial monodromy. We like to mention that, in view of the global KAM theory [6], there exists a proper extension of the non-trivial monodromy in the integrable Lagrangian torus bundle, to the nearly-integrable one. In this sense, we may say that the non-trivial monodromy, that goes with the Hamiltonian Hopf bifurcation and is centered at the thread, survives a small non-integrable perturbation. 2. Hamiltonian Hopf bifurcations of equilibria, periodic and quasi-periodic solutions The above persistence problem is part of a more general study of a quasi-periodic Hamiltonian Hopf bifurcation to be discussed in Section 2.2 below. This quasiperiodic bifurcation can be considered as a natural extention of the Hamiltonian Hopf bifurcation of equilibria [29]. Let us first recall certain facts about the latter. 2.1. The Hamiltonian Hopf bifurcation of equilibria We consider a two-degrees-of-freedom Hamiltonian on R4 = {z1 , z2 , z3 , z4 } with ˆ = H ˆ µ be a family the standard symplectic 2-form dz1 ∧ dz3 + dz2 ∧ dz4 . Let H of Hamiltonian functions with the origin as an equilibrium. Assume that for µ = µ0 , the linearized Hamiltonian system (at the origin) has a double pair of purely imaginary eigenvalues with a non-trivial nilpotent part.1 Moreover, as µ passes ˆ is in generic or (non-semisimple) 1 : −1 resonance this case, we say that the Hamiltonian H at the origin for µ = µ0 . Notice that the linear part at the origin is indefinite. 1 In

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through the value µ0 , the eigenvalues of the linear part (at the origin) behave as follows: as µ increases a complex quartet moves towards the imaginary axis, meeting there for µ = µ0 and splitting into two distinct purely imaginary pairs for µ > µ0 , compare with Figure 2. This bifurcation is referred to as a generic Hamiltonian Hopf bifurcation, provided that a certain generic condition on the higher order terms is met. ˆ with respect to the quadratic More precisely, we normalize the Hamiltonian H ˆ This gives us the Birkhoff normal form part of H. ˆ = (ν1 (µ) + λ0 )S + N + ν2 (µ)M + b1 (µ)M 2 H + b2 (µ)SM + b3 (µ)S 2 + h.o.t.,

(2.1)

where M = 21 (z12 + z22 ), S = z1 z4 − z2 z3 and N = 12 (z32 + z42 ), see [24, 29, 30]. Here λ0 = 0, and ν1 (0) = 0 = ν2 (0). The generic condition now requires that ∂ν2 ∂µ (µ0 ) = 0 and b1 (µ0 ) = 0. We focus on the supercritical case where b1 (µ0 ) > 0. ˆ is invariant under the S1 -action generated by the flow of We may assume that H the semi-simple quadratic part S, since this symmetry can be pushed through the normal form (2.1) up to an arbitrary order [29]. Then, the periodic solutions of ˆ are given by the singularities of the energy-momentum map (H, ˆ S). Following H [29], this map has a (singularity theoretical) normal form (G, S) where G = N + ˆ S) and (G, S) are locally left-right equivalent by δ(µ)M +M 2 . Indeed, the maps (H, 1 an S -equivariant origin-preserving diffeomorphism on R4 and an origin-preserving diffeomorphism on R2 . As a result, the set of critical values C of (G, S), considered as ˆ S), which is a piece the graph over the parameter µ, is diffeomorphic to that of (H, of swallowtail surface [37] in the (δ, S, G)-space. This critical set C determines the local stratification by leaves of (G, S) near the bifurcation point in the parameter space: strata of equilibria, periodic orbits and 2-tori. This stratification corresponds to the situation as sketched in Figure 3, when replacing periodic solutions, 2-tori and 3-tori by equilibria, periodic orbits and 2-tori, respectively. Remarks 2.1. 1. The above discussion also applies for the situation where µ is a multiparameter [30]. ˆ is not S1 -symmetric, one first applies 2. In the case where the Hamiltonian H a Liapounov-Schmidt reduction to obtain an S1 -symmetric Hamiltonian H ˆ up to arbitrary order [29, 30, 42]. This which has the same normal form as H ˆ to that of H in a diffeomorphic reduction relates the periodic solutions of H way. 2.2. The Hamiltonian Hopf bifurcation of (quasi-) periodic solutions Motivated by the persistence problem of the Lagrange top coupled to an oscillator, see Section 1, we consider the quasi-periodic dynamics of Hamiltonian systems with more degrees of freedom near an invariant resonant torus, where our main interest is with the normal 1 : −1 resonance. More precisely, our phase is given  space M  m m 4 dxi ∧ dyi + dzj ∧ by T × R × R = {x, y, z} with symplectic 2-form σ = dzj+2 , where Tm = Rm /(2πZ)m . This space M admits a free Tm -action given by (θ, (x, y, z)) ∈ Tm × M → (θ + x, y, z) ∈ M . A Hamiltonian function is said to be Tm -symmetric or integrable if it is invariant under this Tm -action, compare with

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[12, 26]. We consider a p-parameter family of integrable Hamiltonian functions 1 H(x, y, z; ν) = ω(ν), y + Jz, Ω(ν)z + h.o.t. , 2

(2.2)

where ν ∈ Rp is the parameter, ω(ν) ∈ Rn , Ω(ν) ∈ sp(4, R) and J is the standard m symplectic 4×4-matrix. all higher order terms are x-independent.  By T -symmetry Then the union T = ν Tν ⊆ M × Rp , where Tν = {(x, y, z, ν) : (y, z) = (0, 0)}, is a p-parameter family of invariant m-tori of H parametrized by ν. Let the torus Tν0 be in generic normal 1 : −1 resonance, meaning that the Floquet matrix Ω(ν0 ) at Tν0 has a double pair of purely imaginary eigenvalues with a non-trivial nilpotent part. Note that for m = 0 we arrive at the setting of the equilibria case [29], compare with Section 2.1. For m = 1, the invariant submanifolds Tν are periodic solutions of the Hamiltonian H. For this reason, we speak of the periodic case. Similarly, we refer to our present setting with m ≥ 2 as the quasi-periodic case. Let us consider the local (quasi-)periodic dynamics of the family H with m ≥ 1 near the resonant torus Tν0 . To this end, we first reduce the Hamiltonian H by the Tm -symmetry as follows. For a small fixed value α ∈ Rm , we obtain from the Hamiltonian H a two-degrees-of-freedom Hα defined on the space Mα = (M/Tm ) ∩ {y = α}, where M/Tm denotes the orbit space of the Tm -action. More explicitly, we obtain Hα (z; ν) = H(x, α, z; ν), by identifying Mα with R4 = {z1 , z2 , z3 , z4 }. We may assume that the reduced Hamiltonian Hα is in normal form with respect to the S1 -symmetry generated by the semi-simple part of the quadratic term 1 2 Jz, Ω(ν0 )z [29]. Then, the full Hamiltonian H is invariant under this circleaction. We say that the full Hamiltonian H undergoes a generic periodic (m = 1) or quasi-periodic (m ≥ 2) Hamiltonian Hopf bifurcation at ν = ν0 , if the reduced two-degrees-of-freedom system Hα has a generic Hamiltonian Hopf bifurcation, see Section 2 and Remark 2.1. As before we restrict to the supercritical case. In the following, we focus on the quasi-periodic case where m ≥ 2. For a treatment of the periodic case see [34]. Applying the local analysis of Section 2.1 to the reduced Hamiltonian Hα , we conclude that the local torus foliation of M near the resonant torus Tν0 defines a local stratification in a parameter space by m-, (m + 1)- and (m+2)-tori of the full system H. This stratification is sketched in Figure 5, compare with Figure 3. Notice that the resonant torus Tν0 corresponds to the quasi-periodic Hamiltonian Hopf bifurcation point. Our main goal is to investigate the persistence of the invariant m-, (m + 1)and (m + 2)-tori from the local stratification as in Figure 5(a), when the integrable Hamiltonian H is perturbed into a nearly-integrable (i.e., not necessarily Tm -symmetric) one. Observe that in the example of the Lagrange top coupled with a quasi-periodic oscillator one has m = n + 1, see Section  1. In the sequel we examine the persistence of the p-parameter family T = ν Tν of invariant m-tori. This family consists of elliptic, hyperbolic tori and the resonant torus Tν0 , compare with Figure 5(a). The ‘standard’ KAM theory [2, 12, 26, 31–33, 35, 36, 39] yields persistence only for subfamilies of the family T , containing elliptic or hyperbolic tori. The problem is that the resonant torus Tν0 gives rise to multiple Floquet exponents, compare with Figure 2. To deal with this problem, we develop a normal linear stability theorem [10], as an extension of the ‘standard’ KAM theory. The persistence of the (m + 1)- and (m + 2)-tori will be discussed in Section 4.

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above surface: Lagrangian (m+2)-tori

at surface: elliptic (m+1)-tori

at crease: normally elliptic m-tori at thread: normally hyperbolic m-tori

µ2

quasi-periodic Hamiltonian Hopf bifurcation point (normally resonant 1:-1 torus)

(a) Unperturbed situation

Cantor family of elliptic (m+1)-tori Cantor family of Lagrangian (m+2)-tori

Cantor family of m-tori

(b) Perturbed situation Fig 5. (a) Singular foliation by invariant tori of unperturbed integrable Hamiltonian H near the resonant torus in the supercritical case; (b) sketch of the Cantor families of surviving Diophantine  invariant tori in the singular foliation of the perturbed, nearly-integrable Hamiltonian H.

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3. The quasi-periodic Hamiltonian Hopf bifurcation: persistence of the m-tori In this section we develop the normal linear stability theory, needed for the persistent m-tori [10]. This theory also allows us to obtain a further quasi-periodic normal form of perturbations, necessary for investigation of the non-linear nearly-integrable dynamics. 3.1. Normal linear stability: a part of KAM theory Let us consider a general p-parameter family K = Kν of integrable real-analytic Hamiltonian functions on = Tm × Rm × R2q = {x, y, z} with mthe phase spaceM q the symplectic 2-form i=1 dxi ∧ dyi + j=1 dzj ∧ dzj+2 . We assume that the Hamiltonian vector field X = XK associated with K is of the form ∂ ∂ 2 + [Ω(ν)z + O(|y| , |z| )] , (3.1) ∂x ∂z and Ω(ν) ∈ sp(2q, R). Then, the torus family T = ∪ν Tν , where

Xν (x, y, z) = [ω(ν) + O(|y| , |z|)] where ω(ν) ∈ Rm

Tν = {((x, y, z), ν) ∈ M × Rp : (y, z) = (0, 0)} , is an XK -invariant submanifold. Note that the integrable Hamiltonian H given by (2.2) is a special form of the family K. A typical KAM-stability question is concerned with the persistence of the invariant submanifold T under small perturbation of the integrable Hamiltonian family K. We refer to Ω(ν) as the Floquet (or normal) matrix of the torus Tν . The ‘standard’ KAM theory [2, 12, 26, 27, 31–33, 35, 36, 39] asserts that the ‘majority’ of these invariant tori survives small perturbations, provided that the unperturbed family satisfies the following conditions: (a) the Floquet exponents of the torus Tν0 (i.e., the eigenvalues of Ω(ν0 )) are simple; (b) the matrix Ω(ν0 ) is non-singular; (c) the product map ω×Ω : Rp → Rm ×sp(2q, R) is transversal to the submanifold {ω(ν0 )} × Orbit Ω(ν0 ), where Orbit Ω(ν0 ) denotes the similarity class of Ω(ν0 ) by the linear symplectic group; (d) the internal and normal frequencies satisfy Diophantine conditions. Let us be more specific about these assumptions. The internal frequencies of the invariant torus Tν are given by ω(ν) = (ω1 (ν), . . . , ωm (ν)), and the normal frequencies consist of the positive imaginary parts of the eigenvalues of Ω(ν). Conditions (a) and (b) require that the eigenvalues of Ω0 = Ω(ν0 ) have distinct non-zero eigenvalues. Condition (c) means that the map ω is submersive at ν = ν0 and the map Ω is a versal unfolding of Ω0 (in the sense of [3, 22]) simultaneously. The Hamiltonian K is said to be non-degenerate at the torus Tν0 , if it meets conditions (b) and (c). Remark 3.1. The non-degeneracy has the following geometrical interpretation: ∂ ∂ + Ω(ν)z ∂z is transversal to the conjugacy the normal linear part N Xν = ω(ν) ∂x class of N X0 within the space of normally affine Hamiltonian vector fields as in [3, 10, 22]. Denoted by ω N (ν) = (ω1N (ν), . . . , ωrN (ν)) — called the normal frequencies of the torus Tν — condition (d) is formulated as follows: for a constant τ > m − 1 and a parameter γ > 0, we have that   ω(ν), k + ω N (ν),   ≥ γ |k|−τ , (3.2)

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for all k ∈ Zm \{0} and for all  ∈ Z2 , where |1 |+· · ·+|r | ≤ 2. Due to the simpleness assumption (a), the number r of the normal frequencies is independent of ν, for ν sufficiently close to ν0 , compare with [12, 26]. The map F : ν → (ω(ν), ω N (ν)) is called the frequency map. We denote by Γτ,γ (U ) the set of parameters ν ∈ U such that F(ν) satisfy the non-resonant condition (3.2). Also we need the subset Γτ,γ (U  ), where the set U  ⊂ U is given by
   U  = ν ∈ U : dist. (ω(ν), ω N (ν)), ∂F(U ) > γ . (3.3)

Observe that for γ sufficiently small, the set U  is still an non-empty open neighbourhood of ν0 and that Γτ,γ (U  ) ⊂ U  ⊂ U contains a ‘Cantor set’ of Diophantine frequencies with positive measure. Remarks 3.2. – Condition (a) ensures that the Floquet matrices Ω(ν) are semi-simple and that the normal frequencies (after a suitable reparametrization) depend on parameters in an affine way. – Let Λ be the set of (ω, ω N ) ∈ Rm × Rr satisfying the Diophantine conditions (3.2). Then Λ is a nowhere dense, uncountable union of closed half lines. The intersection Λ ∩ Sm+r−1 with the unit sphere of Rm × Rr is a closed set, which by Cantor-Bendixson theorem [25] is the union of a perfect set P and a countable set. Note that the complement of Λ ∩ Sm+r−1 contains the dense set of resonant vectors (ω, ω N ). Since all points of Λ are separated by the resonant hyperplanes, this perfect set P is totally disconnected and hence a Cantor set. In Sm+r−1 this Cantor set tends to full Lebesgue measure as γ ↓ 0. We refer to the set of frequencies (ω(ν), ω N (ν)) satisfying (3.2) as a ‘Cantor set’ — a foliation of manifolds over a Cantor set. Though condition (a) is generic,2 it excludes certain interesting examples like the quasi-periodic Hamiltonian Hopf bifurcation, as it occurs in the example of the Lagrange top coupled to a quasi-periodic oscillator. Indeed, such systems have a Floquet matrix with multiple eigenvalues, compare with Figure 2. To deal with this problem, we have to drop the simpleness assumption. Instead we impose a H¨older condition on the spectra Spec Ω(ν) of the matrices Ω(ν) as follows. Let U ⊂ Rp be a small neighbourhood of ν0 . Suppose that Ω has a holomorphic extension to the complex domain U + r0 = {˜ ν ∈ Cp : ∃ ν ∈ U such that |ν − ν˜| ≤ r0 } ⊂ Cp

(3.4)

for a certain constant r0 > 0. We say that Ω(ν) is (θ, r0 )-H¨ older, if there exist positive constants θ and L such that the following holds: for any ν˜ ∈ U + r0 , ν ∈ U ˜ ∈ Spec Ω(˜ and for any λ ν ), there exist a λ ∈ Spec Ω(ν) such that     θ ˜ Im λ − Im λ (3.5)  ≤ L |ν − ν˜| , 

where Im denotes the imaginary part. The condition (3.5) holds in particular for matrices with simple eigenvalues. As an extension of the ‘standard’ KAM theory, we have the following. Theorem 3.3 (Normal linear stability [10]). Let K = Kν be a p-parameter real-analytic family of integrable Hamiltonians with the corresponding Hamiltonian 2 The

set of simple matrices is dense and open in the matrix space gl(2q, R).

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vector fields given by (3.1). Suppose that K satisfies the non-degeneracy conditions (b) and (c). Also assume that the matrix family Ω(ν) is (θ, r0 )-H¨ older, see (3.5).  Then, for γ > 0 sufficiently small and for any real-analytic Hamiltonian family K sufficiently close to K in the compact-open topology on complex analytic extensions, there exists a domain U around ν0 ∈ Rp and a map Φ : M × U → M × Rp , defined near the torus Tν0 , such that, i. Φ is a C ∞ -near-the-identity diffeomorphism  onto its image; ii. The image of the Diophantine tori V = ν∈Γτ,γ (U
) ( Tν × {ν} ) under Φ  is K-invariant, and the restriction of Φ on V conjugates the quasi-periodic  motions of K to those of K; iii. The restriction Φ|V is symplectic and preserves the (symplectic) normal linear ∂ ∂ part3 N X = ω(ν) ∂x + Ω(ν)z ∂z of the Hamiltonian vector field X = XK associated with K. We refer to the Diophantine tori V (also its diffeomorphic image Φ(V )) as a Cantor family of invariant m-tori, as it is parametrized over a ‘Cantor set’. The stability Theorem 3.3 includes the cases where the Floquet matrix Ω(ν0 ) is in (nonsemisimple) 1 : −1 resonant, see [10] for details. For the definition of 1 : −1 resonance see Section 2.2. In particular, it is applicable to our persistence problem as formulated in 2.2, regarding the invariant m-tori with a normally 1 : −1 resonant torus, compare with Figure 5(a). Remark 3.4. Theorem 3.3, as is generally the case in the ‘standard’ KAM-theory, requires the invertibility of the Floquet matrix of the central torus Tν0 . We expect that this assumption can be relaxed by using an appropriate transversality condition, compare with Remark 3.1. For recent work in this direction see [9, 43, 44]. 3.2. Persistence of m-tori We return to the setting of Section 2.2. Briefly summarizing, we consider a pparameter real-analytic family H = Hν of Tm -symmetric (or integrable) Hamiltonian functions on the space M = Tm × Rm × R4 given by (2.2), parametrized over  a small neighbourhood U of ν0 . This family has an invariant torus family T = ν Tν , where Tν = {(x, y, z, ν) ∈ M × Rp : (y, z) = (0, 0)} .

Moreover, the Hamiltonian H undergoes a supercritical quasi-periodic Hamiltonian Hopf bifurcation at ν = ν0 . By Tm -symmetry and an application of [29], near the normally resonant torus Tν0 there is a local singular foliation by invariant hyperbolic and elliptic m-tori, elliptic (m + 1)-tori and Lagrangian (m + 2)-tori of the Hamiltonian H. The local stratification associated with this foliation in a suitable parameter space is a piece of swallowtail and is sketched by Figure 5(a). Presently, we are concerned with the persistence of these m-tori from the local foliation under perturbation. By assumption, the central m-torus Tν0 is (normally) generically 1 : −1 resonant. We also assume that the Hamiltonian H is non-degenerate at the invariant tori Tν0 , 3 For

a discussion on symplectic normal linearization, see [12, 26].

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see Section 3.1. As a direct consequence of this non-degeneracy, we need (at least) m + 2 parameters, that is, p ≥ m + 2. Let X = Xν be the family of Hamiltonian vector fields corresponding to the Hamiltonian family H = Hν . By the Inverse Function Theorem and the versality of Ω, after a suitable reparametrization ν → (ω, µ, ρ), the family X takes the shape Xω,µ,ρ = [ω + O(|y| , |z|)]

∂ ∂ 2 + [Ω(µ)z + O(|y| , |z| )] , ∂x ∂z

where µ = (µ1 (ν), µ2 (ν)) ∈ R2 with (µ1 (ν0 ), µ2 (ν0 )) = (0, 0) and where Ω(µ) is given by   1 0 0 −λ0 − µ1  λ0 + µ1  0 0 1 . Ω(µ) =  (3.6)  −µ2 0 0 −λ0 − µ1  0 −µ2 λ0 + µ1 0 This matrix family is a linear centralizer unfolding4 of Ω(0) in the matrix space sp(4, R). This shows that the family X = Xν always has two normal frequencies for all parameters ν sufficiently close to ν0 , that is, ω N (ν) ∈ R2 . A geometric picture of the ‘Cantor set’ Γτ,γ (U ) determined by the Diophantine conditions (3.2) is sketched in Figure 6, where we take ν = (ω, µ, ρ) and ignore the parameter ρ.5 Variation in values of the parameter µ2 gives rise to a quasi-periodic Hamiltonian bifurcation. For this reason it is called the detuning (or distinguished) parameter of the bifurcation. From the normal linear stability Theorem 3.3, we obtain the persistence of the invariant m-torus family that contains a normally 1 : −1 resonant torus. As a corollary of Theorem 3.3, we have Theorem 3.5 (Persistence of Diophantine m-tori). Let H = Hν be a pparameter real-analytic family of integrable Hamiltonians given by (2.2). Suppose that – The family H is non-degenerate at the invariant torus Tν0 ; – The torus Tν0 is normally generically 1 : −1 resonant. Then, for γ sufficiently small and for any p-parameter real-analytic Hamiltonian  on (M, σ) sufficiently close to H in the compact-open topology on complex family H analytic extensions, there exists a neighbourhood U of ν0 ∈ Rp and a map Φ : M × U → M × Rp , defined near the normally resonant tori Tν0 , such that, i. Φ is a C ∞ -smooth diffeomorphism onto its image and is a C ∞ -near the identity map; ii. The image Φ(V ), where V = Tm × {(y, z) = (0, 0)} × Γτ,γ (U  ), is a Cantor  family of H-invariant Diophantine tori, and the restriction of Φ to V induces  a conjugacy between H and H; iii. The restriction Φ|V is symplectic and preserves the (symplectic) normal linear ∂ ∂ + Ω(ν)z ∂z of the Hamiltonian vector field X = XH part N X = ω(ν) ∂x associated with H. 4 This 5 Note

is a linear versal unfolding with minimal number of parameters [3, 22]. that internal as well as normal frequencies are independent of the parameter ρ.

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µ2

ω

m2 ω

µ1

m1

(a) ‘Cantor set’ Γτ,γ (U + )

(b) ‘Cantor set’ Γτ,γ (U − )

µ2 > 0

m2 µ2 = 0

ω

m1 µ2 < 0

(c) union Γτ,γ (U ) = Γτ,γ (U + ) ∪ Γτ,γ (U − )

(d) a vertical section in Γτ,γ (U ).

Fig 6. Sketch of the ‘Cantor sets’ Γτ,γ (U + ) and Γτ,γ (U − ) corresponding to µ2 ≥ 0 and µ2 < 0 respectively. The total ‘Cantor set’ Γτ,γ (U ) is depicted in (c). The half planes in (b) and (c) give continua of invariant m-tori. In (d), a section of the ‘Cantor set’ Γτ,γ (U ), along the µ2 -axis, is singled out: the above grey region corresponds to a half plane given in (c)—a continuum of m-tori.

Fig 7. Stratification of the 10-dimensional matrix space sp(4, R). The cone represents the set of 1 : −1 resonant matrices. The subset of semi-simple matrices is given by the vertex of the cone, while the regular part of the cone contains the generic (or non-semisimple) ones.

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4. The quasi-periodic Hamiltonian Hopf bifurcation: persistence of the (m + 1)- and (m + 2)-tori As mentioned before, see Section 2.2, the local torus foliation of the phase space M near the normally 1 : −1 resonant m-torus Tν0 gives a stratification of a certain parameter space. This local stratification by tori is described by a piece of the swallowtail, see Figure 5(a). The smooth part of this surface corresponds to elliptic (m + 1)-tori, while the open region above the surface (excluding the thread) corresponds to the Lagrangian (m + 2)-tori. The persistence question of these tori can be answered by the ‘standard’ KAM theory [12, 26, 31–33, 35, 36]. 4.1. Elliptic (m + 1)-tori To examine the persistence of the (m + 1)-tori, we first normalize the nearlyintegrable perturbations of H to obtain a proper integrable-approximation. This allows us to investigate the existence of invariant elliptic (m + 1)-tori of the perturbed Hamiltonian. To enable such a normalization, we need Theorem 3.5.  =H  ν be a p-parameter real-analytic family of nearly-integrable HamilLet H tonian functions on M = Tm × Rm × R4 = {x, y, z}. We consider the quasi-periodic  for H  sufficiently close to the integrable family H dynamics of the Hamiltonian H,  possesses a Cantor family given by (2.2). First of all, by Theorem 3.5, the family H  has the same normal linear behaviour of invariant m-tori. Secondly, the family H as the integrable Hamiltonian H. Moreover, by the Inverse Function Theory and  into the form the versality of Ω, a suitable reparametrization brings the family H  ω,µ,ρ (x, y, z) = ω, y + 1 Jz, Ω(µ)z + h.o.t. , H 2

(4.1)

where the parameters (ω, µ, ρ) are restricted to a ‘Cantor set’ and where the Floquet matrix Ω(µ) is given by (3.6). To investigate the existence of the elliptic  we need to consider the higher order (m + 1)-tori in the nearly-integrable family H,  The idea of normalization is to remove terms of the (normalized) Hamiltonian H. non-integrable terms from lower order terms by applying suitable coordinate transformations. These transformation are chosen to be the time-1 flows generated by certain Hamiltonian functions, for details see [7]. Such a normalization leads to the following: Theorem 4.1 (Quasi-periodic normal form [7]). Let H = Hν be the realanalytic family of integrable Hamiltonians given by (2.2). Assume that assumptions of Theorem 3.3 are satisfied and that the parameter ν = (ω, µ, ρ) ∈ Γτ,γ (U  ), see Section 3.1, where U is a small neighbourhood of the fixed parameter ν0 . Then, for  =H  ν sufficiently close to H in the compactany real-analytic Hamiltonian family H open topology on complex analytic extensions, there exists a family of symplectic maps Φ : Tm × Rm × R4 × U → Tm × Rm × R4 × Rp being real-analytic in (x, y, z) and C ∞ -near-the-identity such that: the Hamiltonian  ◦ Φ is decomposed into the integrable part Gint and a remainder R, where G=H Gint = ω, y + (λ0 + µ1 )S + N + µ2 M + 2bM 2 + 2c1 SM + c2 S 2 ,

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with S = z1 z4 − z2 z3 , N = 12 (z32 + z42 ) and M = 21 (z12 + z22 ). The remainder R satisfies ∂ q+|p|+|k| R (x, 0, 0, ω, µ1 , 0) = 0, ∂ q µ2 ∂ p y∂ k z for all indices (q, p, k) ∈ N30 with 2q + 4 |p| + |k| ≤ 4. For a similar non-linear normal form theory for the non-conservative setting see [5, 12] and for the Hamiltonian setting see [23]. A special case of Theorem 4.1 with m = 1 is extensively considered in [34]. Now the integrable truncation Gint contains a family of elliptic (m + 1)-tori determined by the cubic equation S 2 − 4bM 3 − 4µ2 M 2 − 4c1 SM 2 = 0 ,

(4.2)

where M > 0, compare with [17, 24, 29]. By a rescaling argument, the remainder R can be considered as perturbation of the integrable Gint . At this point, our present problem is cast into the form treated in Theorem 2.6 of [11] from which we conclude that most elliptic (m + 1)-tori of the integrable Hamiltonian Gint survive the perturbations by R, and are only slightly deformed. Theorem 4.2 (Persistence of Diophantine elliptic (m + 1)-tori). Let H = Hν be the real-analytic family of integrable Hamiltonian functions given by (2.2) such that assumptions from Theorem 3.5 are satisfied. Then, for any real-analytic  =H  ν sufficiently close to the family H in the compact-open Hamiltonian family H topology, there exists a map Φ : Tm × Rm × R4 × U → Tm+1 × Rm+1 × R2 × Rp defined near the normally resonant torus Tν0 such that: Φ is a C ∞ -near-the-identity  ◦ Φ−1 has a Cantor family diffeomorphism onto its image and the Hamiltonian H of invariant elliptic (m + 1)-tori. 4.2. Lagrangian (m + 2)-tori In this section, we investigate the persistence of the Lagrangian (m + 2)-tori of the integrable Hamiltonian H of the form (2.2). These tori are located in the open region (excluding the thread) above the swallowtail surface, see Figure 5(a). To this end, we apply the classical KAM theory [2, 27, 33] and its global version [6]. The Kolmogorov non-degeneracy condition — required for the KAM theory — on the Lagrangian tori near the resonant torus is guaranteed by the non-triviality of the monodromy, compare with [19, 38, 45]. Reconsider the integrable (i.e., Tm -symmetric) Hamiltonian function H of the form 1 (4.3) H(x, y, z, ν) = ω(ν), y + Jz, Ω(ν)z + F (y, z, ν), 2 on the space M = Tm × Rm × R4 = {x, y, z}, where F denotes the higher order terms. The invariant torus Tν0 given by (y, z, ν) = (0, 0, ν0 ) is generically 1 : −1 resonant, see Section 2.2. We require that the Hessian of the higher order term F with respect to the variable y is non-vanishing at the torus Tν0 . As before we may assume that H is invariant under the free S1 -action generated by the semisimple part of the polynomial 21 Jz, Ω(ν)z , see Section 2.2. By

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this invariance, the Hamiltonian H is Liouville-integrable with the (m + 2) first integrals EM = (H, S, y), where S = z1 z4 − z2 z3 . The Lagrangian (m + 2)-tori of the integrable Hamiltonian H are the regular fibers of the energy-momentum map EM. Our concern is with the persistence of these Lagrangian tori near the thread, see Figure 5(a), when H is perturbed into a nearly-integrable Hamiltonian  In view of the classical KAM theory [2, 27], most of these Lagrangian family H. tori survive the perturbation in Whitney-smooth Cantor families. Here we have to require the Kolmogorov non-degeneracy, which near the thread is a consequence of the non-trivial monodromy. Indeed, non-degeneracy follows by an application of [19, 38, 40, 45] to the reduced two-degrees-of-freedom system Hα and the as2 sumption that det ∂∂yF2 = 0 at the resonant torus Tν0 . We first conclude that, for sufficiently small perturbation, the Lagrangian tori survive in a Whitney-smooth Cantor family of positive measure [2, 33, 35]. Secondly, the corresponding KAMconjugacies, which are only defined on locally trivial sub-bundles, can be glued together to provide a globally Whitney-smooth conjugacy from the integrable to the nearly-integrable Cantor torus family [6]. Theorem 4.3 (Persistence of Diophantine Lagrangian (m + 2)-tori). Let H = Hν be the real-analytic family of Hamiltonians given by (4.3). Suppose that the Hessian of the higher order term F , see (4.3), with respect to y is non-zero at the resonant torus Tν0 . Then, there exists a neighbourhood U ⊂ Rp of ν0 such  sufficiently close to H in the compactthat for any real-analytic Hamiltonian H open topology on complex analytic extensions, the following holds: the perturbed  has a Cantor family of invariant (m + 2)-tori; this family is a Hamiltonian H  ∞ C -near-the-identity diffeomorphic image of T = ν Tν , where ν is restricted to a ‘Cantor set’ (determined by the Diophantine conditions on the internal frequencies);  in these tori, the diffeomorphism conjugates H and H. 5. Concluding remarks We considered a family of Tm -symmetric Hamiltonians on M = Tm × Rm × R4 that has a normally 1 : −1 resonant torus. As the parameter varies, the torus changes from normally hyperbolic into normally elliptic. This generically gives rise to a quasi-periodic Hamiltonian Hopf bifurcation. Near the normally resonant torus the phase space M is foliated by hyperbolic and elliptic m-tori, by elliptic (m + 1)-tori and by Lagrangian (m+2)-tori. This singular torus foliation gives a stratification in a suitable parameter space: the strata are determined by the dimension of the tori. The local geometry of this stratification is a piece of swallowtail catastrophe set: the m-tori are located at the 1-dimensional part of the surface, (m+1)-tori at the regular part of the surface and (m + 2)-tori at the open region above the surface, see Figure 5(a). By KAM-theory [2, 10, 12, 26, 27, 33, 35, 36], these tori survive in Whitneysmooth Cantor families, under small nearly-integrable perturbations, compare with Figure 5(b). We remark that this quasi-periodic stability still holds for the case where the frequencies of the oscillator are kept constant, see [7] for details. In view of the global KAM theory [6], the non-trivial monodromy in the integrable Lagrangian torus bundle can be extended to the nearly-integrable case. This may be of interest for semi-classical quantum mechanics, compare with [14, 15, 18, 20, 21, 41]. An example of the quasi-periodic Hamiltonian Hopf bifurcation is the Lagrange top (near gyroscopic stabilization) coupled to a quasi-periodic oscillator, compare with Section 1.

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Concerning the parameter domains regarding the Diophantine tori of the different dimensions m, m + 1 and m + 2, we expect that these are attached to one another in a Whitney-smooth way, as suggested by the integrable approximation. Following [8] we speak of a Cantor stratification. Here the stratum of the (m + 1)tori consists of density points of (m + 2)-quasiperiodicity of (2m + 4)-dimensional Hausdorff measure. Similarly, the stratum of the m-tori consists of density points of (m + 1)-quasiperiodicity of (2m + 2)-dimensional Hausdorff measure. Also compare with [11]. In a general case where the perturbation destroys the whole Tm -symmetry, the Cantor families of stable KAM-tori always contains continua of tori (the projection of these continua into the parameter space has no Cantor gaps). We refer to these projections as continuous structures in the Cantor family of surviving tori. For special perturbations where a partial symmetry still remains (e.g., perturbations that are independent of certain angle variables), we expect extra continuous structures, for details see [7]. In this paper, we addressed the supercritical case of the quasi-periodic Hamiltonian Hopf bifurcation. We expect similar persistence results for the subcritical case. This is important for a better understanding of the hydrogen atom in crossed electric and magnetic fields [20, 21], where the subcritical bifurcation occurs. The singular torus foliation for this case is described by the tail of the swallowtail surface.6 We expect that our approach for the supercritical case also works for the subcritical case. Acknowledgments The authors are grateful to Floris Takens for his valuable comments. References [1] Abraham, R. and Marsden, J. (1978). Foundations of Mechanics. Benjamin/Cummings Publishing Co. Inc. Advanced Book Program. MR0515141 [2] Arnol d, V. (1963). Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian. Russ. Math. Surv. 18, 5, 9–36. MR0163025 [3] Arnol d, V. (1971). Matrices depending on parameters. Russ. Math. Surv. 26, 2, 29–43. MR0301242 [4] Arnol d, V. (1989). Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, Vol. 60. Springer-Verlag. MR1037020 [5] Braaksma, B. and Broer, H. (1987). On a quasiperiodic Hopf bifurcation. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 4, 2, 115–168. MR0886930 [6] Broer, H., Cushman, R., Fasso, F., and Takens, F. (2004). Geometry of KAM tori for nearly integrable Hamiltonian systems. Preprint, Rijksuniversiteit Groningen. [7] Broer, H., Hanßmann, H., and Hoo, J. (2004). The quasi-periodic Hamiltonian Hopf bifurcation. Preprint, Rijksuniversiteit Groningen. [8] Broer, H., Hanßmann, H., and You, J. (2005). Bifurcations of normally parabolic tori in Hamiltonian systems. Nonlinearity 18, 4, 1735–1769. MR2150353 6 Complementary

to the part of Figure 5(a).

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[9] Broer, H., Hanßmann, H., and You, J. (2006). Umbilical torus bifurcations in Hamiltonian systems. J. Differential Equations 222, 1, 233–262. MR2200752 [10] Broer, H., Hoo, J., and Naudot, V. (2004). Normal linear stability of quasi-periodic tori. Preprint, Rijksuniversiteit Groningen. [11] Broer, H., Huitema, G., and Sevryuk, M. (1996). Quasi-periodic motions in families of dynamical systems. Lecture Notes in Mathematics, Vol. 1645. Springer-Verlag. MR1484969 [12] Broer, H., Huitema, G., and Takens, F. (1990). Unfoldings of quasiperiodic tori. Mem. Amer. Math. Soc. 83, 421, 1–82. MR1041003 [13] Cushman, R. and Bates, L. (1997). Global Aspects of Classical Integrable Systems. Birkh¨ auser-Verlag, Basel. MR1438060 [14] Cushman, R. and Duistermaat, J. (1988). The quantum mechanical spherical pendulum. Bull. Amer. Math. Soc. (N.S.) 19, 2, 475–479. MR0956603 [15] Cushman, R., Dullin, H., Giacobbe, A., Holm, D., Joyeux, M. Lynch, P., Sadovski´i, D., and Zhilinski´i, B. (2004). CO2 molecule as a quantum realization of the 1 : 1 : 2 resonant swing-spring with monodromy. Physical Review Letters 93, 2, 024302(4). ¨ rrer, H. (1985). The energy momentum mapping [16] Cushman, R. and Kno of the Lagrange top. In Differential Geometric Methods in Mathematical Physics (Clausthal, 1983). Lecture Notes in Math., Vol. 1139. Springer, 12–24. MR0820467 [17] Cushman, R. and Meer, J.-C. v. d. (1990). The Hamiltonian Hopf bifurcation in the Lagrange top. In G´eom´etrie symplectique et m´ecanique (La Grande Motte, 1988). Lecture Notes in Math., Vol. 1416. Springer, 26–38. MR1047475 [18] Cushman, R. and Sadovski´ı, D. (2000). Monodromy in the hydrogen atom in crossed fields. Phys. D 142, 1-2, 166–196. MR1763102 ˜ Ngo.c, S. (2004). Vanishing twist near focus-focus points. [19] Dullin, H. and Vu Nonlinearity 17, 1777–1785. MR2086150 [20] Efstathiou, K. (2005). Metamorphoses of Hamiltonian systems with symmetries. Lecture Notes in Mathematics, Vol. 1864. Springer-Verlag. MR2152605 [21] Efstathiou, K., Cushman, R., and Sadovski´i, D. (2004). Hamiltonian Hopf bifurcation of the hydrogen atom in crossed fields. Physic D 194, 250– 274. MR2075655 [22] Gibson, C. (1979). Singular points of smooth mappings. Research Notes in Mathematics, Vol. 25. Pitman (Advanced Publishing Program).MR0533668 [23] Hanßmann, H. (1998). The quasi-periodic centre-saddle bifurcation. J. Differential Equations 142, 2, 305–370. MR1601868 [24] Hanßmann, H. and Meer, J.-C. v. d. (2005). On non-degenerate Hamiltonian Hopf bifurcations in 3DOF systems. In Equadiff 2003 (Hasselt 2003). World Scientific, 476–481. [25] Hausdorff, F. (1962). Set Theory. Second edition. Translated from the German by John R. Aumann et al. Chelsea Publishing Co., New York. MR0141601 [26] Huitema, G. (1988). Unfoldings of quasi-periodic tori. PhD thesis, University of Groningen. [27] Kolmogorov, A. (1954). On conservation of conditionally periodic motions for a small change in Hamilton’s function. Dokl. Akad. Nauk SSSR (N.S.) 98, 527–530. MR0068687 [28] Marsden, J. and Weinstein, A. (1974). Reduction of symplectic manifolds with symmetry. Rep. Mathematical Phys. 5, 1, 121–130. MR0402819 [29] van der Meer, J.-C. (1985). The Hamiltonian Hopf Bifurcation. Lecture

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IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 304–305 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000310

Every compact metric space that supports a positively expansive homeomorphism is finite Ethan M. Coven1 and Michael Keane1 Wesleyan University, USA Abstract: We give a simple proof of the title.

A continuous map f : X → X of a compact metric space, with metric d, is called positively expansive if and only if there exists
> 0 such that if x = y, then d(f k (x), f k (y)) >
for some k ≥ 0. Here exponentiation denotes repeated composition, f 2 = f ◦ f , etc. Any such
> 0 is called an expansive constant. The set of expansive constants depends on the metric, but the existence of an expansive constant does not. Examples include all one-sided shifts and all expanding endomorphisms, e.g., the maps z → z n , n = 0, ±1, of the unit circle. None of these maps is one-to-one, and with good reason. It has been known for more than fifty years that every compact metric space that supports a positively expansive homeomorphism is finite. This was first proved by S. Schwartzman [8] in his 1952 Yale dissertation. The proof appears in [4, Theorem 10.30]. (A mistake in the first edition was corrected in the second edition.) Over the years it has been reproved with increasingly simpler proofs. See [5],[6],[7]. The purpose of this paper is to give another, even simpler, proof of this result. The idea behind our argument is not new. After discovering the proof given in this paper, the authors learned from W. Geller [3] that the basic idea of the proof had been discovered in the late 1980s by M. Boyle, W. Geller, and J. Propp. Their proof appears in [6]. The result also follows from the theorem at the end of Section 3 of [1]. The authors of that paper may have been unaware of this consequence of their theorem, which they called “a curiosity based on the techniques of this work.” That the result follows from the theorem in [1] was recognized by B. F. Bryant and P. Walters [2]. Nonetheless, the fact that this result has a simple proof remains less well-known than it should be, and so publishing it in this Festschrift is appropriate. Theorem. Every compact metric space that supports a continuous, one-to-one, positively expansive map is finite. Remark. Our statement does not assume that the map is onto. Such maps are sometimes called “homeomorphisms into.” We have written the proof so that the fact that f is one-to-one is used only once. 1 Department of Mathematics, Wesleyan University, Middletown, CT 06459, USA, e-mail: [email protected]; [email protected] AMS 2000 subject classifications: primary 37B05; secondary 37B25. Keywords and phrases: topological dynamics, expansive homeomorphisms.

304

Support of a positively expansive homeomorphism is finite

305

Proof. Let X be a compact metric space with metric d, let f be a continuous, oneto-one, positively expansive map of X into (not necessarily onto) itself, and let
> 0 be an expansive constant. Consider the following condition (∗) there exists n ≥ 1 such that if d(f i (x), f i (y)) ≤
for i = 1, 2, . . . , n, then d(x, y) ≤
, too. Suppose that (∗) is not true. Then for every n ≥ 1, there exist xn , yn such that d(f i (xn ), f i (yn )) ≤
for i = 1, 2, . . . , n, but d(xn , yn ) >
. Choose convergent subsequences xnk → x and ynk → y. Then x = y and for every i ≥ 1, d(f i (xnk ), f i (ynk )) → d(f i (x), f i (y)). Now d(f i (xnk ), f i (ynk )) ≤
for i = 1, 2, . . . , nk , so d(f i (x), f i (y)) ≤
for all i ≥ 1. But f is one-to-one, so f (x) = f (y). This contradicts positive expansiveness (with f (x) and f (y) in place of x and y), so the condition holds. Fix k ≥ 0, and apply (∗) consecutively for j = k, k − 1, . . . , 1, 0, with f j (x) and j f (y) in place of x and y. We get (∗∗) for every k ≥ 0, if d(f i (x), f i (y)) ≤
for i = k + 1, k + 2, . . . , k + n, then d(f i (x), f i (y)) ≤
for i = 0, 1, . . . , k, too. Now cover X by finitely many, say N , open sets of the form U (z) := {x ∈ X : d(f i (x), f i (z)) ≤
/2 for i = 1, 2, . . . , n}, where n is as in (∗). If X contains more then N points, consider a finite subset containing N +1 points. For every k ≥ 0, there exist xk = yk in this subset such that f k (xk ) and f k (yk ) lie in the same set U (z). Then by (∗∗), d(f i (xk ), f i (yk )) ≤
for i = 0, 1, . . . , k + n. Since there are only finitely many pairs of these N + 1 points, there exist x = y such that d(f i (x), f i (y)) ≤
for infinitely many i ≥ 0, and hence for all i ≥ 0. This contradicts positive expansiveness. References [1] Adler, R. L., Konheim, A. G., and McAndrew, M. H. (1965). Topological entropy. Trans. Amer. Math. Soc. 114, 309–319. MR175106 [2] Bryant,B. F. and Walters, P. (1969) Asymptotic properties of expansive homeomorphisms, Math. Systems Theory 3, 60–66. MR243505 [3] Geller, W. Personal communication. [4] Gottschalk, W. H. and Hedlund, G. A. (1955). Topological Dynamics. Amer. Math. Soc. Colloq Publ. 36, Providence, RI. MR0074810 [5] Keynes, H. B. and Robertson, J. B. (1969). Generators for topological entropy and expansiveness. Math. Systems Theory 3, 51–59. MR247031 [6] King, J. L. (1990). A map with topological minimal self-joinings in the sense of del Junco. Ergodic Theory Dynam. Systems 10, 4, 745–761. MR1091424 [7] Richeson, D. and Wiseman, J. (2004). Positively expansive homeomorphisms of compact metric spaces, Int. J. Math. Math. Sci. 54, 2907–2910. MR2145368 [8] Schwartzman, S. (1952). On transformation groups. Dissertation, Yale University.

IMS Lecture Notes–Monograph Series Dynamics & Stochastics Vol. 48 (2006) 306–316 c Institute of Mathematical Statistics, 2006
DOI: 10.1214/074921706000000329

On g-functions for subshifts Wolfgang Krieger1 University of Heidelberg Abstract: A necessary and sufficient condition is given for a subshift presentation to have a continuous g-function. An invariant necessary and sufficient condition is formulated for a subshift to posses a presentation that has a continuous g-function.

1. Introduction Let Σ be a finite alphabet, and let S denote the shift on ΣZ , S((xi )i∈Z ) = ((xi+1 )i∈Z ),

(xi )i∈Z ∈ Z.

A closed S-invariant set X ⊂ ΣZ with the restriction of S acting on it, is called a subshift. A finite word is said to be admissible for a subshift if it appears in a point of the subshift. A subshift is uniquely determined by its set of admissible words. A subshift is said to be of finite type if its admissible words are defined by excluding finitely many words from appearing as subwords in them. Subshifts are studied in symbolic dynamics. For an introduction to symbolic dynamics see [7] and [10]. We introduce notation. Given a subshift X ⊂ ΣZ we set x ∈ X, i, k ∈ Z, i ≤ k,

x[i,k] = (xj )i≤j≤k , and

X[i,k] = {x[i,k] : x ∈ X}. We use similar notation also for blocks, b[i
,k
] = (bj )i
≤j≤k
,

b ∈ X[i,k] , i ≤ i ≤ k ≤ k,

and also if indices range in semi-infinite intervals. Blocks also stand for the words they carry. We denote − − Γ+ n (x ) = {b ∈ X[1,n] : (x , b) ∈ X(−∞,n] },
− Γ+ (x− ) = Γ+ n (x ),

n ∈ N,

n∈N

− Γ+ ∞ (x )

and

+

= {x ∈ X[1,∞) : (x , x+ ) ∈ X}, −

Γ+ n (a) = {b ∈ X[1,n] : (a, b) ∈ X(−k,n] },
Γ+ Γ+ (a) = n (a),

x− ∈ X(−∞,0] n ∈ N,

n∈N

1 Institute for Applied Mathematics, University of Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany, e-mail: [email protected] AMS 2000 subject classifications: primary 37B10. Keywords and phrases: subshift, g-function.

306

On g-functions for subshifts + + Γ+ ∞ (a) = {x ∈ X[1,∞) : (a, x ) ∈ X(−k,∞) },

307

a ∈ X(−k,0] , k ∈ Z+ ,

Γ− has the time symmetric meaning. We denote  ωn+ (a) = {b ∈ X[1,n] : (x− , a, b) ∈ X(−∞,n] }, x− ∈Γ− ∞ (a)

ω + (a) =




ωn+ (a),

a ∈ X(−k,0] , k ∈ Z+ .

n∈N

The notions of g-function and g-measure go back to Mike Keane’s papers [5], [6]. Subsequently a substantial theory of g-functions and g-measures developed with contributions from many sides (see e.g. [1],[4],[14],[16],[17] and the references given there. For the origin of these notions see also [2]). These notions have formulations for general subshifts (see [11, p. 24]). We are interested in continuous g-functions and therefore introduce a g-function for a subshift X ⊂ ΣZ as a continuous mapping − g : {(x− , σ) ∈ X(−∞,0] × Σ : σ ∈ Γ+ 1 (x )} → [0, 1]

such that



g(x− , α) = 1,

x− ∈ X(−∞,0] ,

− α∈Γ+ 1 (x )

and a g-measure as an invariant probability measure µ of the subshift X such that   µ {x ∈ X : x[−k,1] = (a, α)}  = g(x− , α)dµ, (a, α) ∈ X[−k,1] , k ∈ N. {x∈X:x[−k,0] =a}

(Note that we have reversed the time direction.) We show in Section 2 that a subshift that has a strictly positive g-function is of finite type. Denote for x− ∈ X(−∞.0] − ∆+ 1 (x ) =




ω1+ (x− [−n,0] ).

n∈N

In Section 2 we prove that a subshift X ⊂ ΣZ has a g-function if and only if for all − x− ∈ X(−∞.0] , ∆+ 1 (x ) = ∅. We refer to this property of a subshift presentation as property g. A directed graph with vertex set M and edges carrying labels taken from a finite alphabet Σ is called a Shannon graph if the labeling is 1-right resolving in the sense that for all µ ∈ M and σ ∈ Σ there is at most one edge leaving µ that carries the label σ. Denote here the set of initial vertices of the edges that carry the label σ by M(σ), and for µ ∈ M(σ) denote by τσ (µ) the final vertex of the edge that leaves µ and carries the label σ. The Shannon graph M is determined by the transition rules (τσ )σ∈Σ . A Shannon graph is said to present a subshift X ⊂ ΣZ if every vertex has an edge leaving it and an edge entering it, and if the set of admissible words of the subshift coincides with the set of label sequences of finite paths in the graph. For a finite alphabet Σ denote by M(Σ) the set of probability measures on ΣN with its weak *-topology. With the notation  {(xi )i∈N ∈ ΣN : ai = xi }, (ai )1≤i≤n ∈ ΣN , n ∈ N, C(a) = 1≤i≤n

W. Krieger

308

M(Σ)(σ) = {µ ∈ M(Σ) : µ(C(σ)) > 0},

σ ∈ Σ,

let for µ ∈ M(Σ)(σ),τσ (µ) be equal to the conditional measure of µ given C(σ), τσ (µ)(C(b)) =

µ(C(σ, b)) , µ(C(σ))

b ∈ ΣN , N ∈ N.

In this way M(Σ) has been turned into a Shannon graph with the transition rules (τσ )σ∈Σ . The Shannon graph M(Σ) is accompanied by another Shannon graph  with vertex set N ∈Z+ MN (Σ), where for N ∈ N, MN (Σ) is the set of probability vectors on ΣN , and where M0 = {∅}. With the notation CN (σ) = {(ai )1≤i≤n ∈ ΣN : a1 = σ}, MN (Σ)(σ) = {µ ∈ MN (Σ) : µ(CN (σ)) > 0}, one sets for σ ∈ Σ, µ ∈ MN (Σ)(σ), N > 1, τσ (µ) equal to the probability vector ν ∈ MN −1 (Σ) that is given by ν(b) =

µ(σ, b) , µ(C(σ))

b ∈ ΣN −1 ,

 and one sets τσ (µ) = {∅} for σ ∈ Σ, µ ∈ M1 (Σ). In this way N ∈Z+ MN has been turned into a Shannon graph that one can equip further with the restriction mapping ι that assigns to µ ∈ MN (Σ), N > 1, its marginal vector in MN −1 (Σ), and that assigns to a µ ∈ M1 (Σ) the empty set. The mapping ι commutes with the transition rules of the Shannon graph. Call a set M ⊂ M(σ) transition complete if for σ ∈ Σ, µ ∈ M ∩ M(Σ) implies that also τσ (µ) ∈ M. Call a set M ⊂ M(σ) in-complete if for all µ ∈ M there is a ν ∈ M that is the initial vertex of an edge ending in µ. Every transition complete and in-complete set M ⊂ M(σ) determines a Shannon graph with transition rules that are inherited from the Shannon graph M(Σ).  These sub-Shannon graphs M ⊂ M(σ) are accompanied by sub-Shannon  graphs N ∈Z+ MN ⊂ N ∈Z+ MN (Σ) where MN contains the probability vectors that are given by the marginals of the measures in M, and where the transition rules  and the mapping ι are passed down from N ∈Z+ MN (Σ). In [12] Kengo Matsumoto introduced a class of structures that he called λ-graph systems. λ-graph systems have the form of a Bratteli diagram, that is, they have a finite number of vertices at  each level. In the structures N ∈Z+ MN the sets MN , N ∈ N, are not necessarily finite, but otherwise these structures have all the attributes of a λ-graph system. We will refer to them as measure λ-graph systems. We say that a measure λ-graph system presents a subshift if the set of admissible words of the subshift coincides with the set of label sequences of finite paths in the measure λ-graph system. In Section 3 we are concerned with the measure λ-graph system that is generated by a g-function g of a subshift X ⊂ ΣZ . The continuity of the g-function translates into a property of the generated measure λ-graph system that we call contractivity. Every contractive measure λ-graph system determines a g-function of the subshift that it presents and it is in turn generated by this g-function. A subshift that is presented by a contractive measure λ-graph system has a property that we call property (D). In Section 4 we prove the invariance of property (D) under topological conjugacy, and point to some classes of subshifts that have property (D) and that have presentations with property g. Every subshift that has property (D) and that has a presentation with property g admits a presentation by a contractive measure λ-graph system.

On g-functions for subshifts

309

2. g-functions of subshifts Lemma 2.1. Let X ⊂ ΣZ be a subshift, let Y − ⊂ X(−∞,0] be dense in X(−∞,0] , and let − g : {(x− , σ) ∈ Y − × Σ : σ ∈ Γ+ 1 (x )} → [0, 1] be a continuous mapping such that − y− ∈ Y − , α ∈ / Γ+ 1 (y ),

g(y − , α) = 0, and such that



g(y − , β) = 1,

y− ∈ Y − .

(1)

− β∈Γ+ 1 (y )

Let x− ∈ Y − ,

+ − − α ∈ Γ+ 1 (x ) \ ∆1 (x ).

(2)

Then g(x− , α) = 0.

(3)

Proof. By (2) there are Mn , ∈ N, n ∈ N, and a(n) ∈ X[−Mn ,0] , such that a(n)[−n,0] = x[−n,0] and

α∈ / Γ+ 1 (a(n)),

(4)

n ∈ N.

(5)

Since Y − is dense in X(−∞,0] one can find y − (n) ∈ Y − such that y − (n)[−Mn ,0] = a(n), By (4) and (6)

n ∈ N.

lim y − (n) = x− ,

n→∞

and from (5) and (6)

− α∈ / Γ+ 1 (y (n)),

n ∈ N.

(6) (7) (8)

Choose an increasing sequence nk , k ∈ N, and a set Γ ⊂ Σ such that − Γ+ 1 (y (nk )) = Γ,

k ∈ N.

(9)

Then by (7) and by compactness of X − Γ ⊂ Γ+ 1 (x ),

(10)

and from (1) and by the continuity of g  g(x− , γ) = 1, γ∈Γ

and then (3) follows from (8) and (9). Corollary 2.2. Let the subshift X ⊂ ΣZ have a strictly positive g-function. Then X is of finite type.

W. Krieger

310

Proof. For α ∈ Σ one has the closed set − Xα− = {x− ∈ X(−∞,0] : α ∈ Γ+ 1 (x )}.

It follows from Lemma 2.1 that + − − Γ+ 1 (x ) = ∆1 (x ),

x− ∈ X(−∞,0] .

Therefore the increasing sequence Xα− (n) = {x− ∈ X(−∞,0] : α ∈ ω1+ (x− [−n,0] )}, n ∈ N, of open subsets of X(−∞,0] is a cover of Xα− . One has therefore an nα ∈ N such that Xα− = Xα− (nα ). With N = max {nα : α ∈ Σ}, one has

+ − − Γ+ 1 (x ) ⊂ ω1 (x[−N,0] ),

x− ∈ X(−∞,0] ,

which means that the subshift X is determined by its set of admissible words of length N + 2. Lemma 2.3. Let X ⊂ ΣZ be a subshift such that − {x− ∈ X(−∞,0] : ∆+  ∅. 1 (x ) = ∅} =

Then there exists a continuous mapping − g : {x− ∈ X(−∞,0] : ∆+ 1 (x ) = ∅} × Σ → [0, 1]

such that g(x− , α) = 0,

− α∈ / ∆+ 1 (x ),

g(x− , α) > 0,

− α ∈ ∆+ 1 (x ).

and − Proof. For x− ∈ X(−∞,0] and α ∈ ∆+ 1 (x ) set

n(x− , α) = min {n ∈ N : α ∈ ω + (x− [−n,0] )}, and

n(x− ) =

min

− α∈∆+ 1 (x )

n(x− , α).

− For x− ∈ X(−∞,0) , and γ ∈ Γ+ 1 (x ), set  0, − g(x , γ) =  (n(x− ,γ)−n(x− ))−1  − ((n(x β∈∆+ (x− )

,β)−n(x− ))−1

− if γ ∈ / ∆+ 1 (x ),

,

− if γ ∈ ∆+ 1 (x ).

To prove continuity of the mapping g at a point (x− , α), , x− ∈ X(−∞,0] , α ∈ ∆ (x− ), let N (x− ) = max n(x− , β), +

− β∈∆+ 1 (x )

On g-functions for subshifts

311

and let y − (k) ∈ X(−∞,0] , k ∈ N, be such that lim y − (k) = x− .

k→∞

For M ∈ N, let k◦ ∈ N be such that y − (k)[−M −N [x− ),0] = x− [−M −N [x− ),0] , Then and

α ∈ ∆+ (y − (k)),

k ≥ k◦ .

k ≥ k◦ ,

|g(y − (k), α) − g(x− , α)| < M −1 |Σ|g(x− , α),

k ≥ k◦ .

− To prove continuity of the mapping g at a point (x− , α), x− ∈ X(−∞,0] , ∆+ 1 (x ) = − − ∅, α ∈ / ∆+ 1 (x ), let y (k) ∈ X, k ∈ N, be such that

lim y − (k) = x− ,

k→∞

and such that

− α ∈ ∆+ 1 (y (k)).

For M ∈ N, let k◦ ∈ N be such that y − (k)[−M −n(x− ),0] = x− [−M −n(x− ),0] ,

k ≥ k◦ .

Then

1 , k ≥ k◦ . M − A g-function of a subshift such that g(x− , α) > 0 for x− ∈ X(−∞,0] , α ∈ ∆+ 1 (x ) we will call a strict g-function. g(y − (k), α) <

Theorem 2.4. The following are equivalent for a subshift X ⊂ ΣZ : (a) X has a g-function. (b) X has property g. (c) X has a strict g-function. Proof. That (a) implies (b) follows from Lemma 2.1. That (b) implies (c) follows from Lemma 2.3. 3. Presentations of subshifts and property (D) Given a g-function of the subshift X ⊂ ΣZ we define inductively probability vectors µn (x− ) ∈ Mn (Σ), n ∈ N, x− ∈ X−∞,0] , by setting µn (x− )(a) equal to zero, if − a ∈ ΣN is not in Γ+ N (x ), and by setting − µN (a) = g(x− , a[1,k) , ak ), a ∈ Γ+ (11) N (x ), N ∈ N, 1≤k≤N

and we let µ(x− ) ∈ M(Σ) be the probability measure that has as marginal measures those that are given by the probability vectors µN (x− ), N ∈ N. We set MN (X, g) = {µN (x− ) : x− ∈ X(−∞,0] },

N ∈ N,

M(X, g) = {µ(x− ) : x− ∈ X(−∞,0] }.   Here N ∈N MN (X, g) ⊂ N ∈N MN is the measure λ-graph system that accompanies the compact transition complete and in-complete sub-Shannon graph

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312

M(X, g) of M(Σ). Entities like the mapping that assigns to a point x− ∈ X(−∞,0] for a subshift X ⊂ ΣZ the measure µ(x− ) ∈ M(Σ), or the inverse image under this mapping of a single measure, appear prominently within a theory that was put forward by James Crutchfield et al (see e.g.[15]). We set for a given transition complete and in-complete Shannon graph M  ⊂ M(Σ) and for its accompanying measure λ-graph system λ-graph system N ∈Z+ MN inductively 0 < m ≤ n, a ∈ X[−n,0] , n ∈ N.

τa (µ) = τa−m (τa(−m,0] (µ)),

Call a compact transition complete Shannon graph M ⊂ M(Σ) contractive if, with X ⊂ ΣZ the subshift that is presented by M, one has for all x− ∈ X(−∞,0] that the limits lim τx− (µ), µ ∈ M, k→∞

[−n,0)

 exist. Call a measure λ-graph system N ∈Z+ MN contractive if, with X ⊂ ΣZ the subshift that it presents one has that for all x− ∈ X(−∞,0] , that lim diam(

k→∞




µ∈Mn

τx−

[−n,0)

(µ)) = 0.

To a contractive Shannon graph there corresponds a contractive measure λ-graph system and vice versa. Due to this one-to-one correspondence between contractive measure λ-graph systems and contractive Shannon graphs one can formulate here arguments and results in terms of one or the other. We will express ourselves in terms of the contractive measure λ-graph systems, the motivation being, that the constituent elements of the contractive measure λ-graph systems are sequentially generated by the g-function according to (11). Note that the theory of contractive measure λ-graph systems and Matsumoto’s theory of λ-graph systems intersect in the theory of topological Markov shifts.  Proposition 3.1. Let n∈Z+ MN be a contractive measure λ-graph system. The  subshift that is presented by n∈Z+ MN has a g-function g such that MN = MN (X, g), N ∈ Z+ .  Proof. Every contractive measure λ-graph system n∈Z+ Mn defines a g-function g of the subshift X ⊂ ΣZ that it presents by 
(µ), x− ∈ X(−∞,0] . (12) τx− (g(x− , α))σ∈Γ+ (x− ) ∈ 1

n∈N µ∈Mn

[−n,0)

 One uses the hypothesis that every vertex in n∈Z+ MN has a predecessor to show  that the g-function that is associated to n∈Z+ MN according to (12) has the stated property. We say that a subshift X ⊂ ΣZ has property (D) if for all admissible words bσ of X there exists a word a ∈ Γ− (b) of X such that σ ∈ ω1+ (ab). Lemma 3.2. A subshift X ⊂ ΣZ that admits a presentation by a contractive λgraph system n∈Z+ MN has property (D).

Proof. Let b be an admissible word of the subshift X and let σ ∈ Γ+ 1 (b). Denote the length of b by K, and let µ ∈ MK+1 be a vertex with a path leaving it that has

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313

label sequence b. Set ν = τb (µ). Then ν(σ) > 0. Let then x− ∈ X(−∞,0) be such that  − x− n∈Z+ MN [−K,0] = bσ, and such that x(−∞,K) is the label sequence of a path in that  leads into the vertex µ. Then it follows for the g-function that is associated to n∈Z+ MN according to (12) that g(x− , σ) = ν(σ) > 0. Apply Lemma 2.1 to conclude the proof. Lemma 3.3. The following are equivalent for a subshift X ⊂ ΣZ : (a) X has property (D). (b) For all admissible words a and c of X such that c ∈ Γ+ (a) there exists an admissible word b of X such that ac ∈ ω + (b). Proof. We prove that (a) implies (b). For this let a and c = (cl )1≤l≤k , k ∈ N, be admissible words of the subshift X such that c ∈ Γ+ (a). Choose inductively words bl , 1 ≤ l ≤ k, such that cl ∈ ω1+ ((bm )l≥m≥1 , ac[1,l) ), Then set

1 ≤ l ≤ k.

b = (bl )k≥l≥1 .

Lemma 3.4. Let X ⊂ ΣZ be a subshift with properties g and (D), and let g be a strict g-function of X. Then the measure λ-graph system n∈Z+ Mn (X, g) presents X. Proof. Let a and b be admissible words of the subshift X such that a ∈ ω + (b). Let N denote the length of a, and let K denote the length of b. Since the g-function g is assumed strict it follows that for x− ∈ X(−∞,0] such that x− = b one has  [−K,0] − that µN (x )(a) > 0. This implies that there is a path in n∈Z+ Mn (X, f ) with label sequence a that leads into the vertex ∅. Theorem 3.5. The presentation of a subshift by a contractive measure λ-graph system has property g. A subshift that admits a presentation by a contractive measure λ-graph system has property (D). Proof. The presentation of a subshift by a contractive measure λ-graph system has property g by Proposition 3.1 and a subshift that admits such a presentation has property (D) by Lemma 3.2. Conversely, if a subshift presentation has property g, then by Lemma 2.3 it has a strict g-function and by Lemma 3.4 this strict g-function generates a contractive measure λ-graph system that presents the subshift. 4. Invariance ¯ ⊂Σ ¯ Z and a continuous shift-commuting We recall that, given subshifts X ⊂ ΣZ , X ¯ there is for some L ∈ Z+ a block mapping map ϕ : X → X ¯ Φ : X[−L,L] → Σ such that ϕ(x) = (Φ(x[i−L,i+L] ))i∈Z . We say then that ϕ is implemented by Φ, and we write Φ(a) = (Φ(a[j−L,j+L] )i+L≤j≤k−L ),

a ∈ X[i,k] ,

k − i ≥ 2L,

and use similar notation if indices range in semi-infinite intervals. Recall that the n-block system of a subshift X ∈ ΣZ is its image in (X[1,n] )Z under the mapping x → (x(i,i+n] )i∈Z , x ∈ X. Call a subshift X ⊂ ΣZ right instantaneous [8] if for all σ ∈ Σ, ω1+ (σ) = ∅.

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Proposition 4.1. A subshift X ⊂ ΣZ has a g-function if and only if one of its n-block systems is right-instantaneous. − Proof. For n ∈ N and σ ∈ Σ one has the sets Xn,σ = {x− ∈ X(−∞,0] : σ ∈ + ω1 (x[−n.0] )}, that are open in X(−∞,0| . By property g these open sets cover X(−∞,0] . − There is a finite subcover {Xn(σ):σ , σ ∈ Σ}. With n = max {n(σ) : σ ∈ Σ} one has that the n-block system of X is right-instantaneous.

For a subshift X ⊂ ΣZ , for L ∈ Z+ , and for mappings Ψ(r) : X[−L,L] → X[1,L+1] one formulates a condition − (RIa) : Ψ(r) (a) ∈ Γ+ L+1 (x , a[−L,0] ),

a ∈ X[−L,L] , x− ∈ Γ− ∞ (a).

If a mapping Ψ(r) : X[−L,L] → X[1,L+1] satisfies condition (RIa) then for 0 ≤ n < L, (r) and for b(r) ∈ X[−L−n,L] , the words an,b(r) that are given by (r)

(r)

(r)

an,b(r) = (b[−L−n,0] , Ψ(r) (b[−L,L] )(0,L−n] ) are in X[−L−n,L−n] , and it is meaningful to impose on Ψ(r) a further condition (r)

(r)

(RIb) : Ψ(r) (an,b(r) ) = Ψ(r) (b[−L−n,L−n] ), b(r) ∈ X[−L−n,L] , 0 ≤ n < L. We say that a mapping Ψ(r) : X[−L,L] → X[1,L+1] that satisfies condition (RIa) and also satisfies condition (RIb) is an RI-mapping, and we say that a subshift that has an RI-mapping has property RI. Proposition 4.2. A subshift admits a presentation that has property g if and only if it has property IR. Proof. A subshift X ⊂ ΣZ admits a right-instantaneous presentation if and only if it has property IR [8]. Apply Proposition 4.1.

⊂Σ

Z be topologically conjugate subshifts, Proposition 4.3. Let X ⊂ ΣZ and X Z

⊂Σ

have property (D). Then the subshift X ⊂ ΣZ also has and let the subshift X property (D). Proof. A subshift has property (D) if and only if one of its n-block systems has property (D). To prove the proposition it is therefore sufficient to consider the

that is implemented situation that there is given a topological conjugacy ϕ : X → X −1

by a 1-block map Φ : Σ → Σ, with ϕ implemented for some L ∈ Z+ , by a block

with coding window [−L, L]. Let there be given aσ ∈ X[−I,0] , I ≥ 2L. One map Φ has to find a b ∈ X[−I−J,−I) , such that b ∈ Γ− (a), For this, let be such that

σ ∈ ω1+ (ba).

(13)

[−L,L] ,

[−I−L,L] ∩ Γ− (

c∈X a∈X c),

a aσ = Φ( c).

(14)

On g-functions for subshifts

315

and by Lemma 3.3 there exists a By property (D) of X such that

b ∈ X

[−I−J−2L,−I−L) ∩ Γ− ( a),

c ∈ ω + ( b[−I−J−L,−I) , a).

Then set

b, b = Φ( a[−I−L,−I+L) )

(15) (16)

and have by (14), (15) and (16) that (13) holds.

An alternate proof of the invariance under topological conjugacy of property (D) can be based on Nasu’s theorem [13, Theorem 2.4] and on a notion of strong shift equivalence for measure λ-graph systems that is patterned after the notion of strong shift equivalence for λ-graph systems [12]. We describe prototype examples of subshifts with property (D) and their presentations with property g. For this, we consider the Dyck inverse monoid with unit 1 and generating set {αλ , αρ , βλ , βρ , }, with relations αλ αρ = βλ βρ = 1, αλ βρ = βλ αρ = 0. The Dyck shift (on four symbols) is the subshift D2 ⊂ {αλ , αρ , βλ , βρ , }Z that contains all x ∈ {αλ , αρ , βλ , βρ , }Z such that

xi = 0,

I− , I+ ∈ Z, I− < I+ .

(17)

I− ≤i
The Motzkin shift (on five symbols) is the subshift M2 ⊂ {1, αλ , αρ , βλ , βρ , }Z that contains all x ∈ {1, αλ , αρ , βλ , βρ , }Z such that (17) holds. These presentations of the Dyck and Motzkin shifts have property g, and the Dyck and Motzkin shifts have property (D). The Dyck and Motzkin shifts are prototypes of a class of subshifts that was introduced in [3] by giving presentations that were called S-presentations. S-presentations have property g and subshifts that admit an Spresentation have property (D). References [1] Bramson, M. and Kalikow, S. (1993). Nonuniqueness in g-functions. Israel J. Math. 84, 1–2, 153–160. MR1244665 [2] Fortet, R. and Doeblin, W. (1937). Sur des chaˆınes `a liaisons compl`etes. Bull. Soc. Math. France 65, 132–148. MR1505076 [3] T. Hamachi, K. I., and Krieger, W. (2005). Subsystems of finite type and semigroup invariants of subshifts. Preprint. ¨ [4] Johansson, A. and Oberg, A. (2003). Square summability of variations of g-functions and uniqueness of g-measures. Math. Res. Lett. 10, 5–6, 587–601. MR2024717 [5] Keane, M. (1971). Sur les mesures invariantes d’un recouvrement r´egulier. C. R. Acad. Sci. Paris S´er. A–B 272, A585–A587. MR0277687 [6] Keane, M. (1972). Strongly mixing g-measures. Invent. Math. 16, 309–324. MR310193 [7] Kitchens, B. P. (1998). Symbolic Dynamics. Universitext. Springer-Verlag, Berlin. MR1484730

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[8] Krieger, W. (2005). On subshift presentations. Preprint. [9] Krieger, W. and Matsumoto, K. (2002). Shannon graphs, subshifts and lambda-graph systems. J. Math. Soc. Japan 54, 4, 877–899. MR1921090 [10] Lind, D. and Marcus, B. (1995). An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge. MR1369092 [11] Maillard, G. (2003). Chaˆines `a liaisons compl`etes et mesures de Gibbs unidimensionnelles. Th`ese, Universit´e de Rouen. [12] Matsumoto, K. (1999). Presentations of subshifts and their topological conjugacy invariants. Doc. Math. 4, 285–340 (electronic). MR1710375 [13] Nasu, M. (1986). Topological conjugacy for sofic systems. Ergodic Theory Dynam. Systems 6, 2, 265–280. MR857201 [14] Olivier, E. (1999). Multifractal analysis in symbolic dynamics and distribution of pointwise dimension for g-measures. Nonlinearity 12, 6, 1571–1585. MR1726665 [15] Shalizi, C. R. and Crutchfield, J. P. (2001). Computational mechanics: pattern and prediction, structure and simplicity. J. Statist. Phys. 104, 3-4, 817–879. MR1853995 ¨ (2003). Uniqueness in g-measures. Nonlinearity 16, 2, 403–410. [16] Stenflo, O. MR1958606 [17] Walters, P. (2005). Regularity conditions and Bernoulli properties of equilibrium states and g-measures. J. London Math. Soc. (2) 71, 2, 379–396. MR2122435