III
Progress in Mathematics Volume 180
Series Editors Hyman Bass Joseph Oesterle Alan Weinstein
Gabriel P. Paternain
Geodesic Flows
Birkhauser Boston Basel Berlin
Gabriel P. Patemain Centro de Matemdtica Facultad de Ciencias 11400 Montevideo. Uruguay
Library of Congress Cataloging-in-Publication Data Patemain, Gabriel P. (Gabriel Pedro), 1964Geodesic flows / Gabriel P. Paternain. p. cm. - (Progress in mathematics
:
v.
180)
Includes bibliographical references and index.
ISBN 0-8176-4144-0 (alk. paper). -ISBN 3.7643-41440 (alk. paper) 1. Geodesic flows.
I. Title. 11. Series: Progress in mathematics (Boston, Mass.) : vol. 180. QA614.82.P38 514'.74-dc21
1999 99-38332
CIP
AMS Subject Classifications: 58F17, 58F05, 54C70, 58FI5, 58F11, 53022
Printed on acid-free paper.
®1999 Birkhauser Boston
Birkhi user
qlh}'
All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser Boston, do Springer-Verlag New York. Inc., 175 Fifth Avenue, New York, NY 10010, USA). except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 0-8176-4144-0
ISBN 3-7643-4144-0 Reformatted from author's disk by TTXniques, Inc., Cambridge, MA. Printed and bound by Hamilton Printing, Rensselaer, NY. Printed in the United States of America.
987654321
To Graciela
Contents
Preface
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0
Introduction
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Introduction to Geodesic Flows 1.1
1
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Geodesic flow of a complete Riemannian manifold
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Euler-Lagrange flows ................ .... Symplectic and contact manifolds 1.2.1 Sympiectic manifolds ................... 1.1.1
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Contact manifolds .....................
The geometry of the tangent bundle . . 1.3.1 Vertical and horizontal subbundles 1.3.2 The symplectic structure of TM . 1.3.3 The contact form . . . . . . . . . .
Jacobi fields and the differential of the geodesic flow . . . . . .
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The cotangent bundle T'M .......... ........... .
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The asymptotic cycle and the stable norm ............. 21 1.6.1
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The asymptotic cycle of an invariant measure The stable norm and the Schwartzman ball .
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The Geodesic Flow Acting on Lagrangian Subspaces
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Twist properties ........................... 32
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Riccati equations
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37
viii
Contents
2.3
The Grassmannian bundle of Lagrangian subspaces
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The Maslov index . . . . . . . . . . . . . The Maslov class of a pair (X, E) 2.4.1 2.4.2 . . 2.4.3 Lagrangian submanifolds .
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Hyperbolic sets ....................... 42 43 ..
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The geodesic flow acting at the level of Lagrangian subspaces . 2.5.1 The Maslov index of a pseudo-geodesic and recurrence . . Continuous invariant Lagrangian subbundles in SM . . . . Birkhoff's second theorem for geodesic flows . . . . . . . . . .
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Geodesic Arcs, Counting Functions and Topological Entropy 3.1
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The counting functions . . . . . . . 3.1.1 Growth of n(T) for naturally reductive
homogeneous spaces . . Entropies and Yomdin's theorem
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Topological entropy .................... 58 Yomdin's theorem .. .. 60
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Entropy of an invariant measure
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Lyapunov exponents and entropy
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3.2.5 Examples of geodesic flows with positive entropy . Geodesic arcs and topological entropy . . . . . . . . .
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Another proof of Theorem 3.32 using Theorem 3.44
4 Mane's Formula for Geodesic Flows and Convex Billiards 4.1
Time shifts that avoid the vertical ..
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Mane's formula for geodesic flows
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Changes of variables .................... 83 Proof of the Main Theorem ................. 88
Mane's formula for convex billiards
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Further results and problems on the subject ............ 102 4.6.1
Topological pressure
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Topological Entropy and Loop Space Homology
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4.2.2 Manifolds without conjugate points . . . . . . . . . . . . . A formula for the topological entropy for manifolds of positive . . . sectional curvature . . . . . . . . . . . . . . . . . 4.5.1
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Rationally elliptic and rationally hyperbolic manifolds 5.1.1 The characteristic zero homology of H-spaces . 5.1.2 The radius of convergence .. . . . . . . . . . Morse theory of the loop space 5.2.1 Serre's theorem . . . . 5.2.2 Gromov's theorem . .
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ix
Contents
5.3
Topological conditions that ensure positive entropy ........ 119 5.3.1
5.3.2 5.3.3 5.3.4 5.4
Growth of finitely generated groups . . . . Dinaburg's Theorem . . Arbitrary fundamental group . . . .
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119 120
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Proof of Theorem 5.20 ................... 122
Entropies of manifolds ....................... 126 Simplicial volume ..................... 126 5.4.1 127 .. .. 5.4.2 Minimal volume .
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Further results and problems on the subject
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Hints and Answers
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References
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Index
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Preface
The aim of this book is to present the fundamental concepts and properties of the geodesic flow of a closed Riemannian manifold. The topics covered are close to my research interests. An important goal here is to describe properties of the geodesic flow which do not require curvature assumptions. A typical example of such a property and a central result in this work is Mafid's formula that relates the topological entropy of the geodesic flow with the exponential growth rate of the average numbers of geodesic arcs between two points in the manifold. The material here can be reasonably covered in a one-semester course. I have
in mind an audience with prior exposure to the fundamentals of Riemannian geometry and dynamical systems. I am very grateful for the assistance and criticism of several people in preparing the text. In particular, I wish to thank Leonardo Macarini and Nelson Moller who
helped me with the writing of the first two chapters and the figures. Gonzalo Tornaria caught several errors and contributed with helpful suggestions. Pablo Spallan7ani wrote solutions to several of the exercises. I have used his solutions to write many of the hints and answers. I also wish to thank the referee for a very careful reading of the manuscript and for a large number of comments with corrections and suggestions for improvement.
This book grew out of lectures which I gave at Bah(a Blanca and Cordoba (Argentina) in 1996, IMPA (Rio de Janeiro, Brasil) in 1997 and Montevideo (Uruguay) in 1998. Part of the text was written while I was visiting IMPA during the second semester of 1997 and the ICTP in Trieste during the first two months of 1998. 1 wish to thank them for their hospitality.
xii
Preface
Finally, my thanks go to Professor Alan Weinstein for his kind interest in the
manuscript and to Ann Kostant and Tom Grasso at Birkhauser for their help throughout the various stages of publication.
Gabriel Pedro Paternain Montevideo, March 1999
Geodesic Flows
0 Introduction
Let M be a closed connected manifold and g a Riemannian metric on M. Let y(x,v)(t) be the unique geodesic with the following initial conditions: J
Y(x.u)(0) = x;
l
Y(x,v)(0) = v.
For a given t e IR, we define a diffeomorphism of the tangent bundle TM
of:TM -p TM, as follows
O,(x, v) := (Y(x.v)(t), Y(x,v)(t)) . The family of diffeomorphisms Of is in fact aglow, that is, it satisfies of+s = Oroos
This last property is an easy consequence of the uniqueness of the geodesic with respect to the initial conditions. Let SM be the unit tangent bundle of M, that is, the subset of TM given by those pairs (x, v) such that v has norm one. Since geodesics travel with constant speed we see that Of leaves SM invariant, that is, given (x, v) a SM, for all t E IR we have that ¢, (x, v) a SM. The restriction of Of to SM is called the geodesic flow of g. Chapter 1 describes the basic properties of the geodesic flow. We begin by recalling that geodesics can be obtained as solutions of the Euler-Lagrange equation of a Lagrangian given by the kinetic energy. We define symplectic and contact manifolds and we set up the basic geometry of the tangent bundle: we introduce the connection map, horizontal and vertical subbundles, the Sasaki metric, the
2
0. Introduction
symplectic form and the contact form. We describe the main properties of these objects and we show that the geodesic flow is a Hamiltonian flow. Also, when we restrict the geodesic flow to the unit sphere bundle of the manifold, we obtain a contact flow. The contact form naturally induces a probability measure that is invariant under the geodesic flow and is called the Liouville measure. Next we write the differential of the geodesic flow in terms of Jacobi fields. In the last section of the chapter we define the asymptotic cycle of an invariant probability measure and the stable norm. We show that the asymptotic cycle of the Liouville measure vanishes and that the same holds for the measure of maximal entropy if the latter is unique. Finally we show that the unit ball of the stable norm coincides with the set of asymptotic cycles of all invariant probability measures. Chapter 2 describes how the geodesic flow acts on Lagrangian subspaces. We introduce Lagrangian subspaces and Lagrangian submanifolds and we show an important property of the vertical subbundle that we call the twist property of the vertical subbundle. This property reflects the fact that the geodesic flow arises from a second order differential equation on TM. Next we derive the Riccati equations, after which we introduce the Grassmannian bundle of Lagrangian subspaces and we show how to attach an index, the Maslov index, to every closed curve of Lagrangian subspaces. The definition we use of the Maslov index follows Mane in [Mall and it is particularly adapted to the Riccati equations. This allows us to show that the lift of the geodesic flow to the Grassmannian bundle of Lagrangian subspaces is transverse to the Maslov cycle. This important property reflects the convexity of the unit spheres in tangent spaces. Using these tools we show two results, one motivated by hyperbolic sets and
the other by KAM tori. We show a theorem of Mane [Mal] that states that if there exists a continuous invariant Lagrangian subbundle E defined on SM, then E is transverse to the vertical subbundle and M does not have conjugate points. When the geodesic flow of M is Anosov, the stable and unstable bundles are continuous invariant Lagrangian subbundles, hence we deduce that if M is a closed Riemannian manifold whose geodesic flow is Anosov, then M does not have conjugate points. This last result was first proved by W. Klingenberg using different techniques [Kll]. Finally we show the following result due to L. Polterovich [Poll] (see also [BP]) which can be seen as the higher dimensional autonomous version of a result of G. Birkhoff that asserts that an invariant circle of a twist map in the cylinder T*SI which is homologous to the zero section must be a graph. Consider a Riemannian metric on the n-torus T. Suppose that P is a Lagrangian torus contained in S1" which is homologous to the zero section of T. Suppose that the set of non-wandering points of the geodesic flow restricted to P coincides with
all P. Then P is a graph, that is, the restriction to P of the natural projection n : ST" -> it is a diffeomorphism. In Chapter 3 we introduce the counting functions and we relate them to the topological entropy hrop(g) of the geodesic flow of g. Given x and y in M and T > 0, define nT(x, y) as the number of geodesic arcs joining x and y with length < T. Already in 1962 M. Berger and R. Bott [BB] observed that there are
0. Introduction
3
significant relationships between integrals of this function and Jacobi fields. We
show that, for each T > 0, the counting function nT(x, y) is finite and locally constant on an open full measure subset of M x M, and integrable on M x M. More generally, if N is a compact smooth submanifold of M, define nT(N, y) to be the number of geodesic arcs with length < T that join a point in N to y and are initially orthogonal to N. The function nT(N, y) enjoys properties similar to those of nT(x, y). If g is CO°, it was shown in [PP] that Yomdin's theorem [Y] can be used to prove that
lim sup< hrop(g) T-.oo T log
It was also shown in [PP] that when N is the diagonal in the product manifold M x M, the above inequality reduces to
IimsupIlogJ T-+,c T
nT(x, y)dxdy
(2)
M <M
Taking N to be the single point x in (1) gives the following inequality, which was first proved by the author inr[P1]: lim sup
I log J
T-oo T
nT(x, y) d y < hrop(g)
for all x E M.
(3)
M
Following Mart in [Ma2] we use the Borel-Cantelli Lemma to show that for every
x E M, one has
r I log nT(x, y) < lim sup I log nT(x, y') d y' T-.oo T Too T J
lim sup
for a.e. y E M. (4)
M
It is immediate from (3) and (4) that for all x E M, one has lim sup
1
T-.oo T
109 nT (x, y) < h,op(g)
for a.e. Y E M.
(5)
These inequalities generalize the work of A. Manning. Suppose x is a lift of a point x E M to the universal cover M of M and B(x, T) is the ball of radius T about z in M (with the metric lifted from M). We shall see that nT(x, y)dy > Vol B(z, T),
(6)
IM
with equality if M has no conjugate points. Manning showed that, for any x E M,
T-IVol B(z, T) converges to a Iimit A that is independent of x. From (3) and (6), it is easy to obtain the inequality h,op(g) > A for any Riemannian manifold, which was first proved by Manning in [Manl]. All this material is covered in Chapter 3.
Maffe gives in [Ma2] another proof of (2) based on a uniform version of Yomdin's theorem. We explain this at the end of Chapter 3. This uniform version
4
0. Introduction
of Yomdin's theorem is useful since it can be applied to other situations, as for instance, convex billiards. Chapter 3 also contains a section with a brief description of the concepts of topological entropy, entropy of an invariant probability measure and their relationship. Chapter 4 presents a proof of Maid's formula for geodesic flows and convex billiards. Maid shows in [Ma2]rthat for any C' Riemannian metric (r > 3), lim inf
1 log
T-oo T
JMxM
nT(x, y) dxdy ? hrop(g)
(7)
Maid thereby obtains the first purely Riemannian characterization of hrop(g) for an arbitrary C°O Riemannian metric. Combining (2) and (7) gives Ma>{e's formula lim 1 T-*oo T
log f
nT(x, y) dxdy = h,op(g).
(8)
MxM
For metrics with no conjugate points we shall prove that lim
1
T-.oo T
lognT(x, y) = h,op(g) = A
for all x, y e M.
The equality h,op(g) =1l if M has no conjugate points was first proved by Freire
and Maid in [FM]. Chapter 4 also contains a formula for h,op(g) in terms of the horizontal subbundle when M has positive sectional curvature and a proof of Maid's formula for convex billiards. The chapter concludes with a section about other related results and problems on the subject. Besides the natural appeal of a formula like (8), there are other reasons to be interested in relations between the topological entropy of the geodesic flow and the growth rate of the average number of geodesic arcs between two points on the manifold. The function nT(x, y) also counts the number of critical points of the energy functional on the path space c2T (x, y) given by all the curves joining x and y with length < T. Therefore using Morse theory, nT(x, y) can be bounded from below by the sum of the Betti numbers of S2T (x, y) (provided of course that x and y are not conjugate). By averaging over M, one can obtain in this fashion remarkable relations between the topology of M and h,op(g). This is the content of Chapter 5.
The chapter begins with the definitions of rationally elliptic and rationally hyperbolic manifolds and with a summary of various properties and characterizations of rationally elliptic manifolds. A manifold X is said to be rationally elliptic if the total rational homotopy,r.(X) ® Q is finite dimensional, i.e., there exists a positive integer io such that for all i > io, r (X) ®Q = 0. The manifold X is said to be rationally hyperbolic if it is not rationally elliptic (see [FHT, FHT2, GHI] and references therein and the first section of Chapter 5). Afterwards, we discuss
results of J.P. Serre [Sel] and M. Gromov [Grl] which allows us to relate the growth of nT (x, y) with the topology of M via Morse theory. Using these ideas we show the result in [P3] that says that if M is a closed manifold that fibres over
0. Introduction
5
a closed simply connected rationally hyperbolic manifold, then for any C°° Riemannian metric g on M, ht p(g) > 0. The more classical result of E. Dinaburg [D] which says that if iri (M) grows exponentially, then 0 for any g, is also discussed here. The chapter concludes with various definitions of entropies of manifolds. We connect them with other notions like Gromov's minimal volume and simplicial volume and we propose various related problems. There are all too many other topics which we have not mentioned here. Most notably, the notes cover almost no material on the vast theory of geodesic flows on manifolds of nonpositive curvature, particularly the rigidity theory of negatively curved manifolds. Fortunately, there are various lectures notes and surveys on this topic (e.g., [Ba, BGS, BCG2, El, E2, EHS]).
1
Introduction to Geodesic Flows
Our aim in this chapter is to introduce the geodesic flow on the tangent bundle of a complete Riemannian manifold from several points of view. Geodesic flows have the remarkable property of being at the intersection of various branches in mathematics; this gives them a rich structure and makes them an exciting subject of research with a long tradition. In Section 1.1 we define the geodesic flow of a complete Riemannian manifold. We also recall that geodesics can be obtained as solutions of the Euler-Lagrange equation of a Lagrangian given by the kinetic energy. In Section 1.2 we define symplectic and contact manifolds. In Section 1.3 we set up the basic geometry of the tangent bundle: we introduce the connection map, horizontal and vertical subbundles, the Sasaki metric, the symplectic form and the contact form. We describe the main properties of these objects, and we show that the geodesic flow is a Hamiltonian flow and that when we restrict it to the unit sphere bundle of the manifold we then obtain a contact flow. The contact form naturally induces a probability measure that is invariant under the geodesic flow and is called the Liouville measure. In Section 1.4 we describe the canonical symplectic form of the cotangent bundle and, using the musical isomorphisms, we shall describe its relations with the symplectic form defined in Section 1.3. In Section 1.5 we write the differential of the geodesic flow in terms of Jacobi fields. In the last section we define the asymptotic cycle of an invariant probability measure and the stable norm. We show that the asymptotic cycle of the Liouville measure vanishes and that the same holds for the measure of maximal entropy if the latter is unique. Finally we show that the unit ball of the stable norm coincides with the set of asymptotic cycles of all invariant probability measures.
1. Introduction to Geodesic Flows
8
Chapter I of Besse's book [Be], Chapter 3 of Klingenberg's book [Kl I], Chapter II of Sakai's book [Sal and Chapter IV in Ballmann's lecture notes [Ba] also contain introductions to the geometry of the tangent and unit tangent bundles as well as some basic facts about geodesic flows.
1.1
Geodesic flow of a complete Riemannian manifold
Let M be a complete Riemannian manifold and let y(x.,)(t) be the unique geodesic with initial conditions as follows:
1
Y(x.u)(0) = x; Y(x.u)(0) = V.
Definition I.I. For a given t E R, we define a diffeomorphism of the tangent bundle TM
0,: TM -TM, as follows Of (X, v) := (Y(x.u)(t), Y(x.u)(t))
The family of diffeomorphisms 0, is in fact a flow, that is, it satisfies 0,+s = Ot.os. This last property is an easy consequence of the uniqueness of the geodesic
with respect to the initial conditions. Let SM be the unit tangent bundle of M, that is, the subset of TM given by those pairs (x, v) such that v has norm one. Since geodesics travel with constant speed, we see that Ot leaves SM invariant, that is, given (x, v) E SM, for all t E R we have that /t (x, v) E SM.
1.1.1
Euler-Lagrange flows
Let L : TM -+ R be a smooth function and let Stxy be the space
Stay :_ (u : [0, 1] -+ M, piecewise differentiable and u(0) = x, u(1) = y}. The action A of L over a path from x to y is the map, A
s R,
:
A(u) :=
f L(u(t), ti(t))dt.
Let us try to find the critical points or extremals of A. Consider a variation s 1+ us E Stxy with s E (-e, e) such that ULA(s)Is=o = 0. If we set W(t) a (t)is=o, then a computation in local coordinates shows that u is a critical point if and only if 1
f0
1
d
8 (u, u) - I L (u, u)I (W)dt = 0. 7x
TV
1.2 Symplectic and contact manifolds
9
If we assume that this equation is satisfied for all variational vector fields W(t) arising from variations with u = up, we have
a(u,u)-d
aU(u.u)=0.
This is known as the Euler-Lagrange equation.
There is a class of Lagrangians that has received lots of attention in recent years. We shall say that a Lagrangian L is convex and superlinear if the following two properties are satisfied.
1. Convexity. We require that LITM : TM -+ R has positive definite Hessian for all x E M. This condition is usually known as Legendre's condition. In local coordinates this means that a2L
auiau;
is positive definite.
2. Superlinearity. There exists a Riemannian metric such that limlvl-+.
L(x, u) = +oo, lvi
uniformly on x.
If M is compact, the extremals of A give rise to a complete flow Or : TM -TM called the Euler-Lagrange flow of the Lagrangian. A very interesting aspect of the dynamics of the Euler-Lagrange flows is given by those orbits or invariant measures that satisfy some global variational properties. Research on the dynamics of these special orbits and measures goes back to M. Morse and G.A. Hedlund and has reappeared in recent years in the work of J. Mather, trying to generalize to higher dimensions the theory of twist maps on the annulus. See [Mat, Fa] and references therein for an account of this theory.
It is well-known that geodesics can be seen as the solutions of the EulerLagrange equation of the following convex and superlinear Lagrangian: I
L(x, v) := 2-g' (V, v) where g denotes the Riemannian metric of M.
1.2
Symplectic and contact manifolds
1.2.1
Symplectic manifolds
Definition 1.2. A 2-form w is said to be symplectic if w is:
1. Introduction to Geodesic Flows
10
closed, dw = 0; nondegenerate, that is, if wp(X, Y) = 0 for all Y E TpM then X = 0. A pair (M, w) of a smooth manifold and a symplectic form is called a symplectic manifold.
Remark 1.3. The existence of a symplectic form in a manifold M implies that M is even dimensional.
Definition 1.4. Let (M, w) be a symplectic manifold and H : M -- R a given C' function. The vector field XH determined by the condition w (X H , Y) = d H (Y) or equivalently ixx w = d H is called the Hamiltonian vector field associated with H or the symplectic gradient of H. The flow apt of the vector field X H is called the Hamiltonian flow of H.
The nondegeneracy of w ensures that XH exists and that it is a C'-1 vector field. In the next lemma we shall see that the Hamiltonian flow of H preserves the sympletic form w. Let us denote by Lx,, w the Lie derivative of w with respect to XH.
Lemma1.5. Proof. Using Cartan's formula
Lxyw = dixxw + and the fact that d w = 0 and d i x w = ddH = 0, we get LX, w = 0. Exercise 1.6. Show that Lxs w = 0 if and only if for all t e R, v, *w = w, where rp, is the flow of XH.
1.2.2
Contact manifolds
Definition 1.7. A 1-form a on a (2n - 1)-dimensional orientable manifold M
is called a contact form if the (2n - 1)-form a n (da)"-t never vanishes. A pair (M, a) of a smooth odd-dimensional manifold and a contact form is called a contact manifold. A contact flow is a flow on M that preserves the contact form on M. Unlike the symplectic manifolds, which admit a variety of Hamiltonian vector fields, a contact manifold comes with a canonical vector field X which is defined
by the conditions ixa = 1 and ixda = 0. The first condition says that X points along the unique null direction of the form dot and the second condition normalizes X (see for example [MS]).
1.3 The geometry of the tangent bundle
I1
Exercise 1.8. Show that X is unique. Show that a 1-form a on a (2n - 1)dimensional orientable manifold M is a contact form if for all x E M the restriction of dax to the kernel of a at x is nondegenerate.
Definition 1.9. The vector field X is called the characteristic vector field (also called the Reeb vector field) and its flow is called the characteristic flow. Note that this flow preserves a since Lxa = 0 by the definition of X. In the next section we shall describe the basic geometry of TM and SM and we shall prove that the geodesic flow is a Hamiltonian flow and a characteristic flow in TM and SM respectively.
1.3
The geometry of the tangent bundle
1.3.1
Vertical and horizontal subbundles
Let n : TM -+ M be the canonical projection, i.e., if 0 = (x, v) E TM then a(9) = X. Definition 1.10. There exists a canonical subbundle of TTM called the vertical subbundle whose fiber at 0 is given by the tangent vectors of curves a : (-e, e) -> TM of the form: a(t) = (x, v + tw), where 9 = (x, v) E TM and W E TxM. In other words, V(O) = ker(dotr).
Geometrically, V(9) is the tangent space to the fiber TM C TM at the point 0 (see Figure 1.1). Remark 1.11. Note that there is no canonical complementary "horizontal" subbundle. In fact to construct such a subbundle, we shall use a Riemannian metric on M. Definition 1.12. Suppose that we endow M with a Riemannian metric. We shall define the connection map
K: TTM -+TM, as follows. Let $ E TOTM and z : (-e, e) -+ TM be a curve adapted to l:, that is, with initial conditions as follows: Z(O) = 8;
i(0) = t. Such a curve gives rise to a curve a : (-e, e) -+ M, a := troz, and a vector field Z along a, equivalently, z(t) = (a(t), Z(t)). Define
KB (D) := (DaZ) (0).
12
1. Introduction to Geodesic Flows
v(e)
TM
e
den
1
M t
x Figure 1.1:
The vertical subbundle
The horizontal subbundle is the subbundle of TTM whose fibre at 9 is given by H(9) := ker KB. Lemma 1.13. K9 has the following properties: 1. KB is well defined;
2. KB is linear.
Exercise 1.14. Prove the lemma. Another equivalent way of constructing the horizontal subbundle is by means of the horizontal lift
L9: TxM -+TBTM,
which is defined as follows (9 = (x, v)). Given V' E TM and a : (-e, e) - M an adapted curve to v', let Z(t) be the parallel transport of v along a. Let a (-e, e) -+ TM be the curve a(t) = (a(t), Z(t)). Then LB(v') := 0(0) E TBTM.
It is immediate from the definition of parallel transport that Ke(L9(v')) = 0, for all v' E T, M.
1.3 The geometry of the tangent bundle
13
Lemma 1.15. Lo has the following properties: 1. LB is well defined;
2. LB is linear;
3. ker(KB) = im(Lo);
4. do,r,Le = IdT,M; 5. The maps d1trlH(B) : H(9) - TxM and KOIV(B) linear isomorphisms.
:
V(9) - TxM are
Exercise 1.16. Prove the lemma. From the lemma we conclude that
TBTM = H(9) ® V(B),
and that the map je : TBTM - TxM x TxM given by
je(t) = (den(4), K9(4)), is a linear isomorphism. Figure 1.2 on the next page summarizes in a diagram the information given by Lemma 1.15. From now on, whenever we write _ (l h, i;,) we mean that we identify $ with
je(s), where h = den(y) and v = KO (y). Definition 1.17. Using the decomposition TO TM = H(6) ® V(9), we can define in a natural way a Riemannian metric on TM that makes H(9) and V(8) orthogonal. This metric is called the Sasaki metric and is given by Ke(n)),r(e) .
((s. n))B :_ (deny, denn),i(e) +
Exercise 1.18. Show that if we endow TM with the Sasaki metric, the map n becomes a Riemannian submersion from TM into M with totally geodesic fibres.
-
To finish this section, we show that the geodesic vector field has a very simple expression in terms of the identification JB. The geodesic vector field G : TM TTM is given by
G(9) :=
a
at
L-0 Of(6) =
a I
at r=O
(ye(t)(t)),
where ye is, as usual, the unique geodesic with initial condition 6 = (x, v). But, note that t Y6(t) is the parallel transport of v along ye. Therefore, G(B) _ Le(v), or equivalently, G(B) = LB(v) = (v, 0) using the identification je.
14
1. Introduction to Geodesic Flows
TM
M
V (0)
x 0
I
I
den
Ke
H(9)
Id
LO
ZO
M x Figure 1.2:
1.3.2
Splitting of ToTM
The symplectic structure of TM
Based on the splitting TeTM = H(9) ® V(O) and the identification Je given in the previous section, we can introduce an almost complex structure in TM. Given 0 E TM we define JO : TBT M - TeTM, by setting
Definition 1.19. Using the Sasaki metric and the almost complex structure, we can define the symplectic form by
2e($, h) := ((Jet, n))s =
Ke(p)) - (Ke(y), dojr(t!)).
Exercise 1.20. Show that SZB is antisymmetric and nondegenerate. Show that for each 9, Jo is a linear isometry of the Sasaki metric. Show that JO is skew symmetric relative to the Sasaki metric, JB = -Id and that Je interchanges the subspaces H(9) and V(B). In the next subsection we shall see that S2 is exact; in particular it is closed and therefore it defines a symplectic form.
1.3 The geometry of the tangent bundle
15
The following result shows that the geodesic vector field is the Hamiltonian (v, v)5 with respect to the symvector field of the energy function H(x, v) plectic form n. Proposition 1.21. d H = iG S2 or equivalently, for all e E TM and all l; E TB T M we have
no (G (0). ).
Proof. Let z : (-e,e) - TM be a curve adapted to , and write z(t) _ (a(t), Z(t)). Then at
1t=o H(z(t)) = iii lr=o
(Z(t), Z(t))«(t)
= ((VaZ)(O), Z(O)) = (K9(t'), v) On the other hand,
Qo(G(B), t)
= =
(de,r(G(B)), Ke(y))
(detr(Le(v)), Ke(y)) = (v, Ke(y)).
Corollary 1.22. The geodesic flow preserves the symplectic form n.
1.3.3
The contact form
In many situations, it is more convenient to work with the geodesic flow restricted
to the unit tangent bundle SM since the latter is compact when M is compact. Thus, it is natural to ask if there exists, like the symplectic form in TM, some structure in SM preserved by the geodesic flow. We shall define in this section a form a in TM such that, when restricted to SM, it becomes a contact form whose characteristic flow is the geodesic flow restricted to SM.
Definition 1.23. The one-form a of TM is defined as
G(B))) =
v),.
Observe that a annihilates V (O). The form a and the symplectic form are related by: Proposition 1.24.
9 = -da. To prove this proposition we will use the following lemma that has its own interest.
16
1. Introduction to Geodesic Flows
Lemma 1.25. Let V/ denote the Riemannian connection of the Sasaki metric. Then for all ?7 E H(8) we have that V/,,G E V (O).
Proof. Let (E,,... ,
be an orthonormal frame field that is geodesic at x = 7r (0) and defined in a neighborhood U of x. This means VE,Ej(x) = 0, for all i and j. Let Xi (y, w) := (Ei(y), 0) be the horizontal lift of the vector field Ei, in other words, X-(y, w) = L(y. ) (Ei(y)). The vector fields {X1..... X} are orthonormal relative to the Sasaki metric and they span the horizontal subbundle (where defined) in T M. Note that it suffices to show that V/x, G belongs to V (8) for all j. The geodesic vector field can be written as
G(y, w) _
(Ei(y), w) Xi, i=1
hence n
V/x,G =
n
JXj (Ei(y),w)Xi+ E(Ei(y).w) V/,xfXi. i=1
i=1
(ii)
(i)
Since n : TM -> M is a Riemannian submersion, the horizontal component of V/X, Xi (0) equals DES E; (x) = 0, therefore the expression (ii) is vertical. To complete the proof of the lemma we need to show that (i) is vertical. In fact we shall prove that it vanishes.
Let aj : (-e, e) -+ M be a curve adapted to E j (x) and let Z j be the parallel transport of v along a. We have _
Xj(Ei('),-)= ddtt-o I
(Ei(aj),Zj)=(VajE1,v)=0.
Proof of Proposition 1.24. Recall that we can write
da (t1. 2) = 1a(2) - 2a(i) - (Y([1, 2}) Using the definition of a and the symmetry of the connection V/ we obtain,
da(41, 2) = 6 ( ( 2 , G)) - $2 (($1, G)) - (([I, '2} , G))
(1.1)
= ((1;2, 0/h, G)) - (('1, V/f2G))
(1.2)
We keep the notation of the previous lemma. Let us consider the vector fields Y1(y, w) := J(y,w)(Xi(y, w)). For each (y, w), the vector fields
(X1,..., Xn, Y1,..., Yn}
1.3 The geometry of the tangent bundle
17
form an orthonormal basis of T(y,,,,)TM. Note that [Yi, Y1] is vertical since each Y, is tangent to the fibres of TM. Therefore using equation (1.1) and the fact that G is horizontal we get da01V(0)XV(0) =0.
From the lemma and equation (1.2) we obtain d«OI H(0)xH(0) = 0
To end the proof it suffices to show that, for all i and j, we have
dae(X1, Yj) = -n0(Xi, Yj). Exercise 1.26. Show that for alI i and j, [Xi, Yj] = 0. Using the exercise we get
dcro(Xi, Y j) = -Yjct(Xi)(B) = -Yj ((Xi, G)) (0); since t H (x, Ej (x)t + v) is an integral curve of Yj, we get da0(Xi, Yj)
dt
(Ei(x), Ej(x)t + v) = -dij. r=0
On the other hand, from the definition of n we obtain
n0(Xi, Yj) =
Y1)) = ((Y1, Yj)) = 3ij.
From now on we shall consider a only restricted to SM.
Definition 1.27. For 0 E SM we define S(0) as the subspace of TOSM given by kera0. Equivalently, S(0) is the orthogonal complement with respect to the Sasaki metric of the one dimensional subspace spanned by G(0).
Exercise 1.28. Let 0 = (x, v). Show that 1. a vector
E TOT M lies in TOSM if and only if (KO(i ), v) = 0;
2. no(4,G(0)) =0forallt E TOSM; 3. S20(4, J0G(0)) = 0 for all t E S(0) C TOSM;
4. the orthogonal complement of S(0) in TOT M is given by the subspace spanned by G(0) and JOG(0) and therefore S(0) and its orthogonal complement are JO-invariant;
5. show that no IS(O) is nondegenerate. More generally, show that the restriction of 00 to any subspace of TO SM complementary to the subspace spanned by G(O) is nondegenerate.
18
I.
Introduction to Geodesic Flows
Corollary 1.29. The form a is a contact form in SM. Proof. By Exercise 1.8 it suffices to show that dae restricted to S(O) is nondegenerate. This a consequence of the last proposition and the last exercise. Lemma 1.30. The geodesic flow in SM is the characteristic flow of a. Proof. Take 0 E SM. From the definitions, we obtain
ae(G(9)) = (dotr(G(0)), v) = (v, v) = 1. and, for
E TOSM, we have
iGdae(f) = dao(G(0), $) _ - 79(G(0), t;) _ -doH(4) = 0, because H is constant on SM. Corollary 1.31. The geodesic flow in SM preserves the contact form a.
Exercise 1.32. Using that r : SM -a M is a Riemannian submersion show that the volume of SM equals the volume of M times the volume of the n - 1 dimensional sphere in R" with the canonical metric.
Exercise 1.33. Show that volume form on SM induced by the Sasaki metric co-
incides (up to sign) with "17a A (da)"-1, where n = dim M. When M has finite volume, the last two exercises show that the volume form a A (da)"-t has finite integral over SM and hence it gives rise to a probability measure µt defined on SM called the Liouville measure.
Lemma 1.34. Let 1 : SM -+ SM be the flip given by I (x, v) = (x, -v). Then 1'a = -a and I preserves the Liouville measure.
Proof. Note that no1 = n. Hence 1
(d(x.-v)n(d(x,v)I
-v)
= (d(x.u)nol (l; ), -v)
= (d(x.v) r( ), -v)
= -a(x.v)( )Therefore
1'(a A (dot)"-t) = (-1)"a A (da)"-(, which implies that the Jacobian of I is I and hence I preserves the Liouville measure.
1.4 The cotangent bundle T * M
1.4
19
The cotangent bundle T*M
We showed that TM has a symplectic structure and SM a contact structure but their definitions depend on the Riemannian metric of M. In this section, we will show that the cotangent bundle T*M has an intrinsic symplectic structure, that means that it does not depend on any metric. When M has a metric, we will prove that there exists a canonical relation between the corresponding structures in TM and T' M.
Definition 1.35. Let n : T*M i M be the canonical projection. Given (x, p) E T * M and 4 E T(x, p) T * M we define the canonical one form A in T * M by:
The symplectic form in T* M is cv := -d A.
Definition 1.36. Let M be a Riemannian manifold. The musical isomorphisms,
TM - T*M,
T'M -+TM
are defined as
(v, ) ,
(v. u0)
:= u(v)
The maps b and # are called musical isomorphisms because in classical notation they lower and raise indices.
Lemma 1.37. There exist the following relations:
1. a = b*A; 2. SZ=b*w. Proof Let 9 = (x, v) and b(9) = (x, p) be points in TM and T*M respectively. Then, A(deb(4)) = p(dn(e)iradeb(4))
= p(den(t)) = (den(y), v) = se(t), where in the third equality we used that nib = n. Finally, using that the exterior derivative commutes with the pullback, we obtain
b*w = b*(-dA) = -d(b'A) = -dot = 2, concluding the proof of 2.
0
It follows right away from the last lemma and the fact that SZ is a symplectic form that w is also a symplectic form.
20
1. Introduction to Geodesic Flows
1.5
Jacobi fields and the differential of the geodesic flow
In this section we shall describe an isomorphism between the tangent space TOT M and the Jacobi fields along the geodesic yo. Using the decomposition of TOT M in vertical and horizontal subspaces, we shall give a very simple expression for the differential of the geodesic flow in terms of Jacobi fields.
Recall that a Jacobi vector field along the geodesic Ye is a vector field along yo that is obtained as the variational vector field of a variation of yo through geodesics. It is well known that J is a Jacobi vector field along yO if and only it satisfies the Jacobi equation
J+R(y9,J)ye=0,
(1.3)
where R is the Riemann curvature tensor of M and dots denote covariant derivatives along yo. We recall that R is given by
R(X, Y)Z = VyVXZ - VXDyZ + V1X.YIZ. Let E To TM and z (-e, e) - TM be an adapted curve to . Then the map (s, t) H tr4,(z(s)) gives rise to a variation of ye = ,r ,(O). The curves :
t H Jr0 ,(z(s)) are geodesics and therefore the corresponding variational vector TIJ_osro,(z(s)) is a Jacobi vector field with initial conditions field J4 (t) given by
Jt(0) = Jt (0)
1-ls=on4)r(z(s)) =den(y) = a Ir=O 4Is=o n^Or(Z(S))
=
U
a, I.f=0 Z(s) = KO()
as r=o 'F1 Ir=o
Let us denote by J(y9) the vector space of all solutions of the Jacobi equatio (1.3). It is a 2 dim M dimensional vector space. Let us consider the map iO TOTM -, J(ye) given by ie(
) = Jt
It is obvious that iO is a linear isomorphism.
Definition 1.38. We shall say that a Jacobi field J is normal to the geodesic ye if
(J(t), yg(t)) = 0 for all t E R. Exercise 1.39. Take 0 E SM. Show that iO restricted to S(0) gives an isomorphism between S(0) and all the normal Jacobi fields to ye.
The next lemma describes the differential of the geodesic flow in terms of Jacobi fields and the splitting of T T M into horizontal and vertical subbundles.
Lemma 1.40. Given 0 E T M, do 0,
E TOT M and t E R, we have
(t
4 (t)) .
1.6 The asymptotic cycle and the stable norm
21
Proof.
t(t) = Jt (t)
=
sI
de(n0r)(l;) = dmr(e)aadoOt(l:). n-Or(z(s))
-o
6a
I
s=o n.0t(z(s)) _ ° (s=o
.I,00t(z(s)) = By means of this identification we can write
fmte(do0t(k),dort(rl)) = (-Jew, Jn(t))+(Jt(t), Jn(t)). Since Q is invariant under 0,, the right hand side should be a constant function of t. This can be checked by differentiation and using the Jacobi equation:
((-Jt. ill) + (Jt, 4Y = - (Jt, in) - (Jt, in) + (Jt, Jn) + (Jt. Jn) R(y, Jn)Y) = 0. = (Jn, R(Y, Jt))) Exercise 1.41. Show that 0, : TM -> TM is an isometry of the Sasaki metric for all t E R if and only if M has constant sectional curvature 1.
1.6
The asymptotic cycle and the stable norm
We shall assume in this section that M is compact.
1.6.1
The asymptotic cycle of an invariant measure
Let A be a probability measure defined on the Borel a-algebra of SM. Definition 1.42. The measure µ is said to be invariant under the geodesic flow if, for any Borel set B e SM, we have µ(01(B)) = µ(B) for all t E R. An invariant measure µ is said to be ergodic if any Borel invariant set has measure zero or one. We shall denote by M(O) the set of all Borel invariant probability measures. The set M (¢) is a compact metrizable convex set with the weak topology of measures [W].
Exercise 1.43. Show that the Liouville measure ut is invariant under the geodesic flow.
The asymptotic cycle of an invariant measure µ will be an element p(µ) E Hi (SM, R). Suppose that c is a cohomology class in H t (SM, R). Take a smooth closed one-form rI in SM such that [A] = c. The asymptotic cycle p(A) is defined as the unique element in Ht (SM, R) such that
(P(µ), c) = fs M A(G) dµ. Lemma 1.44. p(µ) is well-defined.
1. Introduction to Geodesic Flows
22
Proof We need to show that our definition does not depend on the choice of the closed one-form A. Equivalently, we need to show that if f is a smooth function on SM, then
df(G)dp=0. SM
Since µ is invariant, Birkhoff's ergodic theorem implies that
JSM
df (G)
du
J,lim
=
JSM
t
(f (0,(8)) - f (8)) } d,4 = 0.
Suppose now that it is ergodic. By Birkhoff's ergodic theorem lim 1
I
fr
00 t 10
A(G(,,,0)) ds =
J
A(G) d;c,
SM
for u-a.e. 8 E SM. Let j,(8) be the oriented orbit segment from 0 to O,8. By definition of integration of /forms 1 r
A(G(0,0)) ds =
o
Jfir(e)
A.
Pick a family of arcs 69i.e2 of bounded length connecting points 01 and 02, for example, a shortest geodesic with respect to the Sasaki metric. Then there exists a constant C > 0 such that < C.
Replace the orbit segment 6,(8) by the closed loop A(8) obtained by joining fit (0) with fi
e. We obtain lim 1 r-+oo '
1_
A=
JSM
A(G) dµ.
Hence P(A) = rllimoo
where denotes homology class. This gives a geometric interpretation of the asymptotic cycle when µ is ergodic. Asymptotic cycles were first introduced by R. Schwartzman [Sch].
The map n : SM -+ M induces a map in cohomology n' : H 1(M, R) -H I (SM, R).
Lemma 1.45. Suppose that M" is orientable and different from the two-torus. Then the map it* is an isomorphism.
1.6 The asymptotic cycle and the stable norm
23
Proof. Using the Gysin sequence (cf. [BT, Proposition 14.33]) of the unit sphere bundle n : SM -+ M, we obtain
0 - Ht (M. R) '* Ht (SM, R)
HZ-"(M, R)
HZ(MR) -
,
where a. is integration along the fibre and Ae is multiplication by the Euler class.
Then if n > 3, H2-"(M, R) vanishes and n` is an isomorphism. If n = 2, since M is not a two-torus, the Euler class e does not vanish and the map Ae : H0(M, R) = R -+ H2(M, R) = R is an isomorphism and hence ,r' is also an isomorphism.
Lemma 1.46. If f3 is any one form in M, then
,r f(G(0)) = P. (v), where B = (x, v). Proof.
n'P(G(B)) = Pa(e)(detr(G(B))) =18 (v).
0
Exercise 1.47. Let M be diffeomorphic to the two-torus. Show that there exists a basis (Ch c2, c3} of H 1 (SM, R) such that ct and c2 can be represented by the pull back of closed one-forms in M and c3 can be represented by a closed one-form A in SM with the following property. There exists a one-form f on M such that ).©(G(0)) = 8. (v),
where 0 = (x, v). Conclude that if a probability measure µ is invariant under the
flip (x, v) H (x, -v), then A(G) dµ = 0. ISM
Definition 1.48. We set PM (9)
n.(p(u)) E H,(M,R),
where n.: Ht (SM, R) -+ H, (M, R) is the map induced in real homology by 7r. The homology class pM (µ) is also called the asymptotic cycle of A. By Lemma 1.45 and Exercise 1.47 we do not lose essentially any information by looking at the cycle in M rather than in SM, and as the next lemma shows, there is a very simple and natural expression for it which does not involve the geodesic vector field explicitly.
1. Introduction to Geodesic Flows
24
Lemma 1.49. pM (µ) is the unique element in HI (M, R) such that for any closed one form w in M we have
(PM(14), [w]) = f
wdµ,
SM
where we think of w as a function w : TM -+ 1R.
Proof. Note that
(n.P(t), [w]) = (P(,s), (n*w]) = f ,r w(G)dµ,
0
and the lemma follows from Lemma 1.46.
Lemma 1.50. If it is invariant under the flip (x, v)
(x, -v), then
PM(IL) = 0.
Proof. By Lemma 1.49 we have
(PM(µ), [w]) = J
SM
wdµ.
Since wJI = -w and p is invariant under I we have
wdµ = 0, SM
and hence pM(it) = 0. Lemma 1.51. p(µt) = 0, where µt is the Liouville measure. Proof Suppose that M is orientable. Since the Liouville measure is invariant un-
der the flip (cf. Lemma 1.34), it follows from Lemma 1.50 that pM(µt) = 0. Using Lemma 1.45 and Exercise 1.47 we conclude that p(µt) = 0. When M is nonorientable, by passing to an orientable double cover, we can obtain right away that p(Al) = 0. We now give another proof that also holds for arbitrary contact flows.
It suffices to show that if x is any closed one-form in SM, then
X (G)a A (da)"-t = 0. SM
But note that
X(G)CI A (da)"-t = iG(a A (da)"-t) AX and that
iG(a A (da)"-t) = (da)'.
1.6 The asymptotic cycle and the stable norm
25
Hence
'SM k(G)a n (da)"
= 'SM
n x.
Using that (da)'- 1 A A = d(a A (da)"-2 AX) and Stokes theorem, we obtain
(dot)"-t n x = 0. JSM
Another interesting case of vanishing asymptotic cycle occurs when the measure of maximal entropy is unique. See Section 3.2 for the definition of measure of maximal entropy.
Lemma 1.52. If there exists a unique measure of maximal entropy to, then PM (uo) = 0.
Proof. Let I be the flip (x, v) r* (x, -v). Since ¢,01 = 1.0-1 and uo is unique, then AO is invariant under I. Now apply Lemma 1.50.
Definition 1.53. We call the set 13 := pM(M(O)) C Ht (M, llt) the Schwartzman ball.
The set B contains all the homologies in M arising from invariant probability measures, and hence it packs a lot of geometric information. We shall see that B is nothing but the unit ball of the stable norm that we define in the next subsection.
Lemma 1.54. The map pM : M(0) -+ Hi (M, R) is affine and continuous. Proof. Clearly pM is an affine map. Take a sequence of measures un -+ it. For any closed one-form w,
ISMILJSM Since Hi (M, R) is a finite dimensional vector space this implies that pM is continuous.
Proposition 1.55. The Schwartzman ball B is a compact convex set symmetric about the origin. Moreover, the origin lies in its interior.
Proof. We follow D. Massart's thesis [Mst]. Since pM : M(O) -+ Ht (M, R) is of lne and continuous, it is clear that B is compact and convex. Let us show that B is symmetric about the origin. As before, let I be the flip (x, v) t-- (x, -v). Since
I. Introduction to Geodesic Flows
26
4>to1 = J4_t it follows that 1 maps invariant measures to invariant measures. Hence
(PM(l.µ), [w]) =
JSM
wd(4p) wol dlt
ISM
wdµ 'SM
= - (PH(IL). [w]) Hence pM(I /z) = -pM(,u) which shows that 8 is symmetric about the origin. Let us prove now that the origin belongs to the interior of B. Recall that every nontrivial homology class in H 1 (M, Z) contains a closed geodesic. Hence there exist closed geodesics with unit speed yi, ... , y k such that (h i , ... , hk ) is a basis of Hi (M, R), where h; = [yi ]. Let ti be the length of yi. Let µi be the probability measure uniformly distributed along yi. That is, µi satisfies
f Ili = fi SM
/'t;ti
If
Yi(t))dt,
for any continuous function f : SM -* R. Clearly p.i is invariant and PM(ILI) = Pi thi, since
(PM(Iti), [w]) = 'SM wdµi =
1J w =
('hi[w]) .
;
Consequently, B contains the convex envelope of
{±t 'hi} which is a convex set containing the origin in its interior.
1.6.2
The stable norm and the Schwartzman ball
In this subsection we define the stable norm and we show that the unit ball of the stable norm coincides with the Schwartzman ball. Our presentation follows D. Massart's thesis [Mst]. Massart mentions in his thesis that this identification is due to A. Fathi. There is also a proof of this fact for the n-torus in [BIK]. The study of the stable norm has attracted great attention in recent years, see for example [Ban, Bu, BIK]. One of the main questions in the subject is: what norms can arise as stable norms of Riemannian metrics?
Let (M, g) be a closed Riemannian manifold and let r be the quotient of Hi (M, Z) by its torsion subgroup; r is a cocompact lattice in Ht (M, R).
1.6 The asymptotic cycle and the stable norm
27
Definition 1.56. For It E r we set
f (h) := inf £(y), where the infimum is taken over all closed piecewise differentiable curves with [y] = h, where £(y) denotes the length of y.
Exercise 1.57. Show that lim,-,, i"h1 exists by showing first that
f(ht + h2) < f(ht)+ f(h2)+2d where d is the diameter of M.
For h E r, we let lim f (nh) n-,00 n
It is immediate to check that Ih1,, has the following properties. 1.
Ihls < f(h);
2. for all A E Z and all h E 17, we have Ixhls = IAIIhIs;
3. forallht,h2 E r we have I h t +h2ls < IhI Is +Ih21s-
Definition 1.58. A homology class h E Ht (M, R) is said to be rational if there exists a positive real number r such that rh E r. We extend h i-i Ih1, to all rational points in HI(M, R) by homogeneity. Since the rational points are dense in Ht (M, IR), we can extend Ih Is by uniform continuity to a function defined in all Ht (M, R). One can easily show that this extension defines a norm in Hi (M, R) called the stable norm (see [Ban, Appendix] for the details). We shall denote by I[w]I5 the dual norm induced in Ht (M, R). Theorem 1.59. The Schwartzman ball is the unit ball of the stable norm. Proof. Let Bs be the unit ball of the stable norm. Let us prove first that B C_ Bs. Let pM(µ) E B and let w be a closed one-form in M. For any closed piecewise differentiable curve a, we have I ([a], [w]) 1 <_ I[w]I51[a]Is <_ 1[0J]Ise(a).
28
I.
Introduction to Geodesic Flows
Given (x, u) E SM and t > 0, let a be the closed curve obtained following the geodesic defined by (x, v) up to time t and then returning to x by a curve of length < d where d is the diameter of M. We have
x Iwx I f t w(0r(x, v))dt < ([a], [w]) +dmXCM 0
I [w] Is (t + d) + d m ax Iwx 1.
Now observe that by Fubini's theorem and the invariance of pt, we have
f
fw(cbr(xv))dtdtt=t f
M0
SM
Hence
t-t f {I[w]Is(t+d)+dmXCM xlwxl} dµ.
(PM(IA),[wl)I
SM
11
If we let t - oo we obtain I (PM(u), [w)) 1:5 I[wlls, which shows that IPM (µ) I s
Now, let us prove that Bs C B. By compactness of B and the definition of the stable norm, it suffices to show that if h 96 0 is a rational point with Ih l < 1, then h E B. Since h is a rational point there exists a positive real number r such that
rh E 1. Let yn be unit speed closed geodesics such that t(yn) = f (nrh) for all nonnegative integers n. As in Proposition 1.55, let µn be the probability measure uniformly distributed along y,,. We have
PM(An) = f(yn)-t nrh =
nrh f(nrh)*
Let tt be an accumulation point of {µn 1. By the continuity of PM we have
rh PM (A) = IrhI5
=
h IhIs
But since IhIs < 1, Proposition 1.55 implies that h E B. Corollary 1.60.
I[w]Is = max
f
SM
wd s.
El
1.6 The asymptotic cycle and the stable norm
29
Proof. Using the theorem we have (w] 1, =
(h, [w])
max (P(u), [w])
tcEM(m)
max
/LEM(O) SM
wdµ.
El
2 The Geodesic Flow Acting on Lagrangian Subspaces
This chapter describes how the geodesic flow acts on Lagrangian subspaces. We introduce Lagrangian subspaces and Lagrangian submanifolds and we show an important property of the vertical subbundle which we call the twist property of the vertical subbundle. This property reflects the fact that the geodesic flow arises
from a second order differential equation on TM. Next we derive the Riccati equations, after which we introduce the Grassmannian bundle of Lagrangian subspaces and show how to attach an index, the Maslov index, to every closed curve of Lagrangian subspaces. The definition used of the Maslov index follows Maid in [Ma I] and it is particularly adapted to the Riccati equations. This allows us to show that the lift of the geodesic flow to the Grassmannian bundle of Lagrangian subspaces is transverse to the Maslov cycle. This important property reflects the convexity of the unit spheres in tangent spaces. Using these tools we show two results, one motivated by hyperbolic sets and the other by KAM tori. We show a theorem of Mane [Ma I ] that says that if there exists a continuous invariant Lagrangian subbundle E defined on SM, then E is transverse to the vertical subbundle and M does not have conjugate points. When the geodesic flow of M is Anosov, the stable and unstable bundles are continuous invariant Lagrangian subbundles; hence we deduce that if M is a closed Riemannian manifold whose geodesic flow is Anosov, then M does not have conjugate points. This last result was first proved by W. Klingenberg using different techniques [Kll].
Finally we show the following result due to L. Polterovich [Poll ] (see also [BP]) which can be seen as the higher dimensional autonomous version of a result of G. Birkhoff that asserts that an invariant circle of a twist map in the cylinder T*SI which is homologous to the zero section must be a graph. Consider a
2. The Geodesic Flow Acting on Lagrangian Subspaces
32
Riemannian metric on the n-torus T. Suppose that P is a Lagrangian torus contained in SV which is homologous to the zero section of TIT". Suppose that the set of nonwandering points of the geodesic flow restricted to P coincides with
all P. Then P is a graph, that is, the restriction to P of the natural projection it : ST -+ T" is a diffeomorphism.
2.1
Twist properties
Definition 2.1. Let (V, S2) be a symplectic vector space. Since V must be even dimensional, we write dim V = 2n for some positive integer n. A subspace E C V is said to be Lagrangian if its dimension is n and the symplectic form satisfies S2I Ex E = 0. A submanifold P of a symplectic manifold is said to be Lagrangian if the tangent space T, P is a Lagrangian subspace for all x E P. It is straightforward to check from the definition of the symplectic form S2 of TM that the subspaces H(O) and V(O) are Lagrangian. Remark 2.2. More generally, a subspace S C V is said to be isotropic if S2ISXS = 0. Because of the nondegeneracy of n, an isotropic subspace has at most dimension n, so Lagrangian subspaces are maximal isotropic subspaces.
Exercise 2.3. Let V be a 2n dimensional linear space and S2 a symplectic form on V. Prove that there exists a basis lei, en+i } 1:5i5, of V such that 12 (ei , e j) _ S2(en+i, en+j) = 0 and S2(ei,
Sij for I < i, j < n.
The basis of the Exercise 2.3 gives a decomposition of V as the direct sum of two Lagrangian subspaces.
Exercise 2.4. A subspace E C ToTM is Lagrangian if JoE = E. Exercise 2.5. Let V be a 2n dimensional real vector space, and let n be a nondegenerate two-form in V. Define an action of GL(2n, R) on the set of nondegenerate two-forms on V by (ac2)(v, w) = S2(av, aw) for all vectors v and w in V. Using Exercise 2.3 prove that GL(2n, R) acts transitively on the set of nondegenerate two-forms on V.
Exercise 2.6. Let V = C", regarded as a 2n dimensional real vector space, and define a two-form 0 on V by S2 (v, w) = Re ((Jv, w)), where v = (v 1, ... , vn) and w = (w1, ... , wn) are arbitrary elements of C", J(vi,...,vn) = (ivt,...,ivn)
and (v, w) 1.
vktUk. Prove
SZ is a nondegenerate two-form;
2. if g(v, w) = Re ((v, w)) is the usual inner product on R2" = C", then S2(v, w) = g(Jv, w);
2.1 Twist properties
33
3. if E C C" is a real subspace with real dimension n, then E is Lagrangian
if and only if (v, w) ER for allvandwinE; 4. if E C C" is a Lagrangian subspace, then a(E) is a Lagrangian subspace for all a in the unitary group U(n); 5. U (n) acts transitively on the set of Lagrangian subspaces of C.
Exercise 2.7. Let H := (a E GL(2n, R) aQ = a). Prove that an element a E GL(2n, R) lies in H if and only if a'Ja = J, where as denotes the real :
transpose of the transformation a relative to the inner product g on R2i. Show that U(n) is a real subgroup of H of dimension n2.
Lemma 2.8. Let N be a submanifold of M and
TN1:=((x,v)ETM : xEN, v1TN) its normal subbundle. The submanifold T N1 is Lagrangian.
(xi(t), vi(t)) E TN'
Proof. Given rii E T(, .,,)TN', i = 1, 2; define paths t
with initial conditions a; (0) = (x, v) and 6i (0) = >ri. T,,(B)M, we have Using the identification, TBTM Qi (0) = n, = (xi (0), (Vx; vi) (0)) .
Extend Xi (0) to a vector field Xi defined in a neighborhood of x such that when we restrict Xi to N, Xi is tangent to N for i = 1, 2. Since ai is a curve in TN-L, we have that for all t, (X I (x2(t)), V2 (0) = 0,
u1(t)) = 0.
Differentiating the last two expressions with respect to t and evaluating at t = 0 we get ((V X2 X 1WI v) + (X 1(0), V
2 v2 (0))
= 0,
((V1 X2(x), v)+(x2(0), V ,V1(0)) =0. Using the definition of c2 and the last two equalities we obtain C2(x.u)(2J1 ,'72)
_ (x1(0), (Ox2 V2)(0)) - (x2(0), (Or, vl)(0))
-(v, (VX2XI)(x))+(v, (VX,X2)(x))
_ (v, (VX1 X2)(x) - (VX2X 1)(x)) = (v, [X I , X2)(x)) But (v, [X1, X2](x)) = 0 since [X1, X2](x) E T1N and v c- TXN1.
34
2. The Geodesic Flow Acting on Lagrangian Subspaces
Exercise 2.9. Given X E N and V E Tx Nl, the shape operator of N at v is the symmetric linear map
A,,:TN-+TN defined as follows. Let V be a C°O normal vector field in a neighborhood of x in N with V (x) = v. Given W E TX N, A, (w) is the orthogonal projection of V , V onto TxN (see for example [Sa]). Show that T(x,,,) T Nl is given by the set of vectors
E Ttx.,,ITM such that d(x,,,)7r(i;) E TN and K(x,,,)(1;) -
E
TC Nl. Use the above, together with the fact that A is a symmetric linearmap, to give another proof of the last lemma.
Remark 2.10. Observe that if E is a Lagrangian subspace, then d6 ,(E) C TT,(o) TM is also Lagrangian since the geodesic flow preserves Q.
Proposition 2.11 (twist property of the vertical subbundle). Let E be a Lagrangian subspace of C TeTM. The subset given by ft E R : doo,(E) n V(4,(9)) 0 {0}} is discrete.
Proof. Suppose that E n v (9) # 10}, we shall prove that there exists a > 0 such that for all t 96 0, t E (-a, a) we have: do0,(E) n v (o,(9)) = (0).
Let p : ToTM -- H(9) be the orthogonal projection.
Claim: The subspace p(E) is the orthogonal complement of Je(E n V(9)) in H(O) with respect to the Sasaki metric where, as usual, Je denotes the almost complex structure in TM. Proof of the claim. Take elements S = p(i;) and Jo q with t: E E and 17 E En V (9). Then
((8,J91)) _7e(q,U=0, because and q belong to the Lagrangian subspace E and S and Jeq are horizontal. This implies the orthogonality of the subspaces. Their dimensions satisfy
dim p(E) + dim Je(E n V(9)) = dim E - dim(E n V (O)) + dim Jo(E n v (o))
= dim E = dim H(9), where in the second equality we used that Jo is an isomorphism.
0
2.1 T\vist properties
35
Let {ill, ... , qm) be a basis of p(E) and let p, T#,(B)TM -> H(O,(B)) be the orthogonal projection. Then if t is near zero, there exists a set of m linearly independent vectors ( t ) ,. ... , )m(t)} contained in p,(de¢,(E)). Take an orthonormal basis (l; I, ... , k) of E n V(0). Then from the claim we get that k + m = dim H(9). :
T a k e Jacobi vector fields Ji,
i = 1, ... , k with initial conditions: (0) = 0;
I Ji(0) = wi,
where w; :=
or li = (0, w,) using the identification of TBTM with
From the expression for the differential of the geodesic flow in terms of Jacobi vector fields, we get
Pt ((Ji(t), ii (m) = (A (t), 0).
(2.1)
Since Ji is a nontrivial Jacobi field that vanishes at zero, we can define unitary vector fields Wi (for ItI small) by Wi(t)
(-Ji(t)/IJi(t)I,0) fort > 0; (Ji(t)/IJi(t)1,0) fort <0;
then
JAW'
IimW1(t)=lim 0 t0 1-.0 IJi(t)ltI = lim (-Ji (t )II Ji (t) I , 0) = (-wi, 0) = J4. t0 Moreover,
P,(d4Ot(E)) D {,1(t), ... , IIm(t). WI (t), ... , Wk(t)) When Itl is small, (171 (t), .... rim (1), W1 (t), ... , Wk(t)} is near
{ q l , ... , qm, Jell, ... , Jerk} which is a basis of H(8). Then, by continuity,
pt(dett(E)) = H(-01(0)).
We shall refer to Proposition 2.11 as the twist property of the vertical subbundle. To study when the horizontal subbundle has the same twist property, we need to introduce the curvature operator.
2. The Geodesic Flow Acting on Lagrangian Subspaces
36
Definition 2.12. Let 0 = (x, v) be a point in TM. The curvature operator,
R(69): (v}1 - (u)1, is the selfadjoint map given by w i-+ R(v, w)u, where R denotes the curvature tensor of the Riemannian manifold M.
From now on, we shall restrict the symplectic form S20 to the subspace S(9) (see Exercise 1.28). The intersection of the vertical and horizontal subspaces with S(6) defines two Lagrangian subspaces of the symplectic vector space (S(0), 0g) which we still denote by V(6) and H(e). Recall that the differential of the geodesic flow deb, takes S(0) to S(4 0).
Proposition 2.13 (twist property of the horizontal subbundle). Suppose that R(4,6) is a positive (negative) definite operator for all t E R and let E C S(0) be a Lagrangian subspace. The subset (t ER: doO1(E)nH(O,(0)):A (0)} is discrete.
Proof. We will indicate the necessary modifications in the proof of Proposition 2.11 above leaving the details as an exercise. Consider the map R(9) : S(O) - S(9) given by
lZ(O)(h, o) = (th, R(0)(tv)). Define an inner product b in S(8) by setting
b(t, n) = ((t, R(B)-t (v))) Let
S(0) -- V (O) be the orthogonal projection. As in the proof of Proposition
2.11, one can check that p(E) is the b-orthogonal complement of R(9)Ja(E f1 H(0)) in V(0). Now take 1!;1_. , . k } an orthonormal basis (with respect to the Sasaki metric)
of 1 (9)JB(E n H(6)). Then i; = (0, R(B)tai), where (w;, 0) E E n H(0). Take J;, i = 1,... , k Jacobi fields with initial conditions
Ji(0)=w;; 1
J; (0) = 0.
Define the vector fields
_ W'
0,-J;/IJ;I) fort>0;
{ 0, ii/IJrI) fort <0.
2.2 Riccati equations
37
Use the Jacobi equation to obtain
limW;(t)=t;.
t-+0
2.2
Riccati equations
Let E C S(O) be a subspace with dim E _ dim S(O). Suppose that En V (O) _ {0}. Then it is possible to see E as a graph over the horizontal subspace. We can express
E = graph U :_ ((w, Uw), w e H(O)} with U : H(9) -> V (O). If we use the identification of H(9) and V (O) with {v}'L, the map U is defined from {v}l into itself.
Lemma 2.14. The subspace E is Lagrangian if the map U is symmetric.
Proof. Let , n be elements in E; if 14 = (w, Uw) and q = (y, Uy) then, because q) = 0, we have of the condition
Assume now that E is a Lagrangian subspace of S(O). Let I = (-6, 6) be an interval with deb, (E) fl V (O, (8)) = (0) for all t e 1. We set deb, (E) = graph U (t) for all t e 1, with U(t) : {Ye(t)}1 -, {Ye(t)}-L. Let E E; the differential of the geodesic flow can be written as de 0,
(jt (t), it (r))
doOt(
(Jt(t), UJt(t))
therefore we can also write
which corresponds to the change of variables j4(t) = U(t)JJ(t). Hence,
Jt = U it + U it UJt + U2Jt. Substituting into the Jacobi equation, we obtain
0=
J4 (t) + R(tbt(0))Jt(t)
(U + U2 + R(0,(0))) Jt(t). Since a;' E E is arbitrary, the family of operators U(t) satisfies
U+U2+R(0,(0))=0. This is the Riccati equation seen from the horizontal subbundle.
38
2. The Geodesic Flow Acting on Lagrangian Subspaces
Remark 2.15. The covariant derivative U can be interpreted as follows. Take a parallel basis of {ye(t)}1 and consider the matrix A of U in this basis. Then U is the linear map whose matrix in the parallel basis coincides with A.
Remark 2.16. When M is a surface, the maps U(t) can be written as U(t)w = u(t)w, with w E (ye(t))-L, where u(t) is a smooth function oft that satisfies the classical scalar Riccati equation
ti+u2+K(t)=0. where K(t) is the Gaussian curvature of the surface at yo (t). The function u(t) is the slope of the line determined by do t(E) in the plane S(0,(8)).
Analogously, we can do the same for the vertical subspace when we have de¢t(E) fl H(¢,(9)) = (0), for all t E (-e, e). Let U(t) : V (ct(9)) -+ H(O,(8)) be a family of linear maps such that
do0t(E) =graph U(t) := ((Uw, w), WE V (¢t(9))). Let l; E E; we have
d9tdr(4) _ (Jt(t),it (t)) = (U(t)Jl(t),.4(t)). which corresponds to the change of variables J{ = U.4 . Taking the derivative of Jl = UJg and using the Jacobi equation, we obtain
Jl = UJI + UJl = U4 + U(-RJ4) = UJl - URUJI. The last equality implies that U(t) satisfies
U-URU-Id=0. This is the Riccati equation as seen from the vertical subbundle.
Remark 2.17. When M is a surface, the maps U(t) can be written as U(t)w = u(t)w, with w E (ye(t))', where u(t) is a smooth function oft that satisfies the scalar equation
ti-K(r)u2-1 =0, where K(t) is the Gaussian curvature of the surface at ye(t).
2.3
The Grassmannian bundle of Lagrangian subspaces
Suppose that (V, n) is a symplectic vector space.
2.4 The Maslov index
39
Definition 2.18. We shall denote by A the set of all Lagrangian subspaces of (V, 0). The set A has a natural manifold structure and is called the Grassmannian manifold of Lagrangian subspaces.
Exercise 2.19. Using Exercise 2.6 prove that A is diffeomorphic to the homogeneous space U(n)/O(n). Hence A is a compact manifold of dimension S ,n ji.1 where 2n = dim V.
Definition 2.20. Fix a Lagrangian subspace F E A. For each integer k between 0 and n, let us denote by Ak the subset of A given by those Lagrangian subspaces whose intersection with F is a subspace of dimension k.
Exercise 2.21. Let F E A be a Lagrangian subspace of V, and let GF(k, n) be the Grassmannian manifold of k dimensional subspaces of F. Define p : Ak --> GF(k, n) by p(E) = E n F. Prove 1.
If W E GF(k, n), then E E Ak lies in p-I (W) if and only if E = W W', where W' is a Lagrangian subspace of (W ® J(W)}l, the orthogonal complement of W ® J(W) in V = C" relative to the standard Hermitian inner product on Cl;
2. Ak is a submanifold of A and p : Ak -+ GF(k, n) is a submersion;
3. At has codimension
kt- in A.
If M is a Riemannian manifold and 0 E SM, we can consider A(9) the Grassmannian manifold of Lagrangian subspaces of S(9).
Definition 2.22. Given any set X C SM we shall denote by A(X) the set given by the union of all A(O) for 0 E X. In particular, when X = SM, A(SM) becomes a fibre bundle over SM called the Grassmannian bundle of Lagrangian subspaces. Observe that the vertical and horizontal subbundles can be seen now as sections
of the bundle A(SM)
F-+
SM. We shall denote by Ak(9) the set of
Lagrangian subspaces E E A(9) whose intersection with V(9) has dimension k and by Ak(SM) the union of all Ak(0) for all 9 E SM. Using Exercise 2.21 we have that
Lemma 2.23. Ak(SM) is a submanifold of A(SM) of codimension k
2.4
.
The Maslov index
We shall describe in this section how to attach an index to every continuous closed
curve a : S1 -+ A(SM). First we shall define the Maslov cycle.
40
2. The Geodesic Flow Acting on tagrangian Subspaces
Definition 2.24. The Maslov cycle is the set given by
AV = U Ao(SM). k>1
By Lemma 2.23, AV is the union of A 1(SM) with submanifolds of codimension > 3. Remark 2.25. When M is a surface, the Maslov cycle becomes a closed submanifold of codimension one in A(SM).
Exercise 2.26. The twist property of the vertical subbundle (Proposition 2.11)
says that given a point (0, E) in the Maslov cycle Ao(SM), there exists e = e(8, E) > O such that for all t E (-e, e) with t 54 0, we have that (0,(0), doo,(E)) does not belong to the Maslov cycle. Show that e(6, E) > 0 in general is not bounded away from zero even if M is compact. Is e(8, E) bounded away from zero when M is a closed surface?
Let S' be the unit circle of the complex plane. We define a function m Ao(SM) U A1(SM) --), S' as follows. If (0, E) E Ao(SM), by Lemma 2.14 we can take a symmetric linear map U : H(0) -). V(6) such that E = graph U and define
m(8, E) =
I - itr(U) I +itr(U)
When (0, E) E A1(SM), set
m(8, E) = -1. Lemma 2.27. The map m : Ao(SM) U A1(SM) - S1 is continuous.
Proof. Suppose that (8, E) a Ao(SM) and that (8,,, E Ao(SM) is a sequence converging to (0, E). We can write E = graph U,,, where U : H(8,,) -> V(88) are symmetric linear maps. We claim that there exists a constant C > 0 such that for all n the number of eigenvalues of U (counted with multiplicities) larger in absolute value than C is < 1. If this property is false, there exist integers n, < n2 < ... , vectors x j, yj, and two real sequences {A} and (z j } such that
(xj,Y1) = 0,
Unjxj = ).jxj,
Ixjl = IYjI = 1;
U.jyj = Ajyj.
for all j. If S j is the subspace spanned by (x1, A j x j) and (y3, ttj y j ), the property IA, I - oo implies that the sequence (Sj ) converges to a two dimensional
subspace of V(8) and since Sj C En,, this two dimensional subspace is contained in E. This contradicts the property (0, E) E Ao(SM) and proves the
2.4 The Maslov index
41
claim. Moreover the spectral radius of Un goes to 0o when n -+ oo because otherwise, since the maps U,, are symmetric, we would have a sequence of integers n i < n2 < < oo and then E would be also a such that sups I
graph of a symmetric linear map (obtained as the limit of the subsequence U,,,), thus implying (8, E) E Ak(SM). Then every U,, has exactly one eigenvalue ,ln with oo and this implies C and IA,, -+ oo. Hence limn-.ooIn(On. En) = -1.
Exercise 2.28. Show that m cannot be continuously extended to all the bundle A(SM) by showing that if (8, E) E Uk2:2 Ak(SM), then given any real number ). there exists, arbitrarily near E, spaces E' E Ak(SM) such that E' is the graph of a symmetric linear map with trace X.
If y : S' -+ Ak(SM) U A i (SM) is a continuous map, we define the integer lc(y) as the degree of the map moy : S' --> S'. Now given a continuous map y : S' - A(SM) we define s(y) as µ(P) where P : S' -- Ak(SM) U A l (SM) is a curve homotopic to y.
Lemma 2.29. The integer µ(y) is well-defined. Proof Observe that since Uk>2 Ak (SM) is a union of submanifolds of codimension > 3, there always exists a curve P : S' -+ Ak(SM) U Ak(SM) Co-close to y and therefore homotopic to it. The integer µ(y) does not depend on the choice
of P since given another curve P : S' -+ Ak(SM) U A i (SM) homotopic to y the curves P and P are homotopic since, again, Uk>2 Ak(SM) is a union of submanifolds of codimension > 3 and the homotopy between P and P can be taken avoiding Uk>_2 Ak(SM).
Definition 2.30. The integer µ(y) is called the Maslov index of the continuous closed curve y : S' -+ A(SM). It defines a cohomology class in H' (A(SM), Z) called the Maslov class of the Grassmannian bundle of Lagrangian subspaces with respect to the vertical subbundle.
2.4.1
The Maslov class of a pair (X, E)
Let X C SM be any closed connected set and let E be a continuous Lagrangian subbundle defined on X, that is, E is a continuous section E : X -+ A(X) of the bundle A(X) -> X. Using the Maslov index defined above, we can attach to every pair (X, E) a cohomology class in the singular cohomology group H'(X, Z) as follows. Given a continuous closed curve a : S' -+ X, we define the Maslov index of a, µ(a), as the Maslov index of the continuous closed curve Ea : S' -).
A(X) C A(SM). The integer µ(a) defines a cohomology class in H'(X,Z) which we shall call the Maslov class of the pair (X, E). There are two main sources of interesting pairs (X. E) that we now describe.
42
2. The Geodesic Flow Acting on Lagrangian Subspaces
2.4.2
Hyperbolic sets
Definition 2.31. A closed subset X C SM is said to be invariant under the geodesic flow if ¢t(X) C X for all t e R. Definition 2.32. A closed invariant set X is said to be hyperbolic if there exist C > 0 and 0 < x < I such that for all 0 E X, there is a splitting
ToSM = E9 (0) ®Es(0) ®E"(0) such that G(0) E E9 (0), dim E9 (0) = 1;
ES "(4,(0)) for all t E R; IIdoo,IE'(o)II < C>`', NO-,IE-(e)II <_ Cpl' for all t > 0.
Definition 2.33. When SM is a hyperbolic set, we say that the geodesic flow is Anosov.
An important result of D.V. Anosov says that when M is a manifold with negative sectional curvature the geodesic flow is Anosov (for a proof, see for example [HK]).
Exercise 2.34. Let M be a closed surface with constant negative curvature. Using the Jacobi or the Riccati equation construct explicitly the stable and unstable bundles ES and E°.
Exercise 2.35. Consider the quadratic form Q : TSM -). R given by K9($)). Show that if M has negative sectional curvature, the Lie derivative
d dt t=0
Q0, (e) (de0t (4 )).
is positive for any 0 0
E S(9). The existence of a continuous quadratic form like Q that increases along the flow implies that the flow is Anosov [L).
Lemma 2.36. Let X C SM be a hyperbolic set. Then, ES(9) ® E°(9) = S(9). The subbundles 0 -+ ES"'(6) are continuous and Lagrangian.
Proof It is well known from the hyperbolic theory of Dynamical Systems (see [HK]) that the subbundles 0 -+ 0"(0) are continuous. Let us show that ES(B) C S(6); the proof for E" is analogous. Since 0, preserves the contact form of given q E V(0), we have ae(q) = a0,(e)(do0t(q))
2.4 The Maslov index
43
But from the definition of the hyperbolic set Ilde0t(q)II -, 0 as t -* +oo, therefore ao(q) = 0 (recall that M is compact) showing that E'(9) C S(9); therefore since E'(9) ® E"(9) and S(O) have the same dimensions, we have E''(9) (D E"(9) = S(9). To show that 0 -> are Lagrangian we proceed in a similar way. Take q and C in E'(9). Since ¢, preserves the symplectic form f2, we have Q9 (q, ) = go,(o)(de0t(q), doOt(C)), and from the definition of the hyperbolic set I Ideot(q)II -+ 0 and I1de0t(C)II --> 0
as t - +oo, and thus
2e(q,C)=0. Hence Es(9) and E"(9) are isotropic subspaces such that Es(9) ® E'(9) = S(9) and therefore they are Lagrangian.
On account of the last lemma, given a hyperbolic set X, we can attach to each pair (X, E') and (X, E") a Maslov class in H1(X, Z).
Exercise 2.37. Are the Maslov classes defined by (X, E') and (X, E") different?
2.4.3
Lagrangian submanifolds
Suppose that P C TM is a Lagrangian submanifold.
Lemma 2.38. If P is in fact contained in SM, then it is invariant under the geodesic flow.
Proof. Recall from part 2 of Exercise 1.28 that if
E TOSM, then
G(9)) = 0. Take 0 E P. If P C SM, then TOP is a subspace of To SM. Since P is a Lagrangian submanifold, the symplectic form 92O restricted to the sum of TOP and
the one dimensional subspace spanned by G(9) must vanish. This implies that G(9) E TOP and therefore P must be invariant under 01. Suppose now that P C SM is a Lagrangian submanifold. The intersection of TOP with S(9) is a Lagrangian subspace of (S(9), S2e1s(o)). Therefore to P we can attach a Maslov class in HI (P, Z).
Exercise 2.39. Show that when M is a surface, any surface P C SM invariant under the geodesic flow is Lagrangian.
The last exercise gives many examples of Lagrangian submanifolds contained in SM. If we take a surface of revolution, the level sets of the Clairaut integral (cf.
[doC]) are Lagrangian submanifolds in SM. KAM tori are the most important source of examples of Lagrangian submanifolds in SM.
44
2. The Geodesic Flow Acting on Lagrangian Subspaces
2.5
The geodesic flow acting at the level of Lagrangian subspaces
Since the differential of the geodesic flow 0, : SM --> SM takes Lagrangian subspaces to Lagrangian subspaces, it lifts naturally to a flow 0, : A(SM) -> A(SM) by setting (8. E) _ (4r(e), d,90, (E)).
We shall denote by G* the vector field associated to 0,*. In this section we shall
describe a remarkable property of the flow 0; with respect to the Maslov cycle Av c A(SM). Recall that the Maslov cycle is the union of AI(SM) with submanifolds of codimension > 3. It is not a smooth submanifold since it has singularities given by Uk>2 Ak(SM) but it is what is called a stratified submanifold. When M is a surface the Maslov cycle is an honest closed submanifold of codimension one. The remarkable property that we mentioned in the introduction is that 0,* is always transverse to the Maslov cycle. In order to simplify our exposition and to avoid the complications arising from the singularities of the Maslov cycle, we shall prove this property in the much simpler case of dim M = 2. Hence from now on and until the end of this section, let us suppose that M is a surface.
In this case dim A(SM) = 4, dim Ay = 3 and Ay is nothing but the image of the section 0 f-s (8, V(0)). Also, the function m from Section 2.3 is defined in all A(SM) and can be written in the following two ways:
J m(B, E) =
if E n v(o) = {0} where E = {(w, uw) : w E H(6)};
l m(B,E)=-1, ifE=V(B), and
{
m(8, E) = u+ , if E n H(8) = {0} where E = {(uw, w) : w E V(8)); m(8, E) = 1, ifE = H(8).
These two ways of writing m show that m is a smooth function and the Maslov
cycle is given by m-I(-I). Proposition 2.40. Consider the differential of m at a point (8, V (8)) E A v, that is,
d(e.v(e))m : T(e.v(e))A(SM) -' T_1S'; then
d(e,v(e))m(G"(8, V(8))) = -2i. This implies that 0, is transverse to the Maslov cycle.
2.5 The geodesic flow acting at the level of Lagrangian subspaces
45
Proof. Note first that t i-+ (¢r(6), d0 f t(V(6))) is an integral curve of the vector field G' such that at t = 0 it passes through (6, V(6)). For tin a neighborhood of 0 we can write
d04,(V(6)) = ((u(t)w, w) : w E V(¢,(6))}, where u(t) is a function that satisfies (cf. Remark 2.17)
u-K(t)u2-1=0, where K(t) is the Gaussian curvature of the surface at ye(t). The function u(t) has initial condition u(0) = 0, therefore using the differential equation we deduce that u(0) = 1. Hence d(e.v(o))m(G`(6, V(6)))
=
d 1,=0
d
u-i
do4t(V (6))) 2iiu 0 _
7It=0 a+i - (u(0)+i)Z
-2i.
O
As we mentioned before, in higher dimensions the Maslov cycle is a stratified submanifold. In Figure 2.1 we attempt to draw the Maslov cycle with A2(SM) the singular point of the cone and a suitable orientation. The flow 0, is represented as the flow of vertical lines that crosses the Maslov cycle transversally to all the strata and always in the 'same direction'
Exercise 2.41. Show that
d(e.H(e))m(G`(6, H(6))) = 2i K(n(O)), where K is the Gaussian curvature of M.
2.5.1
The Maslov index of a pseudo-geodesic and recurrence
Definition 2.42. Let (X, E) be a pair, where X is a closed connected invariant set and E a continuous Lagrangian subbundle. We shall say that E is invariant if for
all 6 E X and t E R we have de,r(E(6)) = E(o1(6)). Definition 2.43. Let (X, E) be a pair with E invariant. A continuous closed curve a : St --> X is a pseudo-geodesic if for all s E Sl for which
E(a(s)) fl V(a(s)) o (0}, there exists e > 0 such that for t E (-e, e) we have
a(e"s) = 0r(a(s)) Obviously, a closed orbit of the geodesic flow contained in X is a pseudogeodesic.
46
2. The Geodesic Flow Acting on Lagrangian Subspaces
Figure 2.1:
The Maslov cycle and the flow ¢!
Lemma 2.44. If a : S' - X is a pseudo-geodesic then µ(a) > 0 and A(a) > 0 if there exists s E St such that
E((Y(s)) fl V(a(s)) # (0}.
Proof Let us denote by P the set of points s E St for which
E(a(s)) fl V(a(s)) 0 {0}. Since a is a pseudo-geodesic and E is invariant, E o a : S' --> A(SM) is differentiable in a neighborhood of any points E P. By Proposition 2.40 the derivative of m o E o a at any s E P is nonzero and preserves the canonical orientation of St. Hence the Maslov index µ(a), which is the degree of m o E o a, is always > 0 and it is equal to the cardinality of P if P is not empty. 0
2.5 The geodesic flow acting at the level of Lagrangian subspaces
47
Remark 2.45. We have proved the lemma only for the case of surfaces but its statement is valid in any dimensions (cf. [BP, Mal ]).
Definition 2.46. We shall say a point 9 E X is a nonwandering point of 01I x if given an open set U of 0, there exists T > I such that OT (U n X) intersects u n X nontrivially. We shall denote by 0 the set of all nonwandering points of Of Ix.
The set S2 is invariant under Of. Note that the definition also implies that if 0 E S2, then given an open set U of 9 there exists a sequence T - oo for which 4'T (U n X) intersects U n X nontrivially. Lemma 2.47. Take 9 E S2 and 01, 02 two points in the orbit of 0 with 02 = 0,01 for s > 0. Then, given neighborhoods Ut and U2 of 91 and 92 respectively there exist q E U2 n X and T > 0 such that OT (q) E U, n X.
Proof First note that since n is invariant, 92 E S2. Since O, (Ut) n U2 is an open set containing 02, there exists q E &,(Ui) n U2 n X and T' > s such that
OT'(q)E¢,(Ut)nU2nX.Let T=T'-s.Then OT (U) E 0_,(0S(U,) n U2nx) c U1 nx.
Lemma 2.48. If M has finite volume then the set of nonwandering points of 0, is all SM. Proof. Take any open set U in SM and suppose by contradiction that for all t > 1, 0,(U) does not intersect U. Then given two different real numbers tl and t2 with
1t2 - ttl > 1, Of, (U) and 0,2(U) do not intersect. But this is absurd since the geodesic flow preserves the Liouville measure which is finite when M has finite volume.
Lemma 2.49. Suppose that X e SM is a locally arcwise connected closed invariant set and E a continuous invariant Lagrangian subbundle defined on X. If there exists 9 E S2 C X such that E(9) n V(0) 96 (0}, then there exists a pseudo-geodesic a with g(a) > 0. Proof. Let 9 E n be a point such that E(0) n V(9) 54 (0). By the twist property of the vertical subbundle (Proposition 2.11) there exists e > 0 such that for all
t E (-2e, Zr) with t 0 0, we have E(0,(9)) n V (0,(9)) = (0). Let U1 be a neighborhood of 0_E(9) such that for all 0' E Ut n X, E(0') n V(9') = (0). Similarly, let U2 be a neighborhood of such that for all 0' E U2 n X, E(9') n V(01) = (0). Since 0 E S2, Lemma 2.47 says that there exist q E U2 n X and T > 0 such that OT(q) E U1 n X (cf. Figure 2.2). Connect now OT(n) and 0_E(0) by a path yt contained in Ut n X and connect 0, (0) and q by a path y2 contained in U2 n X. This is possible since we assumed that X is locally arcwise connected.
48
2. The Geodesic Flow Acting on Lagrangian Subspaces
Figure 2.2:
Pseudo-geodesic with positive Maslov index
The path a obtained by joining (see Figure 2.2) the arcs yl,
y2
and 01(71)I(o,rl is a pseudo-geodesic which has positive Maslov index by Lemma 2.44.
2.6
Continuous invariant Lagrangian subbundles in SM
In this section we shall give a proof, when M is a surface, of the following general result obtained by R. Mafld [Ma! ].
Theorem 2.50. Let M be a closed Riemannian manifold. If there exists a continuous invariant Lagrangian subbundle E defined on SM. then E(8) n V (B) = (0) for all 0 E SM and M does not have conjugate points. In particular, when the geodesic flow of M is Anosov, the bundles Es and E° are continuous invariant Lagrangian subbundles, and hence we obtain: Corollary 2.51. Let M be a closed manifold whose geodesic flow is Anosov. Then M does not have conjugate points.
2.6 Continuous invariant Lagrangian subbundles in SM
49
The corollary was first proved by W. Klingenberg using different techniques
[Kill. For the proof of Theorem 2.50 we shall need the following lemma.
Lemma 2.52. There exists no smooth closed codimension one submanifold in SM transverse to the geodesic flow Proof. Suppose that E is a smooth closed codimension one submanifold in SM transverse to the geodesic flow. By part 5 of Exercise 1.28, given 9 E E, the restriction of S4 to TOE is nondegenerate. If i : E -+ SM is the inclusion map then (E, i*S2) is a symplectic manifold. Therefore i*S2"-t is a volume form in E and thus
i*S"-t # 0. E
But up to a sign, n"'t = (da)"-t = d(a A
(dot)"-2) and since 8E
= 0 Stokes
theorem implies that
i*S"
=0.
JE
This contradiction show that there is no smooth closed codimension one submanifold in SM transverse to the geodesic flow.
Exercise 2.53. Show that the symplectic form of a closed symplectic manifold cannot be exact. Proof of Theorem 2.50. We begin by noticing that even though the section E : SM -. A(SM) is only continuous, the invariance of E under the geodesic flow implies that E is differentiable along the orbits of the geodesic flow and d
dt
E(4,(0)) =
I
t=O
d I
(Or(9),dechr(E(9))) = G*(9, E(9))
dt t=O
Approximate now in the C°-topology the map E : SM -* A(SM) by a smooth map E : SM -+ A(SM) such that 1. the maps 9 r->
y,It-OE(4,(9)), 9 r-> G*(9, E(9)) are CO-close.
Suppose that there exists a point 9 E SM such that E(9) f1 V (O) 34 (0). This is equivalent to saying that E-t (Ay) # 0. By Lemmas 2.48 and 2.49 there exists a pseudo-geodesic a with µ(a) > 0. If we take k sufficiently close to E so that Ea and La are homot'pic, we deduce that the curve of Lagrangian subspaces La also has positive Maslov index and therefore E- I (AV) 76 0. Next observe that Proposition 2.40 and property I above imply that E-I (A v) is a closed nonempty codimension one submanifold transverse to the geodesic flow. This contradicts Lemma 2.52 and therefore for all 9 E SM, E(0) n V (0) = (0). Finally to deduce that M has no conjugate points, note that for every geodesic there exists a Jacobi field orthogonal to it that does not vanish; simply take JO(t) = d9(7r4t)(q) for 0 0 q E E(9). Now apply the following exercise.
2. The Geodesic Flow Acting on Lagrangian Subspaces
50
Exercise 2.54. Let yt and Y2 be two nontrivial solutions of y+K(t)y = 0. Show that yi (t)Y2(t) - ,yi (t)Y2(t) is constant. Use this fact to prove that if there exists to E (0, oo) such that yt (0) = yt (to) = 0, then Y2 must vanish in [0, to].
2.7
Birkhoff's second theorem for geodesic flows
In this section we shall give a proof of the following result obtained by L. Polterovich in [Poll] (see also [BP]) which can be seen as the higher dimensional autonomous version of a result of G. Birkhoff that asserts that an invariant circle of a twist map in the cylinder T`S' which is homologous to the zero section must be a graph. Theorem 2.55. Consider a Riemannian metric on the n-torus T". Suppose that P is a Lagrangian torus contained in ST" which is homologous to the zero section of TT". Suppose that the set of nonwandering points of 4r I p coincides with all
P. Then P is a graph, that is, the restriction to P of the natural projection r ST" -). T" is a diffeomorphism.
The proof of Theorem 2.55 is partially based on the following result of C. Viterbo [V] (see also [Po12]) whose proof is beyond the scope of the present notes.
Theorem 2.56. The Maslov class of a Lagrangian torus in TT" that is homologous to the zero section must vanish. We shall need the following lemma.
Lemma 2.57. Let P be a torus in TT" that is homologous to the zero section. Suppose that the restriction to P of the natural projection r : TT" - T" is a local diffeomorphism. Then the restriction to P of the natural projection is a difeomorphism. Proof Since P is homologous to the zero section, the map it I p : P -+ 'II" must have degree one and since tr I p is a local diffeomorphism it must be a global diffeomorphism. Proof of Theorem 2.55. Observe first that the map n I p : P -+ T" is a local diffeomorphism, if for all 9 E P, we have that To P fl v (9) = (0). For 9 E P, let us denote by E(9) the intersection of TOP with S(9). Since P is contained in ST", TOP fl V (9) = (0) iff E(9) n V (9) = (0). Therefore, on account of Lemma 2.57, to complete the proof of the theorem we need to show that for all 9 E P we have that E(9) n V(9) = (0). Suppose that this is not case, that is, there exists some 9 E P for which E(9)n V(O) 96 (0). Since we are assuming that every point of Ot I p is nonwandering, Lemma 2.49 implies that there exists a pseudo-geodesic a in P with positive Maslov index. Therefore the Maslov class of P is not zero. On the other hand, by Theorem 2.56, the Maslov class of the Lagrangian torus P vanishes, thus reaching a contradiction.
3 Geodesic Arcs, Counting Functions and Topological Entropy
In this chapter we introduce the counting functions and we relate them to the topological entropy h10 (g) of the geodesic flow of g. In all that follows, unless otherwise stated, M will be a compact manifold en-
dowed with a Riemannian metric g and N C M will be a closed submanifold. As usual, we shall denote by exp-'- : T N-L -+ M the normal exponential map, which is given by the restriction of exp : TM -+ M to TN-L. For each T > 0, we introduce the following sets
DNT := {(x, v) E TNl : Jul < T};
SNT := {(x, v) E TNJ- : Jul = T);
SN-L := SNj .
3.1
The counting functions
For each real positive number T > 0 and y E M, we define
nT(N, y) := #
((expl)-i
(y) fl DNT) .
In other words, nT(N, y) counts the number of geodesic arcs joining N to y, initially orthogonal to N and with length < T. An important particular case occurs
52
3. Geodesic Arcs, Counting Functions and Topological Entropy
Figure 3.1:
Geodesic arc from x to y
when N reduces to a point, let us say x. In this case we shall denote by nT(x, y) the corresponding counting function that counts the number of geodesic arcs join-
ing x toy with length < T (cf. Figure 3.1). The main question we would like to address is the following: Question. How does nT(N, y) grow with T?
Let M be a differentiable manifold, possibly noncompact. Let Co(M) be the space of continuous real-valued functions with compact support endowed with the supremum norm: Ilf11
sup If(x)I. xEM
A Radon measure is a linear functional u : Co(M) -+ R such that for any compact subset K C M, there exists a positive number aK such that
lu(f)I
0 for all f. When M is compact, a positive Radon measure is nothing but a nonnegative bounded linear functional on the space of continuous functions; equivalently, a finite Borel measure by the Riesz representation theorem.
3.1 The counting functions
53
Let us recall that every Riemannian manifold carries a positive Radon measure called the Riemannian measure (cf. [Sal). The Riemannian measure has the additional property of being smooth, that is, in any smooth local chart it is given by integrating a smooth positive function with respect to Lebesgue measure in Euclidean space. When M is orientable, the Riemannian measure is the measure induced by the Riemannian volume element. Also recall that for a linear map L : E -+ F between finite dimensional Hilbert spaces of the same dimension, the determinant det(L) of L is well defined up to a sign. Choose (vi, ... , orthonormal basis of E. Consider the matrix A whose (i, j)-entry is given by (L(vi), L(vj)). Then I det(L)I = det(A). We shall endow any submanifold of TM with the Riemannian metric given by the restriction of the Sasaki matric to the submanifold. The next proposition is proved in [BBI when N is a point.
Proposition 3.1. For T fired, the map y - nT(N, y) is measurable and locally constant on an open full measure subset of M. Moreover
f
nT (N, y) dY = fDNT I det de expl I d9,
M
where d y is the Riemannian measure on M and dO is the Riemannian measure of DNT1 .
The proposition is an immediate consequence of the following more general statement. Let M be a Riemannian manifold and let X be a compact Riemannian manifold with boundary and with the same dimension as M. Let f : X -> M be a smooth map.
Lemma 3.2 (Area formula). The map y r-> n(y) :_ #f
(y) is measurable
and locally constant on an open fill measure subset of M. Moreover,
n(y)dy= f IdetdgfIdq. JM
X
Proof Let C be the set of critical points of f IX-ax. By Sard's theorem, f (C) has measure zero. Observe now that the set C U aX is closed and therefore compact, hence W := f (C U aX) = f (OX) U f (C) is a closed set of measure zero. Let Ua Ua be the decomposition in connected components of the open set M - W. We shall show that nI U. is constant. It suffices to show that n is locally constant. Let y E M - W. First of all, note that n(y) < oe. Indeed, the set f -t (y) is discrete and X is compact. Let f - t (y) = ( X I ,--- xk ) C X - aX. Since y is a regular value of f, there exist open sets Vi, ... , V k with xi E V i for i = 1, ... , k and such that
flv;:Vi-f(Vi)
3. Geodesic Arcs, Counting Functions and Topological Entropy
54
is a diffeomorphism for all i = 1, ... , k. Let V= n1 f (V,). Let us consider the open set V' = V - f (X - U1 V1). For all y E V', n(y) has the same value, and therefore n is locally constant on the full measure open set M - W. From this, we also deduce that y r--s n(y) is measurable. Let n,r := n I U.. Clearly
f n(y)dy = f -
MW
M
n(y)dy =
n,,Vol(U0). a
Let Ya= f (U,,). Then
naVol(Ua)= f Idetdgfldq. a
Thus
J n(y)dy= ff M
Idetdgfldq ' (M- W)
Since def t dq f I c = 0, we have that
Idetdgfldq=
f-ax xIdetdgfldq= f-ax-c Idetdgfldq x
To finish the proof of the lemma we only need to show that the set
(X - ax -C)-(f-I(M-W)), has measure zero. For this observe that
(X -8X -C)-(f-t(M-W))= f-'(W)fl(X -aX -C). Since W has measure zero and f I x-ax-c is a local diffeomorphism, the desired result is a consequence of the following exercise taking U = X - aX - C.
Exercise 3.3. Let U be an open set of X - M. Suppose that flu is a local diffeomorphism and let W C M be a set of measure zero. Then f -t (W) fl U has measure zero.
Clearly, Proposition 3.1 follows from the lemma. Just take X = DNT and f = expl I DNT Remark 3.4. Lemma 3.2 is a particular case of a much more general version of the area formula that holds for Lipschitz maps, manifolds with different dimensions and Hausdorff measures [Fe].
Let us define a function AN : SNl x R -> R by AN(O, t) = Idet do(n^O1)IT9sNI
3.1 The counting functions
Remark 3.5. If
55
is an orthonormal basis of TBSN1, then
AN(6, t) =
det ((4 (t), $i
J
Proposition 3.6.
JT())) = J/T dt f NlAN(6,t)dB. Proof. By Proposition 3.1 we have
SI
rDNT
fu nT(N, y) dy = J
I det de exp1 I dB.
Using Fubini's theorem we obtain T
J
det de exp1 I dO = f dt
DNT
I det dB exp1 I dB.
fN,
0
The proposition follows now from the following exercise.
Exercise 3.7. Using the Gauss Lemma and the change of variables
(X, V) H (x,tv) that takes SN1 to SNP, show that
f , Idetdeexplldo=f N
SN l
f(8, t) = ir4t(0). Show that I det d(0,,)f I = AN(O. t).
(*)
Use (*) and the Area formula (applied to f) to give a direct proof of Proposition 3.6.
Let Y e SM be a submanifold. The volume of 0,(Y) can be computed as Vol(Or(Y)) = f Idet(dem,ITOY)IdO, Y
where dO is the Riemannian measure on Y.
3. Geodesic Arcs, Counting Functions and Topological Entropy
56
Corollary 3.9.
rT
fu IT(N, )')dy < J Vol(¢,SN1)dt. 0
Proof. To prove the corollary it suffices to apply the last proposition and to note that AN(O, t) = Idet do(n4t)ITBSNl1 < Idet 4e0tITOSN.L I,
0
since I det don IEI <_ ]for any subspace E C S(O).
Let us consider the function A : SM x IR -+ R given by A(6, t) _ I det do(n.0t)I v(a)I .
(3.1)
Corollary 3.10. T
nT(x, y)dxdy = JMxM
fo
f
dt
A(9, t)d9. SJM
Proof. It follows right away from Proposition 3.6 applied to N = (x) and
0
Fubini's theorem. Let us define
n(T) := J
IT(x, y) dxdy.
MxM
The growth of n(T) can be viewed as a measure of the complexity of the geometry of geodesics. It also has the property of being a significant global geometric invariant. In general, the growth is very hard to predict. On a compact symmetric space the Jacobi equation can be solved easily. The solutions have at most linear growth. Hence A(9, t) can be bounded uniformly in 9 by a polynomial of degree n - I (n = dim M). Then it follows from Corollary 3.10 that n(T) grows at most like T. If the symmetric space has rank one, then all the orthogonal Jacobi fields are bounded, and thus n(T) grows linearly.
For flat manifolds the Jacobi equation reads J = 0. Then A(9, t) = t"-t and hence n(T) grows like T". In the next subsection we shall determine the growth of n(T) for a certain class of homogeneous spaces, the so called naturally reductive.
3.1.1
Growth of n(T) for naturally reductive homogeneous spaces
Let M" = G/K be a homogeneous space and let n : G -* G/K be the canonical projection. Let g be the Lie algebra of G and t the Lie algebra of K. Recall that M
3.1 The counting functions
57
admits a left invariant metric if g admits an Ad(K)-invariant inner product. This always occurs if K is compact. If g admits an Ad(K)-invariant inner product, then G/K is reductive, i.e., there exists a complement p off in g: g = t ® p so that Ad(K) leaves p invariant. Suppose now that G is compact and let us consider a left invariant metric on M. Since G is compact, K is also compact and such a metric exists. Let p be an Ad(K)-invariant complement as before. The metric on T(KIG/K induces a metric on p denoted by ( , ). M is called naturally reductive (with respect to the complement p) if [X, .]p : p - p is skew-symmetric for all X E p. Naturally, reductive spaces include normal homogeneous spaces and hence symmetric spaces. An important property of naturally reductive spaces is that Expn(e) = 7r.exp Ip
(3.2)
where Exp denotes the Riemannian exponential map on M and exp is the Lie group exponential of G. Thus, geodesics in M are images of one-parameter subgroups of G [Z]. Proposition 3.11. Let M" = G/K be a naturally reductive homogeneous space with G compact. Then n(T)/T" is uniformly bounded above for T > 1. Proof Since M is homogeneous, it suffices to show that
r. nT(e, y)dy, does not grow faster than T". We know that the Jacobi field along the geodesic defined by the unit vector v E p with initial conditions J(0) = 0 and J(0) = W E p is given by
J(t) = dtv(EXPn(e))(tW). Using equation (3.2) we get
J(t) =
exp(t W).
Endow G with a bi-invariant metric and denote its norm on tangent spaces by I * Ia. Since G is compact the last equation implies
IJ(t)I
bounded uniformly by (at + b) I WIG for some constants a and b. Since norms in euclidean space are equivalent there exists c such that I WIG < c for any vector W such that (W, W) = 1. Hence from Proposition 3.6 we get T
JM
nT(e, y)dt < (Lc)"
dt fo
J Se
(at +b)"
d9.
0
58
3. Geodesic Arcs, Counting Functions and Topological Entropy
Next, we shall show that certain metrics over non-homogeneous spaces have n(T) also with polynomial growth. To be precise we prove:
Proposition 3.12. Let n : M" -+ Bk be a Riemannian submersion where M =
G/K is a naturally reductive homogeneous space with G compact. Then nB(T)/Tk is uniformly bounded above for T > 1. Proof Take a point x E B and a unit vector v E Tx B. Consider the geodesic yo defined by 0 = (x, v). Let J(t) be a normal Jacobi field along ye with J(0) = 0 and IJ(0)! = 1. Since ;r is a Riemanian submersion, we can lift yo to a horizontal geodesic y in M. Moreover we can lift J to a normal Jacobi field (not necessarily horizontal) J along y with j(0) = 0 and IJ(0)I = 1. Hence IJ(t)1
Ii(t)I
Now recall that in the proof of the previous proposition we proved that I J(t)I is bounded uniformly by a linear polynomial p(t). Thus
r nB(T)
fdt
P(t)k-t.
J sB
Example 3.13. Let G be a compact Lie group endowed with a bi-invariant metric. Recall that G x G acts on Gas follows. Given gi, 82 in G we have (gi, 82).x = g i xg2 1. Hence if H C G x G is a closed subgroup that acts freely on G, we
can consider the Riemannian submersion n : G -> B = G/H. Now the last proposition implies that for all the spaces thus constructed, nB(T) does not grow faster than Tdim B Concrete examples of this situation are the exotic 7-sphere
constructed by Gromoll and Meyer [GM] E7 = Sp(2)/0, and examples of Eschcnburg [Es]. These examples are strongly inhomogeneous, i.e., they do not have the homotopy type of any homogeneous space.
Open Problem. Compute n(T) for the n-dimensional ellipsoid.
3.2
Entropies and Yomdin's theorem
3.2.1
Topological entropy
Let (X, d) be a compact metric space and let ct : X --> X be a continuous flow. For each T > 0 we define a new distance function
dr(x, y) := max d(Ot(x), Ot(y)) o<-rT
3.2 Entropies and Yomdin's theorem
59
Since X is compact, we can consider the minimal number of balls of radius e > 0
in the metric dT that are necessary to cover X. Let us denote this number by N(e, T). We define 1
h(t, e) := lim sup - log N(s, T). T_. T Observe now that the function e H h(¢, e) is monotone decreasing and therefore the following limit exists:
hrop(0) := 1im h(0, e). e-+0
The number h,op(o) thus defined is called the topological entropy of the flow O,. Intuitively, this number measures the orbit complexity of the flow. The positivity of h,op(o) indicates complexity or "chaos" of some kind in the dynamics of Ot. The topological entropy h,,,p(0) may also be defined as h,0p(r¢1) using the entropy of the time one-map, or it may be defined in either of the following ways. All the definitions give the same number h,op(0) which is independent of the choice of metric [HK, W]. A set Y C X is called a (T, e)-separated set if given different points y, y' E Y,
we have dT(y, y') > e. Let S(T, e) denote the maximal cardinality of a (T, e)separated set. Then
hiop(0)
limo lim sup I log S(T, e).
A set Z C X is called a (T, e)-spanning set if, for all x E X, there exists z E Z such that dT(x, z) 5 e. Let M(T, e) denote the minimal cardinality of a (T, e)-spanning set. Then
hrop(0) =
1
Io
lim sup T log M(T, E). T-.oo
Remark 3.14. Given a compact subset K C X (not necessarily invariant) we can define the topological entropy of the flow with respect to the set K, hiop(O, K), simply by considering separated (spanning) sets of K.
Let u,(y) denote the w-limit set of the orbit y. If y C m(y), then y is said to be recurrent. If y is not a critical orbit (i.e., a fixed point or a closed orbit) then it is said to be nontrivially recurrent. The following proposition gives an idea of the dynamical significance of the topological entropy (for proofs see [HK, W]). Proposition 3.15. The topological entropy verifies the following properties: 1. For any two closed subsets Y1, Y2 in X, htop(0, Yt U Y2) = max h,op(0. Yi). i=1,2
60
3. Geodesic Arcs, Counting Functions and Topological Entropy
2. If Yt C Y2 then htp(O. Yt) < htop(q5. Y2).
3. Let 0,' : X; -+ Xi for i = 1, 2 be two flows and let n : XI -+ X2 be a continuous m a p commuting with Oi i.e. 4 . r = 7r.01. If tr is onto, then htop(0t) > htop(02) and if jr is finite-to-one, then htop(Ot) < hrop(02)
4. Let ¢; : Xi -+ Xi for i = 1, 2 be two flows and let *t :_ 01 x 02 be the product flow on X1 x X2. Then h,op(*) = h,op(.0i) + htop(.O2). 5. Suppose that X is separable. 1f ¢t only has trivial recurrence, then h,op(#) _ 0.
6. Given c E R, let cot be the flow given by cat := Oct. Then h,op(co) _ I cl hmp(O)
Next we shall state a useful result of R. Bowen that we will need later.
Theorem 3.16 (Theorem 17 in [Bo]). Let (X, d) and (Y, e) be compact metric spaces and Of : X -+ X, VV, : Y -+ Y continuous flows. Let it : X -+ Y be a surjective continuous map so that s.¢, = *t.n. Then
hmpM < hmpW + sup hrop(O. ,r(y)) Corollary 3.17 (Corollary 18 in [Bo]). Let (X, d) and (Y, e) be compact metric
spaces and Ot : X - X a flow. Suppose rr : X --+ Y is a continuous map such
that no, = it. Then hrop(m) = sup htop(4, tr-I(y)) yey
3.2.2
Yomdin's theorem
Let 0, : X -+ X be a CI flow on a closed manifold X. Fix a Riemannian metric on X. Given a compact submanifold Y C X of dimension k, we can consider its k-dimensional volume. The Riemannian metric on X induces a Riemannian metric on Y and with respect to this metric we measure the volume of Y and we denote it by Vol(Y). We define any := lim sup
too
1
I
t
log Vol(o, (Y)).
Exercise 3.18. Show that ay does not depend on the choice of Riemannian metric in X.
Yomdin's theorem (cf. [Y]) states that for any compact submanifold Y C X, we have ay
htop.
(3.3)
3.2 Entropies and Yomdin's theorem
3.2.3
61
Entropy of an invariant measure
Let (X, d) be a compact metric space and let 4), : X - X be a continuous flow. We shall consider the Borel a-algebra of the metric space (X, d) and µ a Borel probability measure. The measure A is said to be invariant under 0, if for
any Borel set B, µ(4),(B)) = µ(B) for all t E R. Let M(4)) be the set of all q5,-invariant Borel probability measures. To each µ E M(4)) we can associate a nonnegative real number called the entropy hu of the measure A. In this subsection we briefly recall its definition and its relation with the topological entropy. For a complete account of these notions, we refer to [HK, W].
Definition 3.19. A partition is a disjoint collection of Borel sets whose union is X. A finite partition will be denoted by
l: _ (Al, .. , AdGiven two finite partitions
t={AI,...,Ak},
q={B1. .Bn},
the join i; v q is the partition given by
tvq:(A;nBj : 1
t
.... Ad is given by (the expression 0 log 0 is considered to be 0): k
Hµ()
1).(A;)logµ(A;).
Let f := ¢1 be the time one-map of the flow. Given a finite partition t _ JAI,..., Ak}, set
f-nt = {f-'At, ..., f-"Ak}. Let it E M(4)). Definition 3.20. The entropy of m, with respect to t is given by h (4),
)
:
1
n-1
i
',-too n
Finally, the entropy of the flow of with respect to the measure µ is given by
h,. (0) = sup h (¢, t), where the supremum is taken over all finite partitions $.
3. Geodesic Arcs, Counting Functions and Topological Entropy
62
The variational principle [HK, W] relates the entropies of invariant measures with the topological entropy: hrop(4) =
sup µEM(4')
(3.4)
hµ(4).
Definition 3.21. A measure it is said to be of maximal entropy if hiop(O) = hµ(O)
Measures of maximal entropy do not always exist. However if X is a manifold and ', is COO, an important result of S. Newhouse [New] ensures that measures of maximal entropy always exist.
A measure µ E M(¢) is said to be ergodic if any Borel invariant set has measure zero or one. The ergodic measures are the extreme points of the compact
convex set M(0) (cf. [W]). In the variational principle, it suffices to take the supremum of hµ over all ergodic measures to obtain the topological entropy. Also,
when X is a manifold and Ot is C°O, there exist ergodic measures of maximal entropy. Finally, we mention that when it is ergodic, there is an alternative way of char-
acterizing hµ(O) due to A. Katok that uses the metric dT in analogy with the definition of topological entropy (cf. Theorem 1.1 in [K I) and Proposition 1.6 in
[K2]). Let 6 be any number such that 0 < 8 < 1. For each e > 0 let N(e, T, S) denote the minimal number of a-balls in the metric dT needed to cover a set of measure at least 8. Katok's formula states that 1
h,,(0) = lim imof T logN(s, T,3) = u olimsup
1
I
T-+oo T
3.2.4
logN(e,T.3).
Lyapunov exponents and entropy
Let X be a closed manifold and ¢, : X -+ X a flow of class Ct. In this subsection we relate the Lyapunov exponents of the flow with the entropies hu(o). Our reference for the results quoted here is [HK].
Let f := 01 be the time one-map of the flow. We say that x E X is a regular point if there are numbers At (x) > . . . > Aj(x) and a decomposition
TxX = El
(x),
such that 1
lim log I dx f" (v) I = zj (x), n_.too n
f o r v E Ej . (x), v 0 0 and 1 < j < 1 . The numbers I (x) are called the Lyapunov exponents at x. Oseledec's multiplicative ergodic theorem states that for every
3.2 Entropies and Yomdin's theorem
63
µ E X1(4), the set A of regular points is a Borel set with µ(A) = 1. Moreover the maps x i-+ Aj (x) and x H E j (x) are measurable and invariant. If µ is ergodic, then the Als as well as the dimension of the corresponding E '.s are constant 2-a.e. There is a basic result, known as Ruelle inequality that relates the sum of the positive Lyapunov exponents with the entropy h (0). Let
A+(x) := E Aj (x). (j Aj(x)>O)
Then for any it E M(O), Ruelle inequality asserts that
h, (¢) < 1 A+dµ.
(3.5)
x
The importance of this inequality is based on the immediate corollary (using the variational principle (3.4)) that if hrop(o) > 0, then there exists a measure with some of its exponents positive.
3.2.5
Examples of geodesic flows with positive entropy
Given a closed Riemannian manifold (M", g) we can consider h,0p(g), the topological entropy of the geodesic flow of g and hµ, (g), the entropy of the geodesic flow of g with respect to the Liouville measure µt. These numbers are Riemannian invariants of dynamical origin. Space forms of curvature 0 and I have h,op = 0, as the next exercise shows. It will follow from Exercise 3.36 and Theorem 4.23 that compact hyperbolic space forms have hrop = n - 1.
Exercise 3.22. Let (M", g) be a space form with curvaure 0 or 1. Use the Jacobi equation to show that all the Lyapunov exponents are identically zero. Deduce from Ruelle inequality (3.5) and the variational principle (3.4) that h,op(g) = 0. Anosov [An] proved that the geodesic flow of any manifold of negative curvature is ergodic with respect to the Liouville measure and h,., (g) > 0. In [BG, Do] V. Donnay, K. Burns and M. Gerber constructed real analytic metrics on S2 whose geodesic flows are ergodic with respect to Liouville measure and hµ, > 0 (in fact they are Bernoulli) showing that the simple topology of the sphere is not an obstruction for having geodesic flows with complicated dynamics. However all these examples require some negative curvature. In [KW], G. Knieper and H. Weiss showed the existence of real analytic convex (i.e., positively curved) metrics on the two-sphere whose geodesic flow has positive topological entropy. Their examples are obtained from smooth small local perturbations of an ellipsoid with distinct axes. In [P2] we constructed explicit real analytic metrics with positive curvature and positive topological entropy. More
precisely, consider the ellipsoid E, a, + a2 +
= I with a3 > a2 > a, > 0
and let gE denote the canonical metric of E induced by R3. Let r, , r2, r3 be given
64
3. Geodesic Arcs, Counting Functions and Topological Entropy
real numbers. Then for e 96 0 sufficiently small the geodesic flow of (E, g) has positive topological entropy where
I -e(rlx+r2Y+r3z)
8=
a
:
2
gE
(r2 0 0)
ala2a3( ai + zi + _'f) aZ a3
One expects that "generically" the geodesic flow of a Riemannian manifold has positive topological entropy. In Chapter 5 we shall discuss topological conditions that ensure positive topological entropy for any metric on the manifold. We conclude this subsection with the following outstanding open problem:
Open Problem. Does there exist a convex metric g on S2 with httt (g) > 0?
3.3
Geodesic arcs and topological entropy
Given a closed Riemannian manifold (Ma, g) we consider htop(g), the topological entropy of the geodesic flow of g. Our aim will be to relate htop(g) with the counting functions nT(N, y). We begin with the following basic property of htop(g)
Lemma 3.23. Let c be a positive real number. Then
htop(cg) =
htop(g) Y`
Proof. Observe that the geodesic flow of cg restricted to the unit sphere bundle of (M, cg) coincides with the geodesic flow of g restricted to the set of points (x, v) E TM such that lc-Ivl = 1. Now apply property (6) of Propo-
0
sition 3.15. The following lemma will be quite useful in what follows.
Lemma 3.24. Let f : [0, oo) -+ (0, oo) be a continuous function. Then 1.
lim oi
T-
f T log f (T) < im inf T log 7*-oo
f (t) dt; fo
2.
lim sup T-.oo
I
7
r
log f f (t) dt < max I 0, lim sup 0
1
Too T
log f (T)
3.3 Geodesic arcs and topological entropy
65
3. If there exists a constant C > 0 such that for all t > I and all s e 1/2, 1/21 we have f (t + s) > C f (t), then 1
lim sup
f (t) dt >lim sup
log
T- c'o T
I log f (T).
T-+oo T
fo
Exercise 3.25. Prove the lemma. Remark 3.26. The third part of Lemma 3.24 applies, for example, to the following case. Suppose Y c_ SM is a submanifold and that M is compact. The volume of Of (Y) can be computed as
f(t) := Vol(01(Y)) =
f
I det(deOfIT,y)I d9,
Y
where d9 is the Riemannian measure on Y. Clearly
f(s+t) =
f
r
I det(dqoOsIdem,(Tdnll det(doOtITBY)I de.
Since M is compact there exist a positive constant C such that for all 6 E SM, all nontrivial subspaces L of ToSM and all s E [-1/2, 1/2] we have I det(doO, I L) I > C,
hence f (t + s) > C f (t). Another example to which we can apply the lemma is given by the expansion that we shall consider in the next chapter in the context of Mati6's formula. Theorem 3.27. If the Riemannian metric g is Coo lim sup
1 log f T
S htop(g)
AM
T
fm
nT(N, y)dy < f Vol(4,SNl)dt, 0
hence
limsup 1 log f T-.oo T
y)dy
AM
< max {0,
I
Iim SUP
T-oo T
log VoI(OTSNi) = aSNi J
.
66
3. Geodesic Arcs, Counting Functions and Topological Entropy
By Yomdin's theorem aSN-L < htop(g) which combined with the last inequality yields
1 log
lim sup
T-°o T
JM
nT(N. y) dy < h,op(g).
D
Taking N to be a single point x in Theorem 3.27 gives the following corollary:
Corollary 3.28. If the Riemannian metric g is C°O, then for all x E M we have lim sup
I log J
T-oo T
nT(x,Y)dY :5 hrop(g)
M
Let g x g be the product metric in M x M and let A C M x M be the diagonal.
Lemma 3.29. For any T > 0 we have
nT(x, Y) = nT/ f(O, (x, Y))
Proof. Let y : [0, L] -+ M be a geodesic arc connecting x and y and suppose [L/2, L] -+ M x M be the geodesic arc in M x M that y has unit speed. Let given by
Y(t) _ (y(L - t), y(t)) Clearly y(L/2) E A, y(L/2) E Ty(L/2)A' and y(L) = (x, y). One easily checks that the speed of y is f and therefore y is a geodesic arc leaving orthogonally from A and terminating at (x, y) with length L/i. Now suppose that y : [0, L] -+ M x M is a geodesic arc leaving orthogonally from A and terminating at (x, y) and suppose that y has unit speed. Then it follows easily that y has the form (y(t), y(-t)) and thus YIl_L.LI is a geodesic arc connecting x and y with length 12-L. Hence the set of geodesic arcs between x and y with length < T is in one to one correspondence with the set of geodesic arcs leaving orthogonally from A
and terminating at (x, y) with length < T/.. Lemma 3.30.
hrop(g x g) = f hrop(g)
0
3.3 Geodesic arcs and topological entropy
67
Proof. Let f : S(M x M) -+ SI be the function given by f(x1, vi,x2, v2) = (IvlIx,,Iv21x2)
Since the geodesics in M x M are products of geodesics in M, the function f is constant along the orbits of the geodesic flow of M x M. It follows from Corollary 3.17 that
hrop(g x g) = sup
hrop(f(c))
cES1
If we write c = (1, m), then applying properties (4) and (6) from Proposition 3.15 we obtain (check it!):
htop(f -1(c)) = (1 + m)hrop(g)
Thus hrop(g x g) = Jhrop(g) Exercise 3.31. Generalize the lemma by showing that if (MI, gl) and (M2, 92) are two compact Riemannian manifolds and if we endow MI x M2 with the product metric gt X $2. Then hrop(g1 x 92) =
[htop(g1)]2 + [hrop(g2)l2
Theorem 3.32. If the Riemannian metric g is C°O, we have 1
lim sup
T-*oo T
log
JMxM
nT(x, y) dxdy < htop(g).
Proof Using Lemma 3.30 and applying Theorem 3.27 to M x M with the product metric and the diagonal 0, we obtain lim sup
1 log J
T-.oo T
MxM
nT(0, (x, y))dxdy <_ htop(g x g) = f htop(g).
Using Lemma 3.29 we get lim sup
I log J
T,00 T
nT(L, (x, y)) dxdy
MxM 1
_ vlimsup - log T-.oo T
nT(x, y)dxdy, MxM
which gives the desired inequality. The following lemma is an application of the Borel-Cantelli lemma.
3. Geodesic Arcs, Counting Functions and Topological Entropy
68
Lemma 3.33. Let (X, A, tz) be a probability space and fn : X -* (0, +oo) a sequence of integrable functions. Then 1
Tim sup
1
n-+oo n
fn dµ,
log fn (x) < Tim SUP! log
n-oo n
X
for A-a.e. X E X.
Proof. Set or = lim sup 1
n-.oo n
log f fn dµ. x
Define S(n, e) = {x : fn(x) 2: exp(a +E)n}. Then
alim sup
1
n-00 n
Tim sup
log
fn dµ > Tim sup 1 log fn dµ n-.00 n fs (n.E)
JX
I log{µ(S(n, c)) exp(a + E)n)
n-oo n
= a + E + lim sup 1 log µ(S(n, c)).
n-.oo n
Hence
lim sup
1
log /A (S(n, e)) < -E,
n-.oo n
and then
A(S(n, E)) < +oo.
By the Borel-Cantelli Lemma, for a.e. x, there exists m(x) such that x it S(n, E)
for all n > m(x). This means that f" (x) < exp(a + E)n for all n > m(x) and then 1
Tim sup -
n-.oo n
log fn (x) < a + e.
Since this holds for every e > 0, the lemma is proved. If we combine the last lemma and Corollary 3.28, we obtain:
Corollary 3.34. If g is CIO, for every x E M, lim sup
1
T-.+oo T
for a.e.yEM.
lognT(x, y) < hrop(g),
0
3.4 Manning's inequality
3.4
69
Manning's inequality
Let B(x, r) be the ball with center x and radius r in the universal covering M endowed with the induced metric. Set V (x, r) := Vol(B(x, r)). Proposition 3.35 ([Manl]). 3 log V (x, r) converges to a limit X > 0 as r -> 00 and.X is independent of X.
Proof. Choose a compact fundamental domain N in M for the fundamental group try (M) acting by isometries on M and let a be its diameter. Then
B(x,r - a) C B(y,r) C B(x,r + a), for all r > a and all x and y in N, from which
V (x, r - a) < V (y, r) < V (x, r + a),
(3.6)
for all r > a and all x and y in M, since the covering transformations that bring x into y are isometries. Also we have:
B(x, r + s) C U B(y, s). yEB(x.r)
A maximal subset Y of B(x, r) whose points are pairwise b apart has cardinality at most b V (x, r + b/2) where cb = inff V (z, b/2). Since every point of B(x, r) is within b of some y E Y we have
B(x,r + s) C U B(y,s + b). yEY
We may as well assume that V (x, r) is unbounded. Hence we can choose b so that cb = 1. Put A = a + 3b/2 and then
V(x,r+s) 5 V(x,r+b/2)V(x,s+a+b), hence, setting
r' = r - b/2 and s' = s + b/2 V (x, r + s) = V (x, r' + s') < V (x, r)V (x, s + A).
Now if ks < r < (k + I )s. then V (x, r) < V (x, (k + 1)s)
< V(x,ks)V(x,s+A) < 1
r
V(x,s)[V(x,s+A))"`.
log V (x, r) <
Ir log V (x, s) + kr log V (x, s + A)
<
r logV(x,s)+ I logV(x,s+A),
70
3. Geodesic Arcs, Counting Functions and Topological Entropy
lim sup
1
r-oo r
log V (x, r) < 1 log V (x, s + A)
-
s
for all s and so lim sup 1 log V (x, r) < lim inf 1 log V(x, s + A) s-+oo S r-oo r
lim inf
1
3-too S
log V (x, s),
which shows that Iimr_,o 1 log V(x, r) exists and by (3.6) is independent
0
of X.
Exercise 3.36. Show that if Mn has constant sectional curvature K < 0, then
A=(n-1),1WI. Remark 3.37. Let p : M --), M be the covering projection and let r(M) be the injectivity radius of M. Then d(x, y) < r(M) implies that
d(p(x), p(y)) = d(x, y).
Theorem 3.38 (Manning's inequality [Man1]). If g is a metric of class C2,
x < htop(g)
Proof. Without loss of generality we can assume that k > 0. For given small 8 > 0 we consider B(x, r + S/2) - B(x, r). By Proposition 3.35,
exp(x + e)r > V (x. r) > exp(.k - e)r,
for all r > ri = rl (e) and for any small e with l - e > 0. Suppose that there exits ro such that
V (x, r + S/2) - V (x, r) < exp(x - e)r, for all r > ro. Then by induction on N
V (x, ro + N3/2) < A exp ((ro + N8/2)(a - e)} + exp(ro(A + e)), for all integers N > 0, where A is a positive constant such that
exp(-8/2(A - e)) < A/(1 + A). This would contradict the fact that the growth rate of V (x, r) is A. Then,
V (x, r + 8/2) - V (x, r) > exp(x - e)r,
3.4 Manning's inequality
Figure 3.2:
71
Proof of Manning's inequality
for a sequence ri, r2.... of values of r tending to oo. For such a value of r we take a maximal subset Q, of B(x, r + 3/2) - B(x, r) whose points are pairwise 28 apart. Then # Qr
V (x, r + 8/2) - V (x, r) > SuPZEM V(z, 28)
- ca exp(,l - Or,
for a constant ca. Any two points of the complete Riemannian manifold M can
be joined by a minimizing geodesic so each x can be joined to each q E Q, by a geodesic of length between r and r + 8/2. The unit initial vectors of these Leodesics give a set Y, C SM that is (r, 8)-separated for Z, the geodesic flow of
M (cf. Figure 3.2). This is because d(7r ,y, q) < 8/2, d(7r ,y', q') < 8/2 and d(q,q') > 28 imply 41ry) >
S.
On account of Remark 3.37, the set dp(Y,) C SM is (r, 8')-separated, where 8' = 7 min{8, r(M)}. Thus hrop(g) > lim sup
I
n-oo rn
log # Y,, > A - e,
and e is arbitrary so h,,,p(g) > A as required.
0
3. Geodesic Arcs, Counting Functions and Topological Entropy
72
Lemma 3.39. For any metric g of class C2 and all x E M we have
A < liminf
I log JunT(x, y) dy,
T-w T
and if g does not have conjugate points
hr0P(g) > A = Ttoo T log fm nT (x, Y) dY Proof. Let p P(x) = X.
:
if' -+ M be the covering projection and z E M such that
Let B(0, T) denote a ball with center at the origin and radius T in a tangent space. Note first that since Fxp. (B(0, T)) = B(z, T) we have (cf. Proposition 3.1 and Exercise 3.40 below)
IdetdeezpxId9 > V(z,T).
(3.7)
fB(O,T)
If g has no conjugate points, expz is a diffeomorphism and we have equality in (3.7). Next observe that since p is a local isometry, we have
f
JBdetdaIdO= f(O.T) IdetdeexpId9= fM ( O.T)
where the last equality was proved in Proposition 3.1. Hence
nT(x,Y)dy IM
V(x,T),
with equality if g has no conjugate points.
0
Observe that by combining Corollary 3.28 and Lemma 3.39 we also obtain Manning's inequality for C°O geodesic flows. However, Manning's inequality only requires g of class C2. Exercise 3.40. Let M be a complete, possibly noncompact Riemannian manifold. Given x and y in M and T > 0 consider nT(x, y). Let B(x, T) be the closed ball with center at x and radius T. Show that nT(x, y) > 1 if and only if y E B(x, T). Show that
f nT(x,y)dy= f(x.T) B(nT(x,y)dy=fO,T) Idetdo exp:IdO. M
B
Exercise 3.41. Let p : M0 -+ M be any Riemannian covering, where M is complete. Show that given x E M0, we have fmo nT(x, y) dY = fm nT(P(x), Y) dY.
3.5 A uniform version of Yomdin's theorem
73
3.5 A uniform version of Yomdin's theorem In this section we will state the uniform version of Yomdin's theorem that Made
used in [Ma2] and we will give an outline of its proof. The proof is based on Gromov's presentation of Yomdin's theorem in [Gr3]. The section is not selfcontained. In particular the Main Lemma below is stated without proof. This is the hard technical result that is proved approximating f by its Taylor series expansion and using algebraic properties on the associated varieties.
Let N be a closed manifold and f : N -+ N a C' diffeomorphism. We shall consider N embedded in an Euclidean space R' and f extended to a C' map f : U -s U, where U is an open neighborhood of N. Given a C' map g : U -+ R", define
IIdrgII = sup{IIdkgll : x E U,
I < k < r}.
Fix an integer! > 1. If Y C N, we can define the C'-size of Y as an 1-dimensional set to be the infimum of s > 0 such that there exists a Cr map h : [0, 111 -> R'" satisfying
h([0, 1]') D Y
and
Ildrhll
S.
If no such s exists, the Cr size of Y is oo. If Y C N is a submanifold, the C'size of Y will mean the Cr size of Y as a dim Y-dimensional set. One could use other standard manifolds than [0, If (like 1-spheres) to define the C'-size and we obtain equivalent notions.
Exercise 3.42. Show that when Y is a curve in RI the Ct-size of Y is the length of Y. As Gromov points out in [Gr3], the precise geometric meaning of the C' size for max(l, r) > 2 is rather obscure. Exercise 3.43. Show that every set of C'-size < S can be divided into jr subsets of C'-size < S/j for all j = 1, 2, .... Hint: subdivide [0, 111 into jl cubes [0, j _ ], and use the scaling map [0, 1]1 -+ [0, j- ')I for each subdivision.
We shall assume that the distance between f (U) and 8 U is > 1 /f by rescaling the embedding and thus increasing the distance; this is required in the Main Lemma below. The next result is the uniform version of Yomdin's theorem. For simplicity, we shall state it and sketch its proof for the case of diffeomorphisms, although it also holds for flows. For the function f considered above, define
K = K (f) = I + lim sup
1
k-.+oo k
log ll df k II.
(3.8)
Theorem 3.44. Let f : N - N and K = K (f) be as above. For any S, e > 0 and any integers r, I > 1, there exist C > 0 and an integer no > 1 such that, for
74
3. Geodesic Arcs, Counting Functions and Topological Entropy
every l-dimensional submanifold Y Cr N with C' si ze
Vol(f"(Y))
Cexp { (h0P(fIN)
S, wee have
+ e+ 1K )nj
for all n > no.
Proof. By Exercise 3.43 every set of C'-size < S can be divided into ji subsets
of C'-size < S/j for all j = 1, 2, .... Thus it suffices to prove the theorem in the case when Y has C'-size Gromov [Gr3, 3.6 p.233].
I. We shall need the following lemma proved by
Lemma 3.45 (Main Lemma). There exists C = C(1, m, r) independent of f, such that if Yo C N is a C' 1-dimensional submanifold of C'-size < 1 and Qp, = 1, ..., i, are unit cubes contained in the space Rm in which we embedded N, then
Vol (fi
(Yon(hf(Qfl)))) < (CIIdrf Iltl + I)',
for any positive integer i.
Given a continuous map g : X -+ X of a compact metric space, denote by n(8, i, g) the minimal cardinality of a (8, i)-spanning set for g. Assume that the points x5, s = 1, ..., n(1/2, i, f IN), form a (1/2, i)-spanning set for f IN. Then the manifold N can be covered by sets of the form
A,:= n f-f(Q6),
I <s < n(1/2, i, f IN),
where Qp is the unit cube centered at f a(x,). If Yo is as in Lemma 3.45 we have
Vol(f'(Yo)) < E Vol(f'(Yo n As)) < n(1/2, i, f I N)(CIldrf
II11r
+ I)'. (3.9)
Now define fj : jU -> jU, for j > 1, by fj(x) = jf(j-tx). Then fj(jN) = j N. Let Yo be a submanifold of C'-size < 1. Observe that j Yo can be covered by jt-sets, Yo, k = 1, ..., jt, with C'-size < 1. Therefore we can apply inequality (3.9) to each Yo, obtaining
Vol(fj(jYo)) < EVol(fj(Yo)) < jtn(1/2,i, fjIjN)(Clldrfjllt1r+ 1)i. k
(3.10)
But n ( 1 7, n(1/2,i, fjl jN) =n(2-
fIN).
3.5 A uniform version of Yomdin's theorem
75
By the definition of topological entropy, given S > 0, there exist io > I and jo > I such that
n(2l .i.fIN) for all i > io, j > jo. Then for i > io, j > jo, inequality (3.10) implies
Vol(f'(Yo))
=Vol(j-'fj(jYo)) = j-'Vol(fj(jYo)) 5r2( .i,flN)(ClldrfjlI1"'+1)'
(3.11)
5 (exp { (hrop(f IN) + 0 }) . (CII dr fj II'/' + I)'. The definition of K in equation (3.8) allows us to choose k so that
k
Observe that lld' fj 11 = j choose j > jo such that
log(C(211dit II)"' + 1) <
1K
r
1Id' f ll for s = 1, 2, .... We see from this that we can
Ildrf, II
<< 211dfk11
Then if i > io, inequality (3.11) applied to fk with S = ke gives
Vol(fk'(Yo)) 5 (exp{(hrop(fkIN)+ke)i)) . (Clldrf II'j'+ I)' < exp { (khrop(f IN) + ke + log(C(21l df k I I )ri' + 1)) i }
SexP{(hrop(fIN)+e+K)ki r
1 l If n > kio, we can choose i > io such that 0 5 n - ki < k. For n > kio, we have
Vol(fn-k'(fki(Yo)))
Vol((fn(Yo)) =
Ildf
II'cn-kiI
exp
(hjop(f IN) + e +
1K) r
ki } 1111
rI
Jll
where C = Il df 111k. 'T'his completes the proof of the theorem.
3.5. 1
0
Another proof of Theorem 3.32 using Theorem 3.44
We shall see in this subsection that the uniform version of Yomdin's theorem can be used to give another proof of Theorem 3.32. This was the approach that Marie took in [Ma2J.
3. Geodesic Arcs, Counting Functions and Topological Entropy
76
Let us show first
Lemma 3.46. Given an integer r > 1, there exists a constant S > 0 such that the (n - I)-dimensional manifold Sx M has C'-size < S for all x E M. Proof. Since M is compact and SM fibres over M, there exists a finite cover of M by open sets U ; C M, f o r i = I , ... , q such that for each i there exists a C°O diffeomorphism
hi: S'
1
x Ui -+ lr-1(Ui),
with hi (Sn-I x (x)) = SxM for all x E Ui. Set si := IId,hi 11. Then by definition the C'-size of SxM is < si for all x E Ui. Now take S := maxi si. Observe now that as in the proof of Corollary 3.9 we have
I
A((x, v), t)dv < Vol(ot(S,rM)),
SM
where A((x, v), t) was defined in (3.1). By Theorem 3.44, given e > 0 and an integer r > 1, there exist to > 0 and C, > 0 such that Vol(Q,r(SxM))
r
(n - I)K r
I
for all t >_ to and all x E M, since by Lemma 3.46 there exists S > 0 such that
the (n - I)-dimensional manifold SM has C'-size < S for all x E M. Hence there is a constant C; such that, for all large enough T, we have
f
rTr r nT(x, y) dxdy = J
MxM
0
JMJSAM
< C,' expI
(
A(x, v, t) dvdxdt 1htop(g)+e+(n-
\
r
1)K)T'. J
111
Therefore
limsup
! log J T
MxM
nT(x, y) dxdy < hr,,p(g)+e+
(n - 1)K r
concluding the proof of Theorem 3.32, since the above inequality holds for all
e>0andallr> 1.
4 Mane's Formula for Geodesic Flows and Convex Billiards
In this chapter we present a proof of Mafe's formula for geodesic flows and convex billiards. The proof rests on the twist property of the vertical subbundle that we described in Chapter 2, Pesin's theory which enters via Przytycki's inequality and a very clever change of variables which is useful also in other situations (cf. Proposition 4.8). In all that follows M will be a compact connected manifold endowed with a Riemannian metric g.
4.1
Time shifts that avoid the vertical
Definition 4.1. Given subspaces El, E2 of S(B) with dim El = dim E2 = dim S(0)/2, we define the angle between Ei and E2 as
a(E1. E2) = Idet(PIEl)I, where P : S(O) -+ EZ is the orthogonal projection. Clearly a depends continuously on the subspaces and a(Ei, E2) = 0 if and only if Ei fl E2 # (0).
Exercise 4.2. Show that a(E1, E2) = a(E2, E1). The next two lemmas are two very useful consequences of the twist property of the vertical subbundle.
Lemma 4.3. There exist a constant 8 > 0, an integer m > I and an upper semicontinuous function
r:A(SM)xR-.(0,1/m,2/m,...,l)
78
4. Maid's Formula for Geodesic Flows and Convex Billiards
such that after abbreviating r(0, E, T) to r, we have that forall (0, E) E A(SM) and all T, a(dOOT+r(E), V (OT+TO)) > 8.
Proof. We shall show first that there exist a constant 8 > 0 and an integer m > 1 such that, for all (0, E, T) E A(SM) x R, the set given by
Q(0, E. T) :_ (i E Z : 0 < i < m, a(daOT+r/m(E), V (OT+i/m0)) > 8} is not empty. Suppose that this is not the case. Then given any m > 1, there exists a sequence (0m, Em, Tm) E A(SM) x R such that a(do,,,OT,.,+s(Em), V(¢Tf'm+sOm)) < 1/2m,
(4.1)
for all s E A. where
Am:={j/2m: jEZ,
0<j<2m}.
Note that Am C Am+i. Since (A(SM) is compact the sequence {(WT , (Om), Gom4Tm (Em))} has a convergent subsequence
{ (ITmk (Om! ), domk OTmt (E.,)))
that converges to a point (0, E) E A(SM). It follows from (4.1) and the continuity of a that, given any k and any s E Am,, we have
a(da0s(E), V (4s0)) = 0. Hence for all s in [0, 1], we have deos(E) n V (0s0) -0 (0).
This contradicts the twist property of the vertical subbundle. Now define
r(0, E, T) = min{i/m : i E Q(0, E, T)}. Clearly r is upper semicontinuous and given (0, E, T) E A(SM) x R, we have a(doPT+T(E), V (OT+TO)) > 8.
4.1 Time shifts that avoid the vertical
79
d9 OT V(e-)
v (e)
v( $T+t (e)) I
18
d9#T+t(P d9 Op (V(8 I)) I
e
e- ° -p (e)
Figure 4.1:
Time shifts
Remark 4.4. Observe that by the definition of angle, saying that
a(do4T+r(E), V(4'T+rO)) > S is equivalent to saying that Jdet(dm,.+telrls)I > 8
where S := do T+r (E). Recall that Tr : SM - M is the canonical projection. Lemma 4.5. There exist a constant y > 0, an integer n > 1 and an upper semicontinuous function
p : A(SM) - 10, l /n, 2/n, ..., 1) such that after abbreviating p(O, E) to p and setting B_ = 0_pB, we have that for all (0, E) E A(SM),
a(E, de_dp(V(9-))) > y. Proof. The proof is very similar to the proof of the last lemma. We shall show
first that there exist a constant y > 0 and an integer n > I such that for all (0, E) E A(SM), the set given by
Q(0, E) := Ii E Z : 0:5 i < n, a(E,
Y)
is not empty. Suppose that this is not the case. Then given any n > 1 there exists
(0, En) E A(SM) such that a(En , dd_,eOs(V (O-sen))) < 1/2n,
(4.2)
4. Mane's Formula for Geodesic Flows and Convex Billiards
80
for all s E A. Since A(SM) is compact the sequence ((8,,,
has a convergent subsequence that converges to a point (0, E) E A(SM). From (4.2) and the continuity of a, we get that for all s in [0, 1], we have
a(E, dd_,ecs(V (q5-se))) = 0. Hence for all s in [0, 1], we have
deo-s(E) n V(0-.0) 0 (0). This contradicts the twist property of the vertical subbundle. Now define
p(O, E) = min(i/n: i E Q(0, E)). Clearly p is an upper semicontinuous function with the desired properties.
Let X C SM be any closed set and let E be a measurable Lagrangian subbundle, possibly time dependent, defined on X x R+, that is, E is a measurable section
E : X x R+ - A(X) of the bundle A(X) x R+ -* X. Let us suppose now that µ is any finite Borel measure on X. Define F : X x [0, T] -> X x [0, T + I] by
F(9, t) = (0, t + r(0, E(0, t), t)), where r(0, E, t) is given by Lemma 4.3. Since r is upper semicontinuous, F is measurable. In what follows we shall abbreviate r(B, E(0, t), t)) to r. Lemma 4.6. Given any integrable function P : X x [0, T + I ] --> R+, we have
f
Xx[O.T+IJ
dµdt >
1
M + I JXx(O.TJ
('.F)dµdt.
Proof Given an integer 0:5 i < m, define
A(i) _ ((0, t) E X x [0, T] : r(8, E(0, t), t) = i/m}. On each A(i), F is injective, and if we set dv = dµdt, then v(F(S)) = v(S) for every Borel set S C A (i ). Hence for any integrable function QS : X x [0, T + I ] -> R, we have
I
FA(i))
Ndv=J
Suppose now that 4' > 0. Then
f
(4,.F)dv = x[O.TJ
f
A((0oF)dv.
f
($.F)dv = E f
Vi A(i)
_F fF(A(i)) dv<>I -Odv i Xx[O.T+iJ =(m+1)
f
Xx(O.T+IJ
4'dv.
4.1 Time shifts that avoid the vertical
81
Lemma 4.7. There exists a constant C > 0 such that for any pair (X, E) as above and any finite Bore! measure t on X, we have
IX x[O.T+I]
Idet(d0,enado0t)IE(e,t)I dpdt
>CJ
Idet(de-0t)IE(o,t)I dpcdt, x[O.T]
for all T > 0. Proof Let us consider the function c1(0, t) = Idet(d,0,en4e0t)IE(e,t)I Observe that by Lemma 4.3 we have 450F(8, t) = Idet(dd,+,enls)I Idetde¢,+TIE(e.r)j > 8Idetde0,+TIE(6.1)I,
where S := de¢,+T(E(B, t)). Combining the last inequality and Lemma 4.6, we obtain
1Xx[O.T+1]
(D dµdt >
I
m+1
JXx[O.TJ
E
m+1
45.Fdadt
Idetde0,+TIE(o,,)I dpdt. xx(0,T)
By compactness, there exists a number a > 0 such that Idet(do4s)ILI >_ a,
for all s E [0, 1] and all (0, L) a A(SM). Hence Idet de¢,+r I E(e.t) I > a Idet de0t I E(e,t) I ,
which implies Idet(dm,etro0t)IE(e,t)I IXX(O.T+11
>
as
m+ 1
f
dpdt
Idet(do t)IE(e.r)I dµdt. x[0.T]
Hence the proof of the lemma with C := m
is completed.
4. Ma66's Formula for Geodesic Flows and Convex Billiards
82
Proposition 4.8. Let N C M be a closed submanifold Then
CJ T Vol(ep,SN')dt < 0
fT+I fm
nT+1(N,y)dy <
Vol(O SN')dt, 0
where C is the constant from Lenuna 4.7.
Proof. Let us recall that from Proposition 3.6 and Corollary 3.9 we know that T
IM
nT(N, y) dy =
f dt 'SN' I det(dO,o r.doOt)IT9sNi I do T 0
We also know from Lemma 2.8 that, for all 9 E SNl, the subspace TOSN' is Lagrangian, and so we can apply the last lemma to the pair (SN', TOSN') and the Riemannian measure dO on SN1 induced by the restriction of the Sasaki metric to SNl. We obtain
f nT+1(N, y)dy = J M
Idet(dOo7r4Oj)ITesNjI dOdt N'x[O.T+I]
C J Nl x tO. T] Idetd4OtlTesNiI dodt
Cf
T ,Vol(OSNl)dt.
0
0
4.2
Mane's formula for geodesic flows
Let us first recall Przytycki's inequality for the topological entropy of general C2 flows [Pr].
Given a linear map L : E - F between finite dimensional Hilbert spaces, we define its expansion ex(L) by ex(L) = max I det(LIs)I,
where the maximum is taken over all subspaces S C E. Przytycki's inequality says that for a C2 flow 0, : N -> N on a closed Riemannian manifold N,
h,0 (0) 5 lim inf
1
T-oo T
log
L
ex(dxOT) dx.
Let A C M x M be the diagonal and let t be the geodesic flow of M x M endowed with the product metric.
4.2 Matte's formula for geodesic flows
83
Theorem 4.9 (Main Theorem). If the Riemannian metric g is C°O, then
hrop(g) = lim
1
T-++oo T log fm xM
= T-.+oo lim 1T log
f
ex(de r) d9
SM
= lim 1 log T-*+oo T
nT(x, y)dxdy
Idet(daOTI v(e))I d8 fSS
M
T
lim
I
T-+oo T
log
p
Vol(m,(S&l))dt.
Remark 4.10. The first equality in the Main Theorem gives a purely Riemannian characterization of the topological entropy of the geodesic flow in terms of the growth rate of the average number of geodesic arcs between two points. The second equality says that Przytycki's inequality is in fact an equality for COO geodesic flows. Quite recently, O.S. Kozlovski proved in [Ko] that Przytycki's inequality is an equality for arbitrary C°O maps. The third equality tells us that h,op can be computed by taking the exponential growth rate of the average over M of the volume of 0,(SSM). Finally the fourth equality tells us that SAl is the canonical choice of submanifold to "catch" the topological entropy by means of volume growth.
4.2.1
Changes of variables
In this subsection we introduce a change of variables that plays a central role in the proof of Mane's formula. This change of variables has basically two objectives: one is to avoid the conjugate points and the other is to relate the vertical subbundle with the Lagrangian subspace that realizes the maximal expansion. For this we use the time shifts introduced in the previous subsection.
Lemma 4.11. For each 0 E SM and I E It, there is a Lagrangian subspace R, (0) C S(8), which depends measurably on r and 0, and satisfies a) Idet(derp:IR,(e))I = ex(deor).
b) if E is a subspace of S(0) with dim E = dim S(0)/2, then
Idet(deO,IE)I ? a(E, R+ (8))ex(de¢,).
Proof. Consider the polar decomposition
dectt = Ot(8)Lt(8), : S(e) -* S(8) is symmetric and positive and O,(8) S(e) -, S(0,(9)) is a linear isometry, both being C°° functions of 0. But L,(8) _
where L,(8)
:
4. Maab's Formula for Geodesic Flows and Convex Billiards
84
((deb,)*(de¢,))1/2 and (d,901)* is symplectic (because so is de¢t). Hence L,(8) is symplectic and symmetric. Then it follows that if C is an eigenvector of Lt(8) associated to an eigenvalue A, Jet is an eigenvector associated to the eigenvalue A-1 because L,(8)Je = J9Lt(8)-1 and hence Lt(e)Jei; = JoLt(8)-1 C = A-1
Using this property, it is possible to construct for each t an orthonormal basis o Ci_1, Je(1, ... , Jgi_1 }, where Ci is an eigenvector of S(e) of the form 1. Let R, (8) be the subspace spanned by Lt (8) associated to an eigenvalue Ai Clearly R,(8) is Lagrangian. Exercise 4.12. Show that R, (0) satisfies property (a). To prove property (b), observe first that
Idet(de',IE)I = Idet(L,(8)IE)I,
because O,(9) is an isometry. Notice also that L,(8) leaves R,(8) and R,L(8) invariant, because both of these spaces are spanned by eigenvectors of L,(8). Hence L,(8) commutes with the orthogonal projection P : S(O) -+ R,(8), i.e., L,(8) o P = P o Lt (8). Let us suppose that E fl Ri (8) = (0), otherwise there is nothing to prove. Then P(E) = Rt(8) and thus Idet(Lt(B)IR,(e))I Idet(PIE)l = Jdet(PIL,(e)(E))I Idet(Lt(8)IE)J
5 Idet(L,(0)IE)I Hence
ex(deO,)a(E, RL(8)) < Idet(de4:IE)I Finally we show the measurability of R,(8) as a function of t and 8. Let F denote the vector bundle over SM consisting of pairs (8, h), in which B E SM and h : S(8) -+ S(8) is a symmetric linear map. Given positive integers p and li, 1 < i < p, let F(p, 11, ... , l p) be the set of pairs (8, h) E F, where g has p eigenvalues AI < . . . < Ap with multiplicities lt, ... , lp. Then F(p, lt, ..., lp) is a Borel set and so is the subset P(p, 11, ... , lp) of SM x R defined by 1(8, t) : Lt (8) E F(p, 11, ... , l p)}. Now observe that R, (8) can be chosen to be continuous on each set P(p, 11, ..., l p). Since these sets are Borel and there are finitely many of them, the measurability is proved. Exercise 4.13. Prove that F(p, ii..... l p) is a Borel set.
Lemma 4.14. There exist d > 0, an integer m ? I and measurable functions ri : SM x R -- 10, 1/m, 2/m, ..., 1}, i = 1, 2, such that, after abbreviating Ti (8, t) to ri for i = 1, 2 and setting r = r (8, t) = rl + r2, 691 = 0- T, (8).
02 = Ot+r2 (8),
and
Vi = V (8i) for i = 1, 2,
4.2 Mane's formula for geodesic flows
85
we have for all 0 and t:
a)a(de,Or,(VI), R, (0)) > 8 and b) a(dot4t+r(VI ), V2) > 8.
Proof. It suffices to prove that we can find 81, 62 > 0, integers m1, m2 > 1 and measurable functions r; : SM x R -+ (0, 1/m;, 2/mi, ..., 1 } such that properties (a) and (b) of Lemma 4.14 hold with 8 changed to 81 in (a) and to 62 in (b). Then we can easily obtain Lemma 4.14 with m = mlm2 and 8 = min(81, 82). Let y, n and p be given by Lemma 4.5. Set
a1 := y;
ml :=n; r1(0, t) := p(0, RtL (0)).
Since RtL(0) is measurable and p is upper semicontinuous, the function rl is measurable. Lemma 4.5 applied to E = Rl (0) implies that a((do,Or,)Vi, Ri (e)) > 81, so property (a) is proved. Now let 8, m and r be given by Lemma 4.3. Set
82 := 8; M2 := m; r2(0, W= r(01401 Or, (Vl ), t)
Since r2 is a composition of measurable functions, it is measurable. Lemma 4.3 applied to E = do, Or, (VI) implies that a(do,Of+r,+r2(Vl), V2) > 82,
0
so property (b) is proved.
Let us consider the function A : SM x l1P - R given by
A(0, t) _ Idet do(rr4t)I v(e)I
.
From the last two lemmas we shall deduce the following:
Corollary 4.15. There exists C > 0, such that for all t E R and 0 E SM, the functions r1 and r given by Lemma 4.14 satisfy
A(j-r,(o.t)(B), t + r(0, t)) > Cex(de4t).
4. Ma6i 's Formula for Geodesic Flows and Convex Billiards
86
Proof. Set S = (de, ¢r,) Vt . By Lemma 4.11 and property (a) of Lemma 4.14, we have I det(deOtls)I > a(S, Rt'(9))eX(doO,) > Sex(do0t).
(4.3)
Take a > 0 such that I det(d;OsIL)I ? a,
(4.4)
for every L) E A(SM) and S E [0, 11. Set 8 = 0,(9) and S = dev,S. Then equations (4.3) and (4.4) imply
Idet(de,Ot+rIv,)I
= Idet(dd¢r21s)I Idet(deotls)I Idet(de,Or,lv,)I
> a 8ex(d9O,)
(4.5)
Now set S2 = (4 , of+r) Vt . By property (b) of Lemma 4.14 and the definition of a we have respectively a(S2, V2) > S,
I det(de27rls2)I > S,
which, together with inequality (4.5), implies that A(4'-r,(e.t)(9), t + r(9, t)) = I det(do, (n o Ot+r)Ivl )I = Idet(d62als2)I Idet(de,O,+rlv,)I
> 31det(de,01+rIv,)I > S2a2ex(do¢,).
Hence the proof of the corollary with C = 82a22 is completed.
Proposition 4.16 (Main Proposition I). There exists C > 0 such that for all
T>0,
fT+2 f
T
A(9, t) dodt > C 0
M
J0
J
ex(de¢,) dOdt.
SM
Proof Define F : SM x [0, T] -> SM x [0, T + 2] by
F(9, t) = (4-r,(e.t)(0), t + r(9, t))), where ri (9, t) and r(9, t) are defined as in Lemma 4.14. Given integers 0 < i
m, 0 < j < m, define A(i, j) = {(9, t) E SM x [0, T] : rt (9, t) = i/m, r2(9, t) = j/m}.
4.2 Mafle's formula for geodesic flows
87
On each A(i, j), F is injective, and if we set dµ = dOdt, then µ(F(S)) = µ(S) for every Borel set S c A(i, j). Hence for any integrable function 0 : SM x [0, T + 2] --> R, we have
f
fF(A(i.j)) 4,dµ
o F)dµ.
.lA(i.j)
Suppose now that ' > 0. Then
f
((D oF)dtt=J Mx[o.TI
i,j A(i.j)
U,1 A(i.j)
fF(A(i.j)) 4dµ<
(0.F)dµ
f pi j
Odµ
SMx[O.T+21
0dµ.
=(m+1)2 J SM x (0. T+2]
From this inequality and Corollary 4.15, it follows that T+2
J
> J't
(m + ()2 >
A(F(O, t)) d+
1)2 llSMx[O.TI
ISM
JOT ISM A(0-r,(0.,)(0), t + r(0, t)) dOdt ex(deor) dOdt,
C
(m + 1)2
o
ISM
0
thus completing the proof of the proposition.
Lemma 4.17. There exists a constant K > 0 such that if r), B) and V( are as in Lemma 4.14. Then for all B and t, we have Idet do, 0, 1 v, I > Kex(de¢t)
Proof.
Set E = dej 0r, (Vi ). Take a > 0 such that
I det(dc0:IL)I > a. for every ( 1 ; , L) E A (SM) ands E [0, 1 ]. Using part (a) of Lemma 4.11 and part (b) of Lemma 4.14, we have
Idet(de,0,lv,)I = Idet(do t-r,IE)I
Idet(d0,0r,lv,)I
a8 ex(deO,-r,) > a8
eX(de0t) ex(dm,_,,BOr, )
4. Mafl6's Formula for Geodesic Flows and Convex Billiards
88
Take a constant a' such that for all o E SM and all s E [0, 1)
ex(deos) < a'. The proof of the lemma is completed if we take K := n .
Proposition 4.18 (Main Proposition II). There exists a constant D > 0 such that for all t, we have
D/
ex(do0t)do <
I de t(d&OI
v(o))I do.
ISM
JJS
Proof. The proof of this proposition is very similar to the previous one, so we leave the details as an exercise. One uses Lemma 4.17 and the change of variables
F(9) = of = O-r,0.
4.2.2
Proof of the Main Theorem
In this subsection we explain how the Main Theorem can be derived from: Theorem 3.32 from Chapter 3; Przytycki's inequality; The Main Propositions I and II from the last section, and the following lemma:
Lemma 4.19. lim sup 1 log T-.+oo T
f ex(dOOT) dB = lim sup 1 log fs M
T-+. T
J0
f
ex(de¢t) dodt.
SM
Proof Observe that ex(deot) > I and that ex(doot) < ex(d0,+,o0-s) ex(de0t+s)
Since M is compact, there exists a constant C > 0 such that for all 0 E SM and all s E [- 1/2, 1/2], we have
ex(do4s) < 1/C, hence for all t and alI s E
1/2. 1/2] we have, ex(d9O1+s) ? Cex(do0t).
The lemma is now a consequence of Lemma 3.24 in Chapter 3.
4.2 Marie's formula for geodesic flows
89
Since we are assuming that g is C°O, Theorem 3.32 from Chapter 3 tells us that lim sup
- log J
T-.+oo T
nr(x, y)dxdy <_
(4.6)
MxM
Przytycki's inequality says that
hrop(g) < liminf
I
T-.+oo T
ex(diOT)d6.
log
(4.7)
sM
By Corollary 3.10 we know that
r rr nr(x, y)dxdy = JMxM J0
f
A(0, t) dOdt,
SM
and hence from the Main Proposition I. we obtain rT p
fu xM
nT+2(x, y) dxdy > C
J0
J
ex(deoe)d6dt.
(4.8)
SM
Combining Lemma 3.24, (4.6), (4.7) and (4.8) we deduce
I
lim sup log nT(x, y) dxdy <_ h,,p(g) T-.+oo T fm xM
< lim inf 1 log fs e x(doOT) dO T-.+oo T M < lim inf
I
T-.+oo T
log
ex(demr) dodr fo
f5 M
< lint inf 1 log nr(x, y) dxd y. T-.+oo T JMxM Therefore
hrop(g) = lint
I
T+oo T log fu xM lim inf 1 log T
nT(x, y)dxdy
ex(d9OT) dB. fSM
Using the last two equalities. Lemma 4.19 and (4.8), we get 1
hr0p(g) > lim sup I log ex(d4Or) dB, T-+oo T sM and hence we deduced the first two equalities in the Main Theorem. The third equality is now a direct consequence of the Main Proposition 11 and the fact that by definition of the expansion, I det(dA ,jv(e))I f- ex(doO,)
4. MaA 's Formula for Geodesic Flows and Convex Billiards
90
To get the fourth equality in the Main Theorem recall from Corollary 3.9 and Lemma 3.29 that
r
fMxM nT(x, y) dxdy = JMxM nT/f(A,r)dr < fo
T/f V
.')dt.
Hence, using the first equality in the Main Theorem and Yomdin's Theorem, we obtain nT(x, y)dxdy
hrop(g) = Tli+oo I log f liminf I log T-.+ooT fo limsup
T/f
1
T
logT-.+oo
hrop,
which concludes the proof of the Main Theorem.
0
Remark 4.20. It is important to remark that when the metric is only of class C3, it follows from the previous proof, Przytycki's inequality, (4.8) and Main Proposition II that
I log J
liminf
T-.+oo T
nT(x, y) dxdy >_ hrop(g),
MxM
and 1
lim inf - log
T-++oo T
I det(deOrJ v(o))I dO > h,op(g). SM
Exercise 4.21. Show that if g is of class C', then the geodesic flow is of class Cr-1.
4.3
Manifolds without conjugate points
Lemma 4.22. Suppose that g has no conjugate points. Then there exists a constant c > 0 such that for any xl, yl, x2, y2 E M and all T > 0,
nT(xI, yl)
nr+4c(x2, y2)
4.3 Manifolds without conjugate points
91
Proof. We shall prove that, for all x and y in M, we have nT(x, Y) < nT+c(x, x)
(4.9)
nT(x, x) < nT+c(x, Y).
(4.10)
and
These inequalities imply the lemma because given xt, yt, x2, y2 E M using (4.9) and (4.10) twice, we obtain
nT(xi,YI)
Since g has no conjugate points, the exponential map expr : Tz M -> M is a diffeomorphism, where p : M -+ M is the universal covering of M. Let x and y in M be given, and choose x in p-1 (x). Let c be the diameter of M and choose y E p-1(y) with d(z, y) < c. Since P-1(y) = (YY : Y E iri(M)), it follows that
nT(x, y) = #{y E,r,(M) : d(z, yy) < T}; nT(x, x) = # {y E 7ri (M) : d(x, yx) < T). From the triangle inequality and the fact that d(x, y) < c we obtain
d(z,yx)
d(x,y3)
Theorem 4.23. If g has no conjugate points and is of class C3, then lim
1
T-.+oo f
for allx,y EM.
lognT(x, y) = hrop(g) = 1,
92
4. MaM's Formula for Geodesic Flows and Convex Billiards
Proof. Combining Lemma 4.22 and Remark 4.20, we get 1
lim inf lognT(x, Y) ? hrop(g), T-.+oo T
for all x and y in M. On the other hand, combining Lemma 4.22 and Lemma 3.39 in Chapter 3, we obtain lim sup
1
T-.+oo T
lognT(x, Y) 5 A 5 hrop(g),
for all x and y in M and thus the theorem is proved.
The equality of the topological entropy of the geodesic flow and the volume growth rate of the manifold when there are no conjugate points was established by Freire and MaN in [FM] using different techniques from the ones we used here. Their work extended earlier work of Manning in [Man I]. Manning showed
that A = hrop(g) when M has sectional curvatures < 0. Manning's proof for manifolds of non-positive curvature relies on the fact that for such manifolds, any two geodesics y; : R -> M in the universal covering satisfy
d (yi (t), n(t )) 5 d (y1(0). y2(0)) + d(yi (T), n(T )),
for all 0 5 t 5 T. In [FM] Freire and Mafid observed that for manifolds where this property holds with the right term multiplied by a constant independent of the geodesics, Manning's proof can be applied with insignificant changes, thus providing a much simpler proof of their result if the existence of such a constant could be established for manifolds without conjugate points. However the "Dinosaur example" of Ballmann, Brin and Burns [BBB], that appeared four years later, proved that such a constant does not exist.
4.4 A formula for the topological entropy for manifolds of positive sectional curvature Theorem 4.24. If all the sectional curvatures are positive (negative), then
hrop(g) = Tlim00 T log f
I det(delTIH(e))I d6.
Proof. If all the sectional curvatures are positive (negative), the curvature operators R(0) (cf. Definition 2.12) are positive (negative) definite. In this case, we know from Proposition 2.13 that the twist property of the horizontal subbundle holds. This implies that all the arguments that we did in the previous sections can be done replacing the vertical subspace by the horizontal. In particular, we have
4.5 MaiSB's formula for convex billiards
93
the analogue of Lemma 4.17 and hence of the Main Proposition II. It follows that there exists a constant D > 0 such that D
f
ex(de0,)d0 < f I det(de0(IH(e))I d9 < f ex(doO,)dO.
SM
M
M
The theorem now follows right away from the last inequalities and the Main Theorem. 0
4.5
Mane's formula for convex billiards
The billiard ball model describes the free motion of a particle moving in a planar convex domain bounded by a smooth simple closed curve y. The particle moves with constant speed along a straight line and gets reflected at an angle equal to the incidence angle. The dynamics of the sequence of collisions points and angles of incidence at them is described by the so called billiard ball map. Our exposition of the general facts about billiards is based on [HK, Section 9.2].
We shall assume that y is of class Cr with r > 3 and has everywhere positive curvature. Let us orient y counterclockwise. We shall assume that y is parametrized by arc length, and so we can identify it with R/lZ where 1 is the length of y. The billiard ball map is a diffeomorphism f of class Cr-t of the closed annulus A := y x [0, tr] defined as follows. A point (x, (P) E A, together with the orientation of y, determines uniquely an oriented line I which intersects y in two points x and ft (x, W). Then f (x, W)
(fi (x, W), f2(x, (0),
where f2(x, W)) E [0,;r] is the angle uniquely determined by the vector at ft (x, (P) pointing inwards in the direction of the line obtained by reflecting I in the tangent line to y at fi (x, (p) (see Figure 4.2). The map f restricted to aA is the identity and it satisfies an important property: it is a twist map. This means that if (x, W) E A and if we write f (x, W) = (ft (x, W), f2 (X, W)),
then we have
aft (x, W) > c > 0,
(4.11)
for all points (x, rp) in A (see [HK, Section 9.2]).
Exercise 4.25. Show that I.(x, 0) = 'Ll (x, Jr) = l/k(x), where k(x) is the curvature of y at x. Exercise 4.26. Show that f preserves the volume element sin W dxdc.
94
4. Maine's Formula for Geodesic Flows and Convex Billiards
Figure 4.2:
Convex billiards
It is sometimes convenient to use the coordinate r := cos rp instead of rp, because in these coordinates (x, r), the billiard ball map f will preserve the standard area form dxdr. However, in these coordinates the map f fails to be differentiable at the boundary of the annulus, whereas in the coordinates (x, ip) it is differentiable since we are assuming that y is strictly convex.
Given a point (x, rp) E A and N E Z+, set for 0< i< N
(xi, (pi) = f'(x, g) We will call {xi }o
Given two points x and y in y and N E Z+, let us define nN(x, y) as the number of billiard configurations of length < N that begin at x and end at y. In other words, nN(x, y) counts the number of billiard trajectories leaving from x that hit y after at most N collisions with the boundary. We shall relate now, as in the geodesic flow case, the growth of nN(x, y) with the topological entropy of the billiard ball map f. Let tr : A -> y denote the projection onto the first factor of A. For i E Z+, let Fi.x : [0, n ] -+ y be the map Fi.x (co) = n o f' (x, g) = xi.
(4.12)
Lemma 4.27. Given x E y and N E Z+, y '-+ nN(x, y) is finite and locally constant on an open full Lebesgue measure subset of y.
4.5 Mat k''s formula for convex billiards
Proof. Note that nN(x, y) _ F
95
#F.-.XI (y). To obtain the lemma just apply
o
Sard's theorem to the maps Fi,x.
Lemma 4.28.
i.N
n
dF
nN(x,Y)dy=Erp)i=0
0
fy
where dy denotes Lebesgue measure (arc length) on y.
Proof. It suffices to show that
J
#Fi.xl (Y) dy
=
f
"
dFi,(p)
dV.
I
This is a consequence of the Area formula we proved in Chapter 3.
0
Theorem 4.29. If y is C°o, 1
lim sup log J nN(x, y)dy
forallx E y,
where h,op denotes the topological entropy of f. Also we have lim sup
1
N-.oo N
log
JYxY
nN(x, y) dxdy _< hrop.
Proof. Set Sx = {x} x [0, n]. Using Lemma 4.28 we have
f
f
i=N
y
i=O 0
f nN(x,Y)dY = E
IF
arr(fi(x.(0) app
dw <
N
Vol(f'(S:))
i-o
The proof of the first inequality is now a consequence of Yomdin's theorem. For the second inequality, we need Mane's improved version of Yomdin's theorem. Observe first that for any x E y, Sx has the same C'-size. Hence Theorem 3.44 in Chapter 3 applied to the map f implies that given e and an integer r > 1, there exist C > 0 and a positive integer no such that for all i > no, we have
Vol(f' (Sx)) < C exp l (h0 p + e + r
li
111
)
.
96
4. Mafi6's Formula for Geodesic Flows and Convex Billiards
Therefore,
i=N
J nN(x, y)dxdy Xy
i=o
fYVol(f'(S))dx i=N
-no
E
fVo1(f1(Sx))dx+
fvol(f1(sx))dx
io y
i=no+I
i=no r
i=N
r
r/
i=no+t
y
111 \
<'J Vol(f'(Sx))dx+ E C J expjlhrop+e+dx i=o
y
)))
111
Hence lim sup
I
N-oo N
log
Jyxy
nN(x, y)dxdy
If f is C°O the above inequality holds for all e > 0 and all r > 1 and thus lim sup
1
N-.oo N
logJ
yxy
nN(x, y) dxdy < hrop.
0
as desired.
Theorem 4.30. Suppose that y is of class C3. Then given e > 0, there exists an open set U. e y such that for all xrE Us we have
1
lim inf log nN(x, y) dy >- hrop - e. N-+oo N Jy We shall prove this theorem in the next subsection. We state now and prove the following corollary. Corollary 4.31. If y is of class C3 then,
lim inf N-+oo N
log f
yxy
nN(x, y) dxdy ? hrop
Proof. Let ms be the Lebesgue measure of Us. Since Us is open, ms > 0. By Jensen's inequality we have log
fxy
nN(x, y)dxdy >
logf
nN(x, y)dxdy
u.Xy
fu, 1logfy
nN(x,y)dy1dx. 111
4.5 Mailt's formula for convex billiards
97
If we divide by N the previous inequality and we take the lim inf as N -* +oo, we obtain using Fatou's lemma. lim inf
I
N-oo N
log
Jyxy
nN(x, y) dxdy I
J of
{liminf
-
111 N-oo N
log Jr
n
N(xy)dymE dx. 11
Using Theorem 4.30 we get limi nf
1
N.oo N
log(x, y)dxdy > h,ap -s. fnN xy
Since the last inequality holds for all e > 0, the corollary follows.
If we combine Corollary 4.31 with Theorem 4.29 we obtain right away the following corollary, Corollary 4.32 (Mane's formula for convex billiards). If y is C°O, lim I log nN(x, y)dxdy = N-+oo N yxy
J
4.5.1
Proof of Theorem 4.30
Suppose now that f is a diffeomorphism of class C' with r > 2 acting on a compact surface X. We shall state below the results needed from the theory of smooth Dynamical Systems to prove Theorem 4.30.
Suppose that A is a hyperbolic set for f (cf. Section 2.4.2). Recall that an invariant set is said to be transitive if there exists a dense orbit inside it.
Definition 4.33. A basic set is a transitive hyperbolic set that has a dense set of periodic orbits such that it is the maximal invariant set in a neighborhood of it. The basic set is of saddle type if dim Ex = I for all x E A and it is nontrivial if it does not reduce to a single periodic orbit. We shall need the following important result of A. Katok [KI].
Theorem 4.34. Suppose that
0. Then, given e > 0, there exists a non-
trivial basic set of saddle type AE such that htop(AE) > htop - s.
4. Mane's Formula for Geodesic Flows and Convex Billiards
98
4
WS(X)
x V
Figure 4.3:
Proposition 4.35
Suppose that A is a hyperbolic set. For each x E A the stable manifold of x is the set given by Ws(X) := {y E X :
lmod(fn(x), fn(y)) = O}.
A classical result in the theory of hyperbolic dynamics ensures that WI(x) is an immersed curve of class C' invariant under f. We shall also use the following proposition [New, pag. 234).
Proposition 4.35. Let A be a basic hyperbolic set of saddle type and let W'(x) be the stable manifold of a periodic point x E A. Let y : (-e, s) -> X be a curve of class Cr such that y(0) E W1 (x) and y'(0) is transversal to Ty(o)W'(x). Then lim inf
1
n-ioo n
log Vol (fn(y)) > htop(A). -
Suppose now that f is the billiard ball map on the closed annulus A. As before,
let n : A - y be the canonical projection. For each (x, rp) E A, let V(x, rp) be the kernel of d(x,5,)7r. In other words V (x, rp) is the line tangent to Sx at (x, (p).
The twist condition (4.11) off immediately implies that for all (x, (p) E A, we have
d(x.rp)f (V (x, rp)) # V (f (x, (p)).
(4.13)
Given two lines Lt and L2 tangent to A at the same point, we shall denote by a(Ll, L2) the sine of the angle between them.
4.5 Mane's formula for convex billiards
99
Lemma 4.36. There exist a constant S > 0 and an upper semicontinuous function
r : A x Z -+ {0, 1) such that for all (x, rp) E A and all i E Z, we have after abbreviating r(x, rp, i) to r, a(d(x.,,) f
r+i
(V (x, (p)), V (f
i+r (x, co))) > S.
Proof. We shall show first that there exist a constant S > 0 such that for all (x, (p, i) E A x Z, the set given by
Q(x, w, i) :_ (j E (0, 1) : a(d(x.w) f'+t (V (x, w)), V (f i+j (x, (o)))) > S} is not empty. Suppose that this is not the case. Then there exists a sequence (xm, (pm, im) E A x Z such that a(d(xm.cpm)f'm+j(V
(xm, (pm)), V (f-+j (xm, (pm))) < I /m,
(4.14)
for j = 0, 1. Since A is compact, the sequence {(f'm(xm, cpm), d(xm.Nm)f'm(V (xm, rpm)))}
has a convergent subsequence that converges to a point (x, (p, S) where S is a line tangent to A at (x, (p). From (4.14) and the continuity of a, we get that a(d(x.,p)f i (S), V ((pi (x, (p)) = 0,
for j = 0, 1. From j = 0, we obtain that S = V (x, rp) and from j = 1, we obtain that d(x.,)f (V (x, co)) = V (f (x, co)),
which violates the twist property off in (4.13). Now define
r(x,(p,i) =min{j : j E Q(x,cp,i)}. Clearly r is upper semicontinuous and has the desired properties.
Remark 4.37. Observe that by the definition of a saying that
a(d(x.w)f'+'(V (x, (o)), V (f
i+r (x,
(p))) > 8,
is equivalent to saying that Idetdfi+T(x.V)1rId(x.m)f=+i(V(x.e))I > S.
0
4. Matte's Formula for Geodesic Flows and Convex Billiards
100
Proposition 4.38. There exists a constant C > 0 such that for all N E 7G+, we have
nN+I (x, y) d y > C Vol(f N(Sx))
Proof. Let
c(x,rp,i) :=
I
where Fi,x is given by (4.12). Since 4 > 0 and r takes only the values 0 and I we have for all (x, rp, i) E A x Z
c(x, cp,i +r) <- (D(x,(P, i)+4(x,(p,i + l). and therefore
Sr ssx
(x,cp,i+r)dcp< f 4'(x,cp,i)drp+I c(x,wp,i+I)dtp. sx
Hence
i=N+I
i=N
Ef i=0
0(x, rp, i+ r)drp < 2 i=0
x
f
4'(x, gyp, i)drp. x
Therefore using Lemmas 4.28 and 4.36, we obtain i=N+I
fnN+l(xy)dY= > j4(xco.i)dco?
l i=N N
i=0
>-
i=N
2 i=0
>
2
Idetd(x,,e) f r+lI(v(x,,p))I sx
( 4 (x, p, i + r) dx Sx
as i=N 2
VOl(fi(Sx))
i=0
Vol(fN(Sx)),
where a E (0, 1) is such that for all (x, gyp) E A and any line L tangent to A at (x, rp) we have
detd(x.,,)fILI >a.
NowtakeC:=9. Let us explain now how Theorem 4.30 follows from Theorem 4.34, and Propositions 4.35 and 4.38. Suppose that h,op > 0, otherwise there is nothing to prove.
4.5 Mand's formula for convex billiards
101
By Theorem 4.34, given e > 0, there exists a basic set of saddle type Ae C A such that
hmp(Ae) > hn,p - e.
(4.15)
Take a periodic point (xo, (Po) E Ae and consider W'(xo, rpo). We claim that there exists a point (Xi, (Pi) E W''(xo, (oo) such that Sx, is a curve transverse to W'(xo, rpo) at (xi, (pl). Suppose that (xo, rpo) is not such a point, otherwise the claim would be obvious. Then St is tangent to W'(xo, rpo) at (xo, rpo). This means that V(xo, (po) = T(x,,,,p))W'(xo, (po). Set (xi, (pi) f(xo, (PO). By the twist property off (cf. 4.13), we have d(xo.W))f(V(xo,,po)) 0 V(x(, p() Since W'(xo, (po) is invariant under f, we see that d(xo.lp,)f (V (xo, rpo)) = T(xi,jp1)W5 (xo, (po),
and hence Sx, is a curve transverse to W'(xo, (po) at (xi, (pl ). This proves the claim.
Since the vertical segments Sx = (x] x [0, rI foliate the annulus A smoothly, there exists a neighborhood Uf of x1 in y such that for all x E UE, the curve Sx has a point of transverse intersection with W'(xo, (po). Therefore we can apply Proposition 4.35 to deduce that for all x E U£, we have
I
lim inf log Vol(f N(Sx)) ? hr(,p(Ae) N-.co N Combining the last inequality, Proposition 4.38 and (4.15) we obtain that, for all
XEUE,
liminf
I
N-.co N
log[ nN(x, y) dy > y
-e.
The proof of Theorem 4.30 is completed. Remark 4.39. Let f : S1 x [ 0, n] -+ SI x [0, n] be any CO0 map. Given a point
(x,(p)ES1 x(0,7r ]and NeV'set for 0
and N E Z+, we can consider as well a counting function nN(x, y) defined as the number of f -configurations of length < N that begin at x and end at y. If the map f satisfies a twist condition like 4.11 then Theorem 4.30 and Mafid's formula
hold for f. It is not necessary at all to assume that f arises from a convex billiard table.
Remark 4.40. Theorem 4.30 and all the arguments we did in this subsection hold just as well for geodesic flows on closed surfaces.
102
4. Mafl6's Formula for Geodesic Flows and Convex Billiards
4.6
Further results and problems on the subject
We describe in this section a few results and open problems that are connected with Man6's formula. This section contains no proofs. The inequality in Corollary 3.34 led Mar16 to ask in [Ma2] the following question:
Question L Is it true that lim
I
T-oo T
lognT(x, Y) = hrop
for a.e. (x, y) E M x M, at least generically and when M is a surface? It was shown in [BP1] that this is not the case. We constructed an open set of Coo metrics on S2 for which there exists a positive measure set U C M x M, such that for all (x, y) E U,
limsup 1 lognT(x, y) < hrop.
Too T
In the light of this example, the most that one can hope for is an affirmative answer to the following:
Question H. Is there always a set G C M x M of positive Lebesgue measure such that lim I lognT(x, Y) = hip T-oo T whenever (x, y) E G?
In [BP3] we give a positive answer to this question for a class of manifolds that includes COO surfaces, however in higher dimensions the problem remains open. The results in [BP3] generalize related results of M. Pollicott in [Po1 ]. Another related question is:
Question IIL Is it true that lim sup
1
Tyco T
lognT(x, Y) 5 hrop
whenever x and y are not conjugate?
In [BP2] we constructed a COO metric on the two sphere, arbitrarily close to the round metric, with a point p not conjugate to itself for which the number of
geodesic loops based at p with length 5 T grows as fast as we wish with T. In particular, it is possible for nT(p, p) to have superexponential growth. More precisely, suppose we are given a sequence n 1, n2, ... of positive integers. Then we found a C°O Riemannian metric g on S2 with a point p, such that p is not conjugate to itself along any geodesic and there are nk distinct closed geodesics
4.6 Further results and problems on the subject
103
of minimum period 2k-I 2n that pass through p for all integers k >_ 1. This ensures that
liminf
IT lognT(p, p) > liminf lognk. k-+oo 2k-1 .2,r I
Since the sequence nk was arbitrary, the right hand side of this inequality can be made as large as we wish, even infinite. These results give a fairly complete picture of what goes on for surfaces. However in higher dimensions there is still a lot to be done. Related to Question II we also have the following
Question IV. Is there always a set D C M of positive Lebesgue measure such that
Tin T log
JM nT
(x, y) dy = hrop
whenever x E D? Let us mention that the examples that we constructed in [BPI] have in fact the property that there exists a set V C M, of positive Lebesgue measure, such that for all x E V,
limsup 1 log f M T-ioo T
y)dy < htop.
AM
We note that a positive answer to Question II implies a positive answer to Ques-
tion IV. To see this we use a similar argument to the one we used to deduce Corollary 4.31 from Theorem 4.30. If we assume that the answer to Question II is positive, let us take as D the projection of G onto one factor of M x M and let m be the Lebesgue measure of D. For X E D we have by Jensen's inequality that
logJM nT(x, y) dy > log J nT(x, y) dy D
logm + 1 m
lognT(x, y)dy fD
If we divide by T the previous inequality and we take the lim inf as T we obtain using Fatou's lemma,
Ttr of
+oo,
r T
log fM
y) dy
AM
J
m D
lim of T 1ognT(x, y) dy = hrop.
On the other hand, since for all x E fM we always have (cf. Corollary 3.28) lim sup 1 T-+oo T
nT(x, y)dy < htop,
log
M
104
4. Mane's Formula for Geodesic Flows and Convex Billiards
we obtain a positive answer to Question IV. We observe that by a similar argument
to the previous one it follows that Mane's formula can be easily deduced from positive answers to Questions 11 or IV. Dealing with Question IV might be easier than dealing with Question II. Using Proposition 4.8 we can rephrase Question IV in the following equivalent way
Question V. Is there always a set D C M of positive Lebesgue measure such that
rT lim
I
T-.oo T
log
Jo
Vol(q' S:M)dr = htop
whenever x E D? We have seen in Theorem 3.32 that the inequality lim sup
I log J
T-.oo T
nT(x, y) dxdy < h,Op,
MxM
holds for C°O geodesic flows. We do not know any C'-counterexample (r < oo) to this inequality. The C°D regularity is required because of Yomdin's theorem. It is not hard to give examples of maps and submanifolds of class Cr for which the exponential growth rate of the submanifold is strictly bigger than the topological entropy of the map (cf. [Gr3]). One could also ask much harder questions like what sort of asymptotics or error terms arise for the counting functions. In this respect nothing is known in general. If we restrict ourselves to negatively curved manifolds, then the story is quite different. It is beyond the scope of these notes to survey this core of knowledge, so we shall only mention a few results. Margulis proved in [Mar] that when M is negatively curved there exists a constant C(x, y) such that lim
nr(x, y)
_1
C(x, y) eh'OPT
When M is locally symmetric C(x, y) is independent of x and y. For a proof of Margulis's result based on general properties of Anosov flows we refer to M. Pollicott's paper [Po2]. Quite recently, M. Pollicott and R. Sharp in [PS] estimated the error term for negatively curved surfaces. They showed that there exist a constant c < ht°p and D(x, y) > 0 such that I nT (x, y) - C(x, y) ehboPT I < D(x, y) eCT
4.6.1
Topological pressure
Most of what we did in this chapter can be extended for topological pressure.
4.6 Further results and problems on the subject
105
Let f : SM -+ R be a continuous function. We recall the definition of P(f) the topological pressure of the function f with respect to the geodesic flow d,, (cf. [W]). Given T > 0 and a point B E SM, set T
f f(01(B))dt.
fT(B)
0
Recall that a set E c SM is (T, e)-separated if given B(# BZ E E, there exists t E [0, T] for which the distance between 01 (01) and 4,(02) is at least E. Set,
r(T, e, f) := sup I E eh(8) : E is (T, e)-separated } , BEE
JJJ
r(e, f) := lim sup I log r(T, e, f ).
T-oo T
The topological pressure is defined to be
P(f) = l or(e, f). The topological entropy ht0p of ¢, is P(0). The variational principle for topological pressure says that
P(f) =
sup µEM($)
(hµ +
JSM
f d/.c
,
(4.16)
where M(¢) is the set of all ¢,-invariant Borel probability measures and h, is the entropy of the measure µ [W]. The study of the map f r-> P(f) is important since it determines the members of M(O) and when the entropy map u H h,, is upper semicontinuous on M(¢), the knowledge of P(f) for all f is equivalent to the knowledge of M(4,) and h, for all µ E M(4,) [W]. Also, the variational principle gives a natural way of selecting interesting members of M(4,). Using the formula for the topological pressure of a C°O dynamical system recently obtained by 0. Kozlovski in [Ko], it is possible to give a formula for P(f ) in terms of geodesic arcs between two points in M similar to Mafl's formula for h,°P.
Given x and y in M, let yx,y : [0, e(yx,y)] -- M be a unit speed geodesic arc joining x to y with length £(yx,y). We know that given T > 0, the set of all Yx.y with £(Yx,y) 5 T is finite and its cardinality is precisely nT(x, y), which is locally constant for an open full measure subset of M x M. Given any S > 0 and any T > S, let IY..r)
eo
FT,s(x, y) IY..y: T-8
I(Y,.r(r).Y:.r(t))dr
106
4. Mane's Formula for Geodesic Flows and Convex Billiards
Also let for any T > 0 crx.r)
eo
FT (x y) = (Y,,: t(Y,,)<-TI
Theorem 4.41([P4]). If g is of class C3 for any 3 > 0 we have
P(f) < Tim of
1
T
FT,&(x, y)dxdy.
log f
and if g is C°O,
P(f) = lim
I log J
T-.oo T
FT,a(x, y)dxdy.
MxM
If g is C°O and P (f) > Owe also have
P(f) = lim
I
T-oo T
logJ
fm xM
FT (x, y) dxdy.
When f == 0, we recover Mane's formula. Combining Theorem 4.41 with Lemma 3.33 we obtain:
Corollary 4.42. If g is C00, for almost every (x, y) E M x M we have, fot(Yx.y)
lira sup - log
T-.oo T
e
f(Yx.y(i).Yx.)(t))dt
< P(f ).
lrx.). T-d
Theorem 4.41 is particularly appealing when f is the restriction to SM of a R. In this case we have for a
1-form w. We think of w as a function w : TM CO0 metric g,
P(w) = lim
I
T-+oo T
logJ MxM
of
+
ly,.v:T-d
,
') dxdy. /J
The function HI (M, R) a [w] i-+ P(w) E R is closely related to Mather's function a : H I (M, R) r--). R [Mat]. Recall that a is given by
a(w)=-min{fT (Iu12/2-w)dµ:
AEMTM(O)(4.17)
M
where MT M (0) is the set of ¢i-invariant Borel probability measures of TM with compact support. Let SMa be the set given by those points (x, v) E TM such that I vi = a. It is possible to show that the support of a minimizing measure, that is,
4.6 Further results and problems on the subject
107
a measure for which the minimum in (4.17) is achieved, must be contained in the energy level SM (.) [DC1. Hence if µ is minimizing,
2a(w) = f wdµ. TM
Multiplication by the positive real a defines a natural bijection between the set of c6,-Borel probability measures of TM which are supported on SMa and M(4'). Hence
sup f wd1z.
2a(w) =
LEM(O)
M
Note that by Corollary 1.60 we have
From the variational principle (4.16) we obtain
2a(w) < P(m) < hrop + 2a(m). Combining this inequality with Theorem 4.41 yields
Corollary 4.43. If g is C°O and h,op = 0, 1
I[0) 11s = %5W(-(')) = Iim
T-.o° T
log
JMxM
dxdy. lYx.y:l(Y:.y)5T}
The corollary applies, for example, to a COO torus of revolution.
If (M, g) does not have conjugate points, one can show equality in Theorem 4.41 assuming that g is only of class C3. More precisely we have:
Theorem 4.44 ([P41). Suppose that g is of class C3 and (M, g) does not have conjugate points. Then, for any S > 0, we have
P(f) =
lim I log FT,s(x, y)dxdy. T-.oo T JMxM
If w is a closed /form, then for any x and y in M, we have
P(m) = lim
I
T-.oo T
log
E
ef)".y
(Yx,y: t(Y:.y)5T)
Finally, ifs E P, then for any x and y in M, we have hrop - s =
lim
I
T-+oo T
e-$I(Yx.y)
log (y,.,: T-S
108
4. Mafid's Formula for Geodesic Flows and Convex Billiards
Consider those s e C for which the following limit exists and is finite, e-sr(Y,.r) 1 dxdy.
178(s) := lim
T-.ooJMxM
J
We can regard >lg(s) as the "Poincard series" of the Riemannian metric g. Suppose that (x, y) is a pair of non-conjugate points and let exr<ex.y<...<ex.y<
be the lengths of all the geodesic arcs yx.y. Given s E C we set 00
a-al,
rig(s. X. Y) n=1
whenever the series converges. If x = y and z is any point in the universal covering M that projects to x, then rig (s, x, x) is exactly the Poincard series of the action of n1 (M) on M at the point z, provided that M has no conjugate points. One can show (cf. [P4]) that if g is C°O, then rig (s) is a holomorphic function on the half plane Re(s) > hl0p. If g is of class C3 and (M, g) does not have conjugate points, then for any x and y in M, s i-+ ng (s, x, y) is a holomorphic function
on the half plane Re(s) > hrp
5 Topological Entropy and Loop Space Homology
The chapter begins with the definitions of rationally elliptic and rationally hyperbolic manifolds and with a summary of various properties and characterizations of rationally elliptic manifolds. Afterwards, we discuss results of J.P. Serre [Se I] and M. Gromov [Grl] which allows us to relate the growth of nT(x, y) with the topology of M via Morse theory. Using these ideas we show the result in [P3] that says that if M is a closed manifold that fibres over a closed simply connected rationally hyperbolic manifold, then for any C°O Riemannian metric g on M, h,op(g) > 0. The more classical result of E. Dinaburg [D] which says that if Yri (M) grows
exponentially, then h1op(g) > 0 for any g, is also discussed here. The chapter finishes with various definitions of entropies of manifolds. We connect them with other notions, such as Gromov's minimal volume and simplicial volume (see the chain of inequalities (5.7)), and we propose various related problems.
5.1
Rationally elliptic and rationally hyperbolic manifolds
In what follows, X" will be a closed connected and simply connected n dimensional manifold. Definition 5.1. A manifold X is said to be rationally elliptic if the total rational homotopy tr,(X) ® Q is finite dimensional, i.e., there exists a positive integer io such that for all i > to, tri (X) ® Q = 0. The manifold X is said to be rationally hyperbolic if it is not rationally elliptic (cf. [FHT, FHTZ, G1111 and references therein).
5. Topological Entropy and Loop Space Homology
110
It was proved by Y. Feslix and S. Halperin [FH] that the class of n dimensional manifolds X" (or more generally the simply connected topological spaces of the rational homotopy type of a CW-complex) is divided into two subclasses: either
(a) ,ri (X) is finite for all i > 2n - 1, or (b) the integers pi = Fl i dim rrj(X) ® Q grow exponentially in i (i.e, there exist C > I and a positive integer k such that if i > k then pi ? C'). Therefore a manifold in the class (a) is rationally elliptic and a manifold in the class (b) is rationally hyperbolic. The "generic" manifold is rationally hyperbolic; rational ellipticity is a severely restrictive condition as we shall see below. Examples of rationally elliptic manifolds are simply connected homogeneous spaces [Se2], manifolds that admit a codimension one compact action [GH2], Dupin hypersurfaces [GH2] and any known manifold that admits a Riemannian metric of nonnegative sectional curvature. A conjecture attributed to R. Bott states that any compact simply connected manifold that admits a metric of nonnegative sectional curvature must be rationally elliptic (cf. [GHI]). If M is rationally elliptic, we have:
1. dim zr.(X) ®Q < n [FH]; 2. Ek> t 2k dim (7r2k (X) ®Q) < n [FH]; 3.
Lk> 1(2k -I) dim (a2k_ 1(X) ®Q) < 2n -I [FH];
4. dim H,(X,Q) < 2" [GH1, H1]; 5. the Euler-Poincare characteristic XX > 0 [H2];
6. XX > 0 if and only if Hi (X, Q) = 0 for all i odd [H2]. It is very easy to produce plenty of examples of rationally hyperbolic manifolds, as the following three lemmas show.
Lemma 5.2. Let W be any simply connected compact manifold that is not a rational homology sphere. Then the connected sum of a large enough number of copies of W must be rationally hyperbolic. Proof Note first that if W is not a rational homology sphere, then n := dim W > 4 and there exists io 0 0, 1, n - 1, n such that b := dim H;o(W, Q) 0 0
and dim Hi (W, Q) = 0 for 1 < i < io. By the Hurewicz isomorphism theorem b = dim ,r;o(M) ® Q. Write the connected sum W#W as U U V where U and
5.1 Rationally elliptic and rationally hyperbolic manifolds
III
V have the homotopy type of W with an n-disk removed and u n V has the homotopy type of an (n - 1)-sphere . The Mayer-Vietoris sequence gives
-> H;o(UnV)- H;o(U)®H;o(V) H;o(W#W)-+ Hi,_,(UnV)-> Since U n V has the homotopy type of an (n - 1)-sphere and removing an n-disk does not affect homology in dimensions < n - 1, we deduce that dim H10(W#W, Q) = 2b.
Iterating this argument we obtain dim H;o(#k W, Q) = kb.
Now take k such that kb > n. Then the connected sum of k copies of W must be rationally hyperbolic by property I above.
In the case of W = Sk x St, with k, I > 2, (Sk X S1)#(Sk x St) is already rationally hyperbolic [HL]. These manifolds admit an action of the group SO(k) x SO(l) with codimension two.
Lemma 5.3. Let X be any simply connected compact manifold with dimension four or five. If dim H2(X, Q) > 2, then X is rationally hyperbolic. Proof. Observe that by the Hurewicz isomorphism theorem
b := dim H2(X, Q) = dim (7r2(X) (9 Q). By property 2 above, if X is rationally elliptic, we get
2b
For example, connected sums of more than 3 copies of CP2 are rationally hyperbolic. However for 4-manifolds one can be much more precise.
Lemma 5.4 ([P5]). Let M be a compact smooth simply connected manifold of dimension 4. Then M is rationally elliptic if and only if M is homeomorphic to S4, CP2. S2 X S2, CP2#UP2 orCP2#CP2.
Proof. Suppose that M is rationally elliptic. By the previous lemma, b2 < 2. Since M is smooth, the Kirby-Siebenmann obstruction vanishes. Therefore by M. Freedman's theory [Fr], the homeomorphism type of M is completely determined
by the intersection form of M. It follows that if b2 = 0, M is homeomorphic
5. Topological Entropy and Loop Space Homology
112
to S4 and if b2 = 1, M is homeomorphic to CP2. When b2 = 2, the possible intersection forms are
(1 0)' (0 -1) and
I
p
1
These forms correspond to S2 X S2, CP2#CP2 and CP2#CP2 respectively. On the other hand S4, CP2 and S2 X S2 are homogeneous spaces and hence they are rationally elliptic. In [FHT2] it is shown that Poincar6 complexes M such that H*(M, Q) is generated by two elements are rationally elliptic, hence CP2#CP2
and CP2#CP are rationally elliptic.
5.1.1
The characteristic zero homology of H-spaces
In this subsection we shall summarize some relevant results about the homology of an H-space over a field of characteristic zero. It is well known that if X is a simply connected manifold, then nX the based loop space of X is a path connected H-space with unit and homotopy associative multiplication. Our main reference is the appendix of the paper by J. Milnor and J. Moore [MM). Let G be a path connected H-space with unit and homotopy associative multiplication. In this case H.(G,Q) is a Hopf algebra such that it is the universal enveloping algebra of the Lie algebra of its own primitive elements. Let 7rq (G) stand for the q-th homotopy group of G based at the identity element of C. The Whitehead product, [
,
] :7rp(G) ®7rq(G) -> 7rp+q(G).
is defined as follows. Let us represent x E 7rp(G) and y E 7rq(G) by continuous maps
f:Sp -+ G and g:Sq -+ G, respectively. Consider the map h : SP X Sq - G defined by
h(x, y) = f (x) * g(y) * f1(x) *
g-'(y),
where * denotes the product in the H-space G. Since the restriction of h to SP v Sq is homotopically trivial, h induces a map Sp+q = SP A Sq - G whose homotopy class is [x, y]. We recall that the wedge Sp V Sq is the union of SP and Sq with one point identified and the smashed product SP A Sq is Sp X Sq/Sp V Sq. The Whitehead product has the following properties: 1.
if x E 7rp(G) and y E 7rq(G), then [x, y] = (_l)pq+l[y, x);
2. if x E 7rp(G), y E 7rq(G). Z E 7r,(G), then
(- l)pr [x, [y, Z]] + (-1)pq [y, [Z, x]] + (_ l)qr [Z, [x, y]] = 0;
5.1 Rationally elliptic and rationally hyperbolic manifolds
113
3. if x E 7rp(G), y E 7rq(G) and
Xn : 7rn(G) - Hn(G), is the Hurewicz morphism, then Ap+q[X, Y] = Ap(X) Aq(Y) - (-l)pq)Lq(Y) Ap(X).
Consequently, the graded vector space 7r.(G) ® Q becomes a Lie algebra over Q and the induced morphism x : 7r.(G) ® Q -+ H.(G, Q) is a morphism of Lie algebras. Details concerning the preceding may be found in [Wit]. Let U(7r.(G) 0 Q) denote the universal enveloping Lie algebra of 7r. (G) 0 Q. We have the following result [MM]. Theorem 5.5. Let A : 7r. (G) (9 Q -)- H. (G, Q) be the Hurewicz morphism of Lie algebras. The induced morphism
1 : U (7f. (G) ®Q) - H. (G, Q), is an isomorphism of Hopf algebras. In particular, it follows from the theorem that A must be a monomorphism. Let X be a closed simply connected manifold with based loop space S2X. Since 7ri+1(X) = 7ri(S X), we obtain that for all nonnegative integers i dim 7ri+t (X) ®Q < bi (S2X, Q) := dim Hi (QX, Q).
(5.1)
For simply connected manifolds X, the Lie algebra 7r.(X) ® Q plays a role similar to the fundamental group for aspherical manifolds. Proposition 5.6. If X is rationally hyperbolic the integers
l.i = >bj(S2X,Q)
j<
grow exponentially in i.
Proof. If X is rationally hyperbolic, the integers pi = F j
On the other hand, if X is rationally elliptic, 7r.(X) 0 Q is finite dimensional and therefore H.(StX, Q) is the universal enveloping algebra of a finite dimensional Lie algebra. It is not hard to check that in this case µi grows at most polynomially in i. Therefore for a closed simply connected manifold, /y can only grow exponentially or polynomially and these two types of growth also characterize rationally elliptic and rationally hyperbolic manifolds. Another interesting characterization of rationally elliptic manifolds has been obtained by S. Halperin in [H3]. He shows that X is rationally elliptic if and only if 7ri(X) is finite for
114
5. 7bpological Entropy and Loop Space Homology
2n < i < 3n - 2. Since H.(92X, Q) is the universal enveloping algebra of 7r. (X) ® Q, the integers dim (trk(X) ® Q) for k < i can be computed from the integers bk(S2X, Q) fork < i, and conversely. Thus the calculation of bk(QX, Q)
for 1 < k < 3n - 2 suffices to determine whether X is rationally elliptic or hyperbolic.
5.1.2
The radius of convergence
Let Rx be the radius of convergence of the series
Edim(tri(X)0Q)t', i>_2
and let R be the radius of convergence of the Poincard series
Pax(t)
Ebk(2X,Q)t' i>o
The theorem of Poincard-Birkhoff-Witt (cf. [MM]) gives the following formal equality of series: (I
Pstx(t) _ i>1
+ (I - t2i)°2,
whereas := dim (ni+1(X)®Q). I. Babenko showed in [Ball that if X is rationally
hyperbolic, then Rx = R. Note that X is rationally hyperbolic if and only if
Rx<1. A manifold X is said to be formal if there exists a morphism of differential graded algebras from the minimal model of X to (H* (X, Q), 0) that induces an isomorphism in cohomology. The interest in this class of spaces lies in the fact that for them, all the rational homotopy invariants of X can be obtained from H'(X, Q). For example H.(CIX, Q) = Q). In [DGMS], it is shown that compact simply connected Kdhler manifolds are formal. In [NMJ is it shown that any manifold with dimension < 6 is formal. We have the following result of Y. Fdlix and J.C. Thomas [FI']:
neorem 5.7. Suppose that X is formal and rationally hyperbolic and let Px be the Poineard polynomial of H.(X, Q). Write Px(t) = ]11=1 (t - zi). Then R < min Izi 1. 1
For more on the dichotomy rationally elliptic/hyperbolic, we refer to [Fell, which contains a detailed account with proofs of many of the facts mentioned here.
5.2 Morse theory of the loop space
5.2
115
Morse theory of the loop space
Let M be a Riemannian manifold. Given a piecewise smooth path c : [a, b] - M we define the energy of c as
fb E(c) := 2
IcI2dt.
If L(c) denotes the length of c, then from the Cauchy-Schwarz inequality we get
L2(c) < 2(b - a)E(c), with equality if and only if c has constant speed (for example if c is a geodesic arc).
Suppose now that M is a complete Riemannian manifold and let N C M be a closed submanifold. Given Y E M we shall denote by S2(N, y) the space of piecewise smooth paths c : [0, 11 --+ M such that c(0) E N and c(l) = y. We shall regard the energy of a curve as a functional
E:12(N,y)-*R. It is well-known that the critical points of E are precisely the geodesic arcs leaving orthogonally from N and ending in y Also, if y is not a focal point of N, that is, if y is not a critical value of expl : TNl -+ M, then the critical points of E are nondegenerate in the sense of Morse theory (cf. [Mi 1, Sal). Given a positive real number T (bigger than 2d (N, y)), set
S2T (N, y) := E-1(-oo, T2/2],
S2T (N, y) := E-1(-oo, T2/2), The counting function nT(N, y) counts precisely the number of critical points of E that belong to S2T (N, y). We choose an r > 0 so that the following holds:
1. For any point x of the compact set B(y, T), B(x, r) is convex; 2. expl is a diffeomorphism restricted to an open set containing
((x, v) E TNl : x E N f1B(y, T), I v I < r). Next fix a subdivision A: 0 = to < tl < < tk = I such that ti+1 - t, < r2/ T2 (i = 0, .... k - I). Now we define a subset r2o(N, y) of E2T (N, y) as the set of c E f2T(N, y) such that: 1.
2.
cllt,.,,+, l are minimal geodesics for i = I , ... , k - 1; is a minimal geodesic joining N to c(t1).
116
5. Topological Entropy and Loop Space Homology
We define S2o-(N, y) in a similar way. It can be shown that the set OAT (N, y) (no-(N, y)) is a strong deformation retract of S2T(N, y) (S2T-(N, y)). Moreover, QTo-(N, y) can be given the structure of a smooth finite dimensional manifold such that E restricted to QA (N, y) is a proper smooth function on it whose critical points are precisely the geodesics leaving orthogonally from N and ending in y with length < T. If y is not a focal point of N, then E restricted to SIT- (N, y) is a Morse function (cf. [MiI, Sa]). Hence the Morse inequalities imply that if y is not a focal point of M, then,
E bi (QT-(N, y)) < nT(N, y),
(5.2)
i>o
where bi (QT-(N, y)) denotes the i-th Betti number of c2T-(N, y) with respect to a given coefficient field. Finally, we recall the "fundamental theorem of Morse theory" as presented by Milnor in [Mi I]. We state it for the special case in which N reduces to a point x.
Theorem 5.8. Let M be a complete Riemannian manifold and let x and y be two points in M which are not conjugate. Then 0 (x, y) has the homotopy type of a countable CW-complex which contains one cell of dimension X for each geodesic from x to y of index x.
5.2.1
Serre's theorem
In this subsection, we shall sketch a proof of the following result of J.P. Sere [Sel] which is the oldest giving some information on the growth of nT(x, y). Theorem 5.9. Let M be a complete Riemannian manifold and let x and y be two points in M which are not conjugate. If M is noncontractible (in particular if M is compact) then
lim nT(x, y) = oo. T-oo
Proof. Suppose that limT,0 nT(x, y) < oo, that is, there are finitely many geodesics connecting x and y. Then, by Theorem 5.8, S2 (x, y) has the homotopy type of a finite CW-complex; in particular, S2(x, y) has finitely many connected components and hence M has a finite fundamental group and without loss of generality we can assume that M is simply connected. Also, there exists a positive integer io such that for all i > io, bi (S2(x, y)) = 0. But if M is non-contractible, there exists an integer k > 2 such that rrk(M) is not trivial. Serre shows in [Sell that if rrk(M) is not trivial for some k > 2, then the sequence (bi (12(x, y))} contains infinitely many nonzero elements for some coefficient field, thus obtaining a contradiction.
5.2 Morse theory of the loop space
5.2.2
117
Gromov's theorem
In this subsection we shall prove the following result due to M. Gromov [Gr11.
Theorem 5.10. Given a metric g on a simply connected closed manifold X, there exists a constant C1 > 0 such that given any pair of points x and y in X and any positive integer i, any element in Hi (f2 (X, x, y)) can be represented by a cycle whose image lies in QCt: (X, x, y). Gromov's original proof in [Grl J is very short. A more detailed proof is given in [Gr2, p. 1021 but it seems to contain some mistakes. The proof we present here is taken from 1P3]. Proof. Let (Va) be a finite covering of X by convex open sets. Recall that a set is convex if any two points in it are joined by a unique geodesic that stays in the set. Let T be a triangulation of X. For each point p E X, let T (p) be the (closed) face
of T of minimum dimension that contains p (so T(p) = (p} if p is a vertex of T) and let 0(p) be the union of all maximal simplices of T that contain p. Note that if q is near p, O(q) C 0(p). We choose a triangulation T of X that is fine enough so that for all p in X, 0(p) lies in one of the Va. Given a positive integer k, we define open subsets f k(X, x, y) of f2(X, x, v) in the following way. We shall say that w E f2k (X, x, y) if for each integer j = 1, 2, ... , 2k, the image under w of each subinterval [(j - ])/2k, j/2k1 lies in one of the sets V. and
0 (w((j - 1)/2*))) U 0 (w(j/2k)) lies in the same Va.
Let Bk(X, x, y) C ftk(X, x, y) be the space of broken geodesics y such that y E Qk(X, x, y) and the restriction of y to each subinterval
[(j - l)/2k,j/2k] is a geodesic parameterized at constant speed. Each y E Bk(X, x, y) determines a sequence (pj = y(j/2k)) which has the properties:
(1)po=xandp2* =y; (2) O (pj - r) U O (pj) lies in a single Va for each j = 1, 2, ... , 2k. Conversely, any sequence (p j ), 0 < j < 2k with these two properties determines a broken geodesic in Bk(X, x, y). Moreover the correspondence between broken geodesics in Bk(X, x, y) and sequences of points is bijective (because the parameterization will distinguish different geodesics with the same trace). Observe that this correspondence induces on Bk(X,.r, y) a cell decomposition: a cell that contains y is given by the cartesian product
T(pi) x T(P2) x ... x T(P2*-1) Thus we can think of Bk(X,x, y) as a finite cell complex. Note that the methods in Milnor's book [Mill show that Bk(X,x, y) is a deformation retract of 12k(X, x, Y)
118
5. Topological Entropy and Loop Space Homology
Given any two vertices in the triangulation, we can connect them by a unique minimizing geodesic arc. The union of these arcs forms a one-dimensional cycle E homotopic to the one-skeleton of the triangulation. Since X is simply connected, there exists a continuous map g : X -> X such that g collapses E to a point and g is homotopic to the identity. Choose a smooth map g : M -r M that is sufficiently CO close to g so that g and g are homotopic
and g maps E into a contractible neighborhood. Let Fr : M - M be a smooth family of smooth maps such that F0 = id and Fl maps i(E) to a point. Set f := Flog. The map f is smooth, it collapses E to a point and it is smoothly homotopic to the identity. Observe that f naturally induces a map
f :0(X,x,Y)-> n(X,f(x),f(Y)) We need the following lemma.
Lemma 5.11. There exists a constant C1 > 0 such that for any integer k > 1, we have
f (i-skeleton of Bk(X, x, y)) C S2c"(X, f (x), f(y)), for all i < dim Bk(X, x, y). Proof. Consider a cell
T(pl) x T(p2) x ... x T(p2k_t) with dimension i < dim Bk(X, x, y). Take a path y in this cell. Then y is a broken geodesic, each leg of which lies in one of the sets Va. Since f sends E to a point we observe that
E(f (y)) < K2 d2 N(y)/2, where K := maxxEx IIdxf II, d is the maximum of the g-diameters of all the convex open sets Va, and N(y) is the number of legs of y that do not lie in E. Since E is made up of geodesic segments, the leg of the broken geodesic y that
joins T (pj) to T (pi+i) must lie in E if 1 < j < 2k - I and dim T (pj) = dim T (pj+i) = 0. Thus the only legs of y that could fail to lie in E are the initial leg, which starts at x, the final leg, which ends at y, and legs that begin or end in a T (pj) with nonzero dimension. We see that
N(y) < 2 + 2i < 4i. If we set Ci = 2Kd, we obtain
E (f(y)) < Cji/2 < Cji2/2, which concludes the proof of the lemma.
0
5.3 Topological conditions that ensure positive entropy
119
We shall show that for all x and y in X, any q E Hi (9 (X, f (x), f (y)) can be represented by a cycle whose image lies in nCii(X, f (x), f (y)), where C1 is the constant given by Lemma 5.11. This implies the theorem since f is a surjective map (it has degree one since it is homotopic to the identity). Observe that c21(X, x, y) C n2 (X, x, y) C s23 (X, X. y) C ... and cc
11(X, x, Y) = U S2k(X, X, Y) k=1
Let q E Hi (Q (X, f (x), f (y)) be given. Since f is homotopic to the identity, there exists µ E Hi (n (X, x, y)) such that f.(µ) = q. If we are given a cycle that represents .t, this cycle will have an image that lies in S2k(X, x, y) for some k. Retract 12k(X, x, y) onto Bk(X, x, y). Then our cycle can be moved by a homotopy into the i-skeleton of Bk(X, x, y). By Lemma 5.11, f maps all points in the i-skeleton of Bk(X, x, y) to points in 92CIi(X, f (x), f (y)) and hence j. (A) = q can be represented by a cycle whose image lies in Q C1 i (X, f (x), f (y)).
5.3
Topological conditions that ensure positive entropy
5.3.1
Growth of finitely generated groups
0
Let r be a finitely generated group and let {$1, ... , gp) be a set of generators for the group r. For each positive integer s let y(s) be the number of distinct group elements which can be expressed as words of length < s in the specified generators and their inverses. Exercise 5.12. Show that if r is free abelian of rank two with specified generators
x and y, then y(s) = 2s2 + 2s + 1. Lemma 5.13. Let {g 1, ... , gp) and let {h 1, ... , by } be two different sets of gen-
erators for the same group and let y(s) and y'(t) be the corresponding growth functions. Then there exist positive constants k and k' so that y(t) < y(kt) for all
t and y(s) < y'(k's) for all s. Proof. If k is large enough so that each hj can be expressed as a word of length
< k in the generators gi and g, 1, then the required inequality y(t) $ y(kt) is clearly satisfied.
0
Lemma 5.14. s log y(s) converges to a limit v as s -> oo and the property of v > 0 is independent of the specified set of generators.
120
5. Topological Entropy and Loop Space Homology
Proof. Note that
Y(s+t)
y(s) < y(kt) < Y(t)k <
Y(1)1+sll
implies that lim sup s-+'
I
logy(s) <
S
I
t
logy(t).
Hence
lim ups log Y(s) < inf (i log Y(t)
< tim inf s log Y(s),
and the sequence converges.
Definition 5.15. We shall call v the exponential growth rate of I' with respect to the specified set of generators. We shall say that the group r has exponential
growth ifv>0.
5.3.2
Dinaburg's Theorem
Let M be a closed Riemannian manifold and let M be its universal covering endowed with the induced metric.
Lemma 5.16. Let N be a compact fundamental domain for the action of it, (M) on M. Consider the set F given by those a e trI (M) such that the translate aN intersects N. Then F is a finite set of generators for nt (M).
Proof. Let b > 0 be the minimum of d(aN, N) as aN ranges over all translates of N which do not intersect N. Fix xp E M. We shall show that if d(xo, aN) <
bt + e for some positive integer t and 0 < s < b, then a can be expressed as a t-fold product a = #i ... 6, with fi , ... , fir E F. Choosey E aN with d(xo, y) < bt + e, and choose points y l, ... , y,+i = y along a minimal geodesic from xo toy so that d(xo, yi) < e and d (y; , yi+i) < b. Each y; belongs to some translate &N where we can choose Si to be I and Sr+1 to be a. Let f; = S IS;+i so that #i ... = a. Since the two points Sf 1y; and Si 1y;+i have distance < b, and belong to N and to 48 N respectively, it follows from the definitions of b and F that f; E F. This proves the lemma. 0
Let B(x, r) be the ball with center x and radius r in M. Now set V (x, r) :_ Vol(B(x, r)). Recall from Proposition 3.35 that r log V (x, r) converges to a limit
x as r - oo and x is independent of x. Let a be the diameter of the compact fundamental domain N.
5.3 Topological conditions that ensure positive entropy
121
Corollary 5.17. A > v/2a, where v is the exponential growth rate of n1 (M) with respect to F.
Proof. Since d(xo, fixo) < 2a for all 14 E F, it follows that B(xo, gas) contains at least y(s) points of the form g(xo), g E n1(M). If e > 0 is chosen so small that g(B(xo, e)) is disjoint from B(xo, e) for all g different from the identity, then B(xo, gas + e) contains at least y(s) disjoint sets of the form g(B(xo, e)), g E ir1(M). Hence y(s)V (xo, e) < V(xo, 2as + e) for all positive integers s and we conclude that v < 2aA.
Theorem 5.18 (Dinaburg's theorem [D]).
hrop(g) > v/2a. Proof. It follows immediately from the corollary and Manning's inequality in Chapter 3.
From Dinaburg's theorem, it follows that if the fundamental group of M has exponential growth, then for any Riemannian metric g on M, htpp(g) > 0. In [Mi2], Milnor shows that the fundamental group of any manifold that admits a metric of negative sectional curvature has exponential growth.
Exercise 5.19. Show that given x E M
A = lim I log #(a E 7r1(M) : a(x) E B(x, r)}. r+°o r Show that .X > 0 if and only if n1 (M) has exponential growth.
5.3.3 Arbitrary fundamental group In the previous subsection we saw how a topological condition, that is, exponential growth of n1(M), forces htop(g) to be positive for any metric g on M. What happens for manifolds for which v = 0? In particular, what happens for simply connected compact manifolds? We shall see in this subsection that the right thing to look at in this case is the growth of the loop space homology.
Theorem 5.20 ([P3]). Suppose M is the total space of a fibre bundle with base a compact simply connected manifold whose loop space homology grows exponentially for a given coefficient field (e.g., a rationally hyperbolic base). Then for any C°D Riemannian metric on M, htop(g) > 0. We shall describe now some consequences of Theorem 5.20.
Corollary 5.21. Let X be a rationally hyperbolic manifold Then for any C°O Riemannian metric on X, h,op(g) > 0. Equivalently, if htop(g) = O for some C00 Riemannian metric g on X, then M is rationally elliptic.
122
5. Topological Entropy and Loop Space Homology
Proof. It follows from Theorem 5.20 and Proposition 5.6. This corollary was pointed out by Gromov in [Gr3, Section 2.7].
Recall that if r is a finitely presented group, then there exists a closed 4-manifold N so that tri (N) = I' [Msy]. Therefore by considering the 8-manifold M = ((S2 x S2)#(S2 X S2)) x N, Theorem 5.20 implies:
Corollary 5.22. Let r be any finitely presented group. Then there exists a closed 8-manifold M with trt (M) = r, so that for any C°O Riemannian metric g on M, htop(g) > 0. Remark 5.23. Note that if M is simply connected and it fibres over a manifold X as in Theorem 5.20, then the loop space homology of M also grows exponentially for some coefficient field. This follows from the spectral sequence of the fibration
S2M -+ S2 (N, y) - N, where N denotes the fibre of M -> X, and 12 (N, y) is the space of paths leaving from a fixed fibre and ending in a fixed point y (if the homology of the loop space of X grows exponentially, the homology of n (N, y) also grows exponentially).
5.3.4
Proof of Theorem 5.20
Let N denote the fibre of the fibration n : M -> X. For a point x E X, let Nx = rr-' (x) = N. For a point x E X and a point p E M, let S2(Nx, p) denote as before the space of piecewise smooth paths defined on [0, 1] leaving from Nx
and ending in p. Also, for x, y E X, let S2(X, x, y) be the space of piecewise smooth paths in X defined on [0, 1] from x to y. Recall that the homotopy type of f2 (X, x, y) is independent of the points x and y. Indeed, if x' and y' are two other points in X and if a and P are paths joining x' to x and y to y' respectively, then the map f2 (X, x, y) -+ SZ(X, x', y'),
yHa*y*0 is a homotopy equivalence. Now let g be an arbitrary C°O Riemannian metric on M and let Eg(a) denote the energy of a path in M. As usual, S2T (Nx, p) and S2T-(N,,, p) will denote the subspaces of paths a such that E8 (a) < T2/2 and Eg (a) < T2/2 respectively. The theorem will be a consequence of the following proposition.
Proposition 5.24. There exists a constant C > 0 such that given any point p E M and given any positive integer i, we have dim H1(c2X) < dim H1(12T (N:, p)),
for any T > Ci.
5.3 Topological conditions that ensure positive entropy
Proof. Observe that n : M - X naturally induces a map fr
:
123
92(Nx, p) ->
92 (X, x, r(p)). To prove the proposition we need the following result:
Proposition 5.25. Let g and gx be metrics on M and X respectively. For any p E M and x E X. there is a continuous map 0 : 52(X, x, 7r (p)) -+ 0(Nx, p)
such that n o 0 is the identity and there is a constant B > 0 such that
Eg(9(w)) 5 BEgx(w) for any w E S2 (X, x, ;r (p)). The constant B depends only on g and gX and is independent of p, x and w. Proof. Choose a metric gm on M such that
tr : (M, gm) -> (X, gx) is a Riemannian submersion. The Riemannian metric gm naturally defines a hor-
izontal distribution and the horizontal lift of a piecewise smooth path on X is uniquely defined provided an initial point in M is chosen. Therefore there exists a continuous map 9 : 52(X, x, 7r (p)) -+ S2(Nx, p) with n o 0 = id. Since M is compact, there exists a constant B > 0 such that if a is any piecewise smooth path in M, then
Eg(a) 5 B EgM(a). Now let w be a path in Q (X, x, tr(p)). Then
Eg (9(w)) 5 B EgM (9(w)) = B Egx(w),
since n : (M, gm) -- (X, gX) is a Riemannian submersion and 0 is the associated horizontal lift.
Let us complete the proof of Proposition 5.24. Take p in M, and let y = tr(p). Let C = Ci f, where Ct is the constant given by Theorem 5.10 and B is given by Proposition 5.25. Set A = S2T-(Nx, p) for T > Ci. The map *A = irlA : A --+ 92 (X, x, y)
naturally induces a morphism
(fA) : Hi (A) -- Hi(S2(X, x, y)). To prove the proposition it suffices to show that this morphism is onto. Take q E Hi (52(X, x, y)). By Theorem 5.10, q can be represented by a cycle whose image lies in t2c'' (X, x, y). By Proposition 5.25, 8 0 is a cycle whose image lies
124
5. Topological Entropy and Loop Space Homology
in Oct Ii' (Nx, p) C OT- (Nx, p) and therefore it defines an element in Hi (A). But (NA).(0.(q)) = q since n o 0 = id which concludes the proof of the proposition.
Let g be an arbitrary Riemannian metric on M, and let p and x be points such that p is not a focal point of the fibre Nx. Let us explain now how Theorem 5.20
can be derived from Proposition 5.24. If the point p is not a focal point with respect to Nx, then inequality (5.2) says that
E bi (S2T (Nx. p))
nr(N1, p).
i>0
Take T = Ck for any positive integer k and combine the last inequality with Proposition 5.24 to obtain k-1
Edim Hi(c2X)
nck(Nx, P)
1=0
Recall that nT(N1, p) is finite for almost all p by Sard's theorem. Integrating the last inequality with respect to p, we get k-1
E dim Hi (S2X) < i=0
Vol 8I(M)
f
nck(Nx, P)dp.
(5.3)
Define
p := lim sup
k-.oo k
dim Hi(OX)
log
.
i=0
Then the last inequality and Theorem 3.27 yield
p < C htop(g)
(5.4)
If for some coefficient field p > 0, then htop(g) > 0, as desired. Let us derive now some consequences of inequality (5.4). Suppose that X is rationally hyperbolic and let R be the radius of convergence of the Poincard series PnX. Then R < I (recall the discussion in Subsection 5.1.2) and inequality (5.4) implies
-log R < C htop(g)
(5.5)
Combining this inequality with Theorem 5.7, we obtain:
Corollary 5.26. Suppose that X is rationally hyperbolic and formal. Write PX (t) _ r["=1 (t - zi). Given a C°° Riemannian metric g on M, we have C(g) htop(g) ? 1ma ^(- log Izi 1). where C(g) is the constant from the proof of Theorem 5.20.
5.3 Topological conditions that ensure positive entropy
125
For a 4-dimensional simply connected manifold X, we have
Px(t) = 1 + b2t2 +
t4,
where b2 is the second Betti number. We know from the first section of this chapter
that X is formal and if b2 > 2, X is rationally hyperbolic. From this observation and the last corollary, we get:
Corollary 5.27. Let X be a compact simply connected 4-manifold with second Betti number b2 > 2. Then, for any C°O Riemannian metric g on M. we have
b2 -
I
-2 log
C(g)hrop(g)
62-4 2
The last two corollaries were pointed out by I. Babenko in [Ba2]. In fact, he obtains in [Ba2] more precise estimates on the constant C(g) as well as other related results.
Remark 5.28. Observe that in the special case in which M coincides with X inequality (5.3) gives k-1
I
Edim Hi(c2M) < Volg(M) i-0
fAtnCk(x,Y)dy.
This can be used to show that compact simply connected homogeneous spaces are rationally elliptic; a result obtained by Serre in [Se2]. Indeed using Proposition 3.11 in Chapter 3 (endow the homogeneous space with a normal metric) we get that JM
nCk (x, Y) dY
grows at most like (Ck)t" M. Hence k-1
E dim H; (S2M)
i-0 grows at most like (Ck)thm M and M is rationally elliptic.
Remark 5.29. For manifolds with an infinite fundamental group, the growth of the homology of any of the components of the loop space does not seem to relate clearly with the growth of the average number of geodesics between two points on the manifold and the arguments we gave to prove Theorem 5.10 are no longer valid.
126
5. Topological Entropy and Loop Space Homology
5.4
Entropies of manifolds
Let M" be a closed connected manifold of dimension n. Given a smooth Riemannian metric g, let htop(g) be, as always, the topological entropy of the geodesic flow of g. Let A(g) be exponential growth rate of volume of balls in the universal covering of M that we introduced in Chapter 3 in the section about Manning's inequality. We would like to study the functionals g -* htop(g) and g t-+ A(g) in the space of all Riemannian metrics, in particular the infimums of such functionals. In order to make the problem meaningful, we need to normalize the functionals appropriately, since by multiplying g by a constant factor we can make htop(g) and A(g) go to zero or to infinity. One possibility is to restrict the functionals to the space of metrics with volume one. Another possibility is to restrict the functionals to the space of metrics with diameter one. This motivates the introduction of the following entropies of manifolds:
hv(M) :=
inf
htop(g).
(g: VOlg(M)=1)
AV(M) :=
inf (g: Vo4 (M)=1)
hd(M) :
(
(g: d
Ad(M) :=
A(g);
M)=t) htop(g):
inf (g: dg(M)=1)
A(g)
These four numbers are differential invariants of M and by Manning's inequal-
ity we have that hv(M) > Av(M) and hd(M) > Ad(M). If 7r1(M) has subexponential growth, then by Exercise 5.19, A(g) = 0 for all g and thus Av(M) = Ad(M) = 0. If nl(M) has exponential growth, Dinaburg's theorem ensures that
hd(M)>Ad(M)>0. We shall say that a metric g with volume one is entropy minimizing if It v (M) _
htop(g)
5.4.1
Simplicial volume
The definitions and results in this subsection are due to Gromov [Gr4]. Let M be a closed manifold. Let us denote by C. the real chain complex of M: a chain c E C. is a finite combination E; r;a; of singular simplices ai in M with real coefficients ri. Define the simplicial 11 -norm in C. by setting Ilcll norm gives rise to a pseudo-norm on the homology H.(M, R) as
Iri I. This
Ilall = inf llzll, where z runs over all singular cycles representing Of E H.(M, R). When M is orientable, define the simplicial volume of M, IIMII, as the simplicial norm of
5.4 Entropies of manifolds
127
the fundamental class. When M is not orientable we pass to a double covering Mo and set IIMII = IlMoll/2. The following properties are shown by Gromov in [Gr4]: 1.
if f : M -- M' is a map of degree d. then I I M I I ? d IIMII. Hence if M admits a self-mapping of degree > 2. then I I M I I = 0. In particular, spheres and tori have zero simplicial volume,
2. if M is a closed oriented surface with XM < 0, then IIMII = 21XMI In fact, any manifold with negative sectional curvature has IIMII > 0;
3. if M1 and M2 are closed manifolds, then there exists a constant C > 0 which depends only on dim(M1 x M2) such that CIIMIIIIIM211 >_ IIM1 X M211 >_ C-11IM1111IM211;
4. for n > 3. the connected sum of n-dimensional manifolds satisfy IIM1#M211= 11M1 11 + IIM211;
Gromov also shows in [Gr4j that
I' (g) Volg(M) > Cnln- IIMII,
(5.6)
where
C"=I
(2) /
rIn2 1
I.
Hence
[Xv(M)]" ? C"I n ! Will
and thus hv(M) > ).v(M) > 0 if IIMII # 0. In particular hv(M) > 0 if M admits a Riemannian metric with negative sectional curvature.
5.4.2
Minimal volume
The minimal volume MinVol(M) of a manifold M is defined as the infimum of Volg(M) over all metrics g such that the sectional curvature K. of g satisfies I Kg I < 1. This differential invariant has been introduced by M. Gromov in [Gr41.
In this subsection we shall relate hv(M) and MinVol(M), but first we shall describe a few properties of the minimal volume. Exercise 5.30. Using the Gauss-Bonnet theorem show that the minimal volume of closed connected surfaces M is given by
MinVol(M) =2nlXM1.
128
5. Topological Entropy and Loop Space Homology
In [Gr4] Gromov gave several examples of manifolds with MinVol(M)= 0. We list them (we assume that M is closed):
I. M admits a flat metric; 2. M admits a locally free S' -action. In particular MinVol(M x S1) = 0, and also MinVol(S2n+') = 0; 3. M admits an F-structure (cf. [Gr4]);
4. M is the product of an arbitrary Mo by a manifold in one of the above examples 1-3;
5. M is odd dimensional and diffeomorphic to a connected sum of manifolds of example 4. For instance connected sums of odd dimensional tori have zero minimal volume (notice that such connected sums admit no nontrivial circle actions) or manifolds like (stn+t x S2k) # (S2-+l x SZ.t)
with n, k > 1. There is no example known of a simply connected odd dimensional manifold M with MinVol(M) i4 0.
Lemma 5.31. Let J be a normal Jacobi field along a geodesic Ye with J(O) = 0 and 1.1(0)1 = 1. Set R(t) = R(0,9), where R(0,0) is curvature operator at 0,0. Then for any l > Owe have
rJ(T)I2 + 1J(T)12 <
1 + 1/l exp 20
J
IIR(t) - l1dlI dtl
/
\\\
.
Proof. Let us consider the function
J(t)) +(J(t), J(t)).
Y (t) = I
Differentiating with respect to t and using the Jacobi equation i(t) = -R(t)J(t) we obtain:
2(i,(lld-R)J). Hence Y < IYI <
IIR - lldlly.
Using that y(O) = 1, we obtain right away that
I + 1/l exp
From the lemma we get immediately:
r
\2f Jo
\
IIR(t) - lldlldt 1 .
0
5.4 Entropies of manifolds
129
Corollary 5.32. For any l > 0, we have I det(de¢Tlv(e))I < (1 + 111)T exp
fT
(_:!.
2f Jo
IIR(t) - ildll dt)
Proposition 5.33. For any Riemannian metric g of class C3 and any !
>/>
.
0, we
have
r
/
I
hfop(g) < liminf 1 log IIR(0,0) -11d11 dtdB. exp I n T-oo T JSM ` 2T fo Proof. From Chapter 4 we know that (cf. Remark 4.20)
hfop(g) < liminf 1 log J I det(deOTI vtel)I dB. T-.oc T SM To prove the proposition it suffices to combine this inequality with the last corollary.
Proposition 5.34 (cf. [Man2]). Let L2 be an upper bound for the modulus of the sectional curvature of M. Then htop(g) 5 (n - 1)L.
Proof For all 0 E SM we have
IIR(O) - L2ldil < 2L2, and the proposition follows from Proposition 5.33 taking l = L2. Proposition 5.35.
[hv(M)J" < MinVol(M).
Proof. Take a sequence of metrics 8k such that Vol(gk) - MinVol(M) and I Kgk I < 1. Set 8k
[Vol(gk)]21n8k
Clearly Vol(gk) = I and K8k = [Vol(gk)]2j"Kgk. Hence IKgklmax 5 [Vol(8k)]21".
130
5. Topological Entropy and Loop Space Homology
Using Proposition 5.34 we obtain hrop(gk) <
IKgkImax < [Vol(gk)]11".
Thus
hv(M) < [Vol(gk)]Il" -+ [MinVol(M)]'/".
Corollary 5.36. hv((S2n+t x S2k) # (S2n+1
x S2k) ) = 0.
Proof. Since
MinVol( (stn+i x s2k) # (S2n+I x S2k)) = 0, the corollary follows from the proposition.
Remark 5.37. Recall that (S2n+1 X S2k) # (S2n+t x S2k) is a rationally hyperbolic manifold, hence for any C°O metric g, hrop(g) > 0, however the corollary shows that there exists a sequence of metrics gn with volume one such that hrop(gn) --+ 0 as n -> oo. In particular the constant C(gn) from inequality (5.4) must tend to infinity. Note that the manifold (S2n+t X Su`) # (S2n+I X S2k) does not admit entropy minimizing metrics.
Observe that for the two-sphere, hv(S2) = 0, but MinVol(S2) = 47r.
5.5
Further results and problems on the subject
We begin with the following two questions:
Question I. Does there exist a manifold M whose fundamental group has subexponential growth and hd(M) > 0? More specifically, does there exist a simply connected manifold M with hd(M) > 0? Question H. Does there exist a manifold M whose fundamental group has subexponential growth and hv(M) > 0? More specifically, does there exist a simply connected manifold M with hv(M) > 0? Suppose that M is a simply connected rationally hyperbolic manifold. Inequal-
ity (5.4) says that 0 < p < C(g) hrop(g), where C(g) is a geometric constant. When d8 (M) = 1, C(g) is given (up to a constant factor) by the Lipschitz constant of a map that contracts a 1-dimensional cycle made up of geodesic arcs to a point. However, it could happen that sup(g: d,(M)=1) C(g) = oo. We saw an
5.5 Further results and problems on the subject
131
example in Remark 5.37 in which M is rationally hyperbolic, hv(M) = 0 and sup(g: Volg(M)=II C(g) = 00. We collect the results from the previous section in the following chain of inequalities:
Cnn
IIMII
(5.7)
[Av(M)]" < [hv(M)]" < MinVol(M).
The only known manifolds with hv(M) > 0 are manifolds with IIMII 0. For these manifolds ni(M) has exponential growth. When M carries a locally symmetric metric of negative curvature we have the celebrated results of G. Besson, G. Courtois and S. Gallot:
Theorem 5.38 ([BCG1]). Suppose that M carries a locally symmetric metric go of negative curvature that we normalize to have volume one. Then
hv(M) = hrnp(go) = A(go) and if there exists a metric g with volume one for which
h,0p(go) then
g is isometric to go when n > 3 or g has constant negative Gaussian curvature when n = 2. Related to the chain of inequalities (5.7) we could also ask:
Question III. Does there exist a manifold M with Av(M) < h v(M)? Finally we ask in general: Question IV. When do entropy minimizing metrics exist? If they exist describe the moduli space of such metrics.
There is a hope of answering this question for simply connected 4-manifolds.
We conjecture that for any simply connected 4-manifold by = 0. In [P5] we showed that there exist metrics with zero topological entropy on S4, CP2, S2 x S2,
CP2#CP2 or CP2#CP2. In fact, the proof of the existence of such metrics is quite straightforward for S4, CP2, SZ X S2, CPZ#CPZ but non-trivial for CP2#CP2. On account of Corollary 5.21 and Lemma 5.4, if our conjecture is true, then the above are the only five manifolds for which there exist entropy minimizing metrics. The problem of describing the moduli space of metrics with zero topological entropy seems formidable even for S2.
Hints and Answers
Hints and/or answers are given for some of the exercises.
1.6. Prove first that for any t E R, dto;
0 using that for any t and s in R,
SOr+s = rG141.
1.8. Show that dax is nondegenerate on the kernel of a, if (dot)" is nonzero on the kernel of ax. If you have difficulties with this see [MS, Proposition 3.4.1 ].
1.18. Prove that a straight line in a tangent space T.,M is a geodesic of the Sasaki metric.
1.26. Let 0 = (x, v). Let a j : (-e, e) - M be an integral curve of Ej through x, and let Vj (t) be the parallel transport of v along a j. Consider the map
Z;j : (-s, e) x (-e, e) -+ TM given by Z; j (t, s) = (a; (t), sEj (a; (t)) + V; (t)). Prove that
az;j
(t s) = Yi0Z;;(t s) , ,
as
az;; at
'
(0, s) = x;ozij (0, s);
134
Hints and Answers
azi; at
(r, 0) = xiozi;(r, 0).
Hence
[Xi, Y;)(f)(B) = xiY;(f)(B) - YA(f)(B) = a a(f°zi')(o,o)- a a(f°zi')(0,0) at
as
as
at
=0. 1.28. For part 5, show that the restriction of no to TOSM has a unique null direction spanned by G(O).
1.32. Since n : SM - M is a Riemannian submersion using Corollary 5.7 in [Sal we can write (assuming that M has finite volume)
Vol(SM) =
=
I
/
du 1 dx = Vol(S"')Vol(M). ///
1.33. Take normal coordinates around a point in M and lift them to TM. Express a in these coordinates. Compute Of A (da)"-t and compare it with the volume form of the Sasaki metric in the same coordinate system.
1.41. If the geodesic flow 0, is an isometry of the Sasaki metric for all t E R, then the map t i-+ (J(t), J(t)) + (J(t), i(t)) is constant for any Jacobi field J. Differentiate with respect tot and use the Jacobi equation to show that M has constant sectional curvature 1.
1.43. Use that the geodesic flow 0, : SM -+ SM preserves the contact form a. 1.47. Suppose that M is a closed oriented surface. Given a vector v e Tx M with norm one let i v be the unique vector such that (v, i v) is a positive orthonormal basis of Tx M. Let 1 be the one-form on SM given by VSO (l;) _ (KO ( ), i v) where
B = (x, v). Prove that }' annihilates G(O) and that it defines a connection on the circle bundle SM -+ M. What is the curvature of this connection? Prove that if M is the two-torus, then there exists a one-form f in M such that k := tL + rr'8 is closed. Take c3 = [A). For the last part, note that if A is invariant under the flip the integral of f is zero.
1.57. Recall that if a sequence {u") is subaditive, i.e., a"+m < u" +um, then u"/n converges. 2.9. For the first part just use the definitions of A. and K(x,,,). For the second part note that if f1, !2 E TOT N1 then (49,r((:1), Au(den(2))) - (den($2).
A is symmetric.
Hints and Answers
135
2.26. Let T3 be the flat 3-torus and let B = (x, v) E ST3. Let u 1 and u2 be vectors in T;T3 such that {v, u 1, u2) is an orthonormal basis. Let tI, 2 E T9ST3 be given by !:1 = (0, a1) and 2 = (u2, -u2/r), where r is any positive real number. Show that the subspace Er spanned by $t and 42 is Lagrangian and (0, ET) E Av(ST3). Show that e(8, ET) < r. Conclude that e(8, E) is not bounded away from zero (r was arbitrary). When M is a closed surface e(8, E) is bounded away from zero, why?
2.28. Let (8, E) E Uk,2 Ak(SM) with 0 = (x, v). Suppose that M has dimension n > 3. Let {v, et,... , be an orthonormal basis of T1M such that
(el, ..., ep) is a basis of derr(E) where p = n - I - dim(E fl V(8)). There exist vectors u1 ,..., u p such that (ei, ui) E E. Write n-I
ui =
u/ej,
p.
j=1
Let A = (u/)t<j. j
11(k),...,'fin-P-1(k) be functions of k E R such that
IAi (k) I - oo ask - oo; tr(A) + F_!_1 -I ki(k) _ X.
Let C(k) be the (n - p - 1) x (n - p - 1) diagonal matrix whose entries are the xi (k). Let U(k) be the symmetric matrix given by:
U(k) =
A
(B
Bt
C(k)
Clearly tr(U(k)) = X. Let E'(k) := graph U(k). Show that E'(k) -+ E as k -00.
2.34. If the curvature is -1, then
E"(e) = It E S(8) : den(t) = KO(t)}.
E'(0) = It E S(8) : den(!i) = -Ke(li)}. 2.35. Use the Jacobi equation.
2.37. No.
2.39. If P is invariant, then G(0) E T9 P. Hence since P C SM, for any t E TOP we have f2o(G(0), i;) = 0. Since P is two-dimensional, TOP is Lagrangian. 2.41. Follow the proof of Proposition 2.40 and replace V (O) by H(8).
136
Hints and Answers
2.53. Let (X2n, 92) be a closed symplectic manifold. Show that if f2 is exact then S2" is also exact and hence by Stokes theorem
0" = 0. JX
This contradicts the fact that n' is a volume form.
2.54. Without loss of generality suppose that y1 (t) # 0 for all t E (0, to). Then the signs of jyl(0) and y1 (to) are different. But since yl(O)y2(0) = yl(to)y2(to), Y2 must vanish in [0, to].
3.3. Let (V.) be a countable cover of U by open sets such that f I v is a diffeomorphism. Then f- I (W) n V. = (f I yj -1(W) has measure zero. Hence f-1(W)f1U=tP 1f-I(W)flV" has measure zero. 3.18. Let g I and g2 be two Riemannian metrics on X. Show, using that X is com-
pact, that there exist positive constants a and b such that for all x E X and all V E TX,
agx(v, v) < gx(v, v) < bg1(v, v). Use this to show a similar relation between volumes of submanifolds corresponding to different metrics. Finally take the log of these inequalities and let t -> oo.
3.25. We prove 3. Take t" - oo such that lim
I
1 log f (tn) = lim sup
T-.oo T
"''0O to
log f (T).
Then
limsup
I
T-.oo T
log foT
f(t)dt > lim sup I log J o nyoo to + 1/2 lim sup
1
nyoo to + 1/2
n
log
f(t)dt f(t)dt
limn+tI1/22 log(Cf On))
= lim sup
I log f (T).
Tyco T
3.31. Just follow the proof of Lemma 3.30 and make the appropriate modifications.
3.36. The volume of a ball in Euclidean space grows polynomially, while in hyperbolic space of constant curvature K it grows exponentially with exponent
(n - 1),I--K. 3.40. Just use the Hopf-Rinow theorem and the Area formula.
Hints and Answers
137
3.41. Use the previous exercise and that p : Mo - M is a local isometry.
4.12. Solve the following linear algebra problem. Let V be a real vector space with dimension 2n endowed with an inner product. Let L : V -+ V be a symmetric linear map. Let (vt,... , be a basis of eigenvectors with eigenvalues such that k; > 1 and A, 1 = k; 1 for I < i < n. Show I ..... k2n that S H I det L I S I achieves its maximum at the subspace spanned by vI, ... , vn.
4.21. If g is of class C', the Christoffel symbols are of class Cr-1. In local coordinates the geodesic vector field can be expressed in terms of the Christoffel symbols.
4.25. Lets := fi(x. rp) - x and let S(s) be the angle between positive tangents at x and fI (x, rp). The curvature of y at x is given by S(s)
k(x) = li m s-.o
-.
S
Note that S(s) = rp + f2(x, rp). Choose Euclidean coordinates whose origin coincides with x and such that the horizontal axis coincides with the tangent line to y at x. Using these cordinates and the fact that k(x) 96 0 show that (x, 0) = 0, hence
lim
f1(x, w) - w = of 1 (x, 0).
co-.OW+'f2(x,(P)
aw
Conclude that
I /k(x) = af1(x, 0). 4.26. This hint is taken from (HK, Section 9.2]. Let (XI, x2) be Euclidean coordinates in R2 and let a be the cyclic angular coordinate. The free motion with unit speed in the plane can be written as:
a=0, XI = cosa, X2 = sins
and hence is divergence free and preserves the volume form dx1dx2da. At a po x E y the coordinate a coincides up to an additive constant with the coordinate
gyp. Thus this form can be written as dx1dx2drp = dt(dx sinVd(p), that is, dt multiplied by the volume form dx sin tpd(p on A, which in turn is invariant under the reflection in the tangent line to y at x. Thus this latter form in invariant under f .
138
Hints and Answers
5.19. Choose a compact fundamental domain N for the action of ,rt (M) on M
such that x lies in the interior of N. Take d > 0 such that B(x, d) C N and R > 0 such that N C B(x, R). Then {B(a(x), d) : a E zrI(M)} is a collection of pairwise disjoint sets, so
#{a E7ri(M) : a(x) E B(x,r)} <
Vol(B(x, r)) Vol(B(x, d))'
On the other hand (B(a(x), R) : a(x) E B(x, r)} is a covering of B(x, r), so
#(a E ni (M) : a(x) E B(x, r)} >
Vol(B(x, r)) Vol(B(x, R))
5.30. Suppose that M is orientable. The nonorientable case follows from the orientable one. The Gauss-Bonnet theorem implies right away that
MinVol(M) > 2JrIXMI A metric of constant curvature ±1 has the property that Vol(M) = 2 n IXM I which shows that if M is different from the two-torus, then
MinVol(M) = 27rIXMI
For the two-torus simply multiply the flat metric by a positive number that approaches zero so that the volume also approaches zero.
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Index
A(6, t), 56 AN (0, t), 54 B(x, r), 69 C'-size, 73
AV, 40 Ak(SM), 39 Ak(8), 39 Al, 18
DNT , 51
920,14
G(8), 13
0,',44
H(9), 12
0"8
hrop(g), 63 hrop(O), 59
r,11 M(0), 21
hp(0), 61 J0, 14 Ke, 11 Le, 12
nT(N, y), 51
almost complex structure, 14 angle between subspaces, 77 Anosov geodesic flow, 42 area formula, 53
nT (x, y), 52 S(8), 17
asymptotic cycle, 21
SM, 8
billiard ball map, 93 configuration of length N, 94
SNl, 51 SNT , 51 TN-L, 33
V(x, r), 69 V(8), 11
a, 15 A(SM), 39 A(8), 39
canonical one-form, 19 characteristic flow, 11
vector field, 11 connection map, 11
148
Index
contact flow, 10 form, 10
manifold, 10 counting functions, 52 curvature operator, 36 differential of the geodesic flow, 20 Dinaburg's theorem, 120
entropies of a manifold, 126 entropy of a finite partition, 61
of an invariant measure, 61 topological, 59 Euler-Lagrange equation, 9 flow, 9 expansion, 82 exponential growth of a
invariant Lagrangian subbundle, 45 isotropic subspace, 32
Jacobi equation, 20 Jacobi field, 20 Lagrangian convex, 9 submanifold, 32 subspace, 32 superlinear, 9 Liouville measure, 18 Lyapunov exponents, 62 manifold rationally elliptic, 109 rationally hyperbolic, 109 without conjugate points, 90 Manning's inequality, 70 Maslov class, 41
group, 120
flip, 18
formal manifold, 114 fundamental theorem of Morse theory, 116 geodesic
class of a pair (X, E), 41 cycle, 39 index, 41 Maild's formula for convex billiards, 97 for geodesic flows, 83 measure ergodic, 21 invariant, 21
flow, 8
vector field, 13 Grassmannian bundle of Lagrangian subspaces, 39 manifold of Lagrangian subspaces, 39 Gromov's theorem, 117
Hamiltonian flow, 10 vector field, 10 homogeneous space naturally reductive, 57 reductive, 57 horizontal lift, 12 horizontal subbundle, 12
of maximal entropy, 62 positive Radon, 52 Riemannian, 53 smooth, 53 minimal volume, 127 musical isomorphisms, 19
nonwandering point, 47 normal Jacobi field, 20 orbit nontrivially recurrent, 59 recurrent, 59 partition, 61
Przytycki's inequality, 82
Index
pseudo-geodesic, 45
Riccati equation from the horizontal subbundle, 37
from the vertical subbundle, 38 Ruelle inequality, 63
Sasaki metric, 13 Schwartzman ball, 25 separated set, 59 Serre's theorem, 116 set basic, 97 hyperbolic, 42 invariant, 42 transitive, 97 simplicial volume, 126 spanning set, 59 stable manifold, 98
149
stable norm, 27 symplectic form, 9
form of TM, 14 manifold, 10 time shifts, 77 topological pressure, 104 twist map, 93 twist property
of the horizontal subbundle, 36 of the vertical subbundle, 35 unit tangent bundle, 8
variational principle, 62 vertical subbundle, 11 Yomdin's theorem, 60 uniform version of, 74
Progress in Mathematics Edited by: Hyman Bass
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A. Weinstein
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100 TAYLOR. Pseudodifferential Operators and Nonlinear PDE 101 BARKER/SALLY (eds). Harmonic Analysis on Reductive Groups 102 DAVID (ed). Seminaire de Th6orie des Nombres. Paris 1989-90 103 ANGER/PORTENIER. Radon Integrals 104 ADAMS/BARBASCHNOGAN. The
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106 BusER. Geometry and Spectra of Compact Riemann Surfaces 107 BRYLINSKI. Loop Spaces, Characteristic Classes and Geometric Quantization
108 DAVID (ed). S6minaire de Th6orie des Nombres. Paris 1990-91 109 EYSSEITh/GAWGO (eds). Computational Algebraic Geometry 110 Luszno. Introduction to Quantum Groups I I I SCHWARZ. Morse Homology 112 DONG/LEPOWSKY. Generalized
Vertex Algebras and Relative Vertex Operators 113 MOEGLIN/WALDSPURGER.
Decomposition spectrale et series d'Eisenstein
114 BERENSTEIN/GAY/VIDRAS/YGER.
Residue Currents and Bezout Identities 115 BABELON/CARTIER/KOSMANNScHWARZBACH (eds). Integrable
Systems, The Verdict Memorial Conference: Actes du Colloque International de Luminy 116 DAVID (ed). S6minaire de Th6orie des Nombres, Paris 1991-92 1 17 AUDIN/LaFONTAINE (eds). Holomor-
phic Curves in Symplectic Geometry 118 VAISMAN. Lectures on the Geometry of Poisson Manifolds 119 JOSEPH/ MEURAT/MIGNON/PRUM/ RENTSCH1ER (eds). First European
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AEBISCHERBORER/KALIN/LEUENBERGERI RE!MANN (eds). Symplectic Geometry
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LuBGTLKY. Discrete Groups. Expanding Graphs and Invariant Measures RIFSEt.. Prime Numbers and Computer Methods for Factorization
126
HORMANDER. Notions of Convexity 128 SCHMIDT. Dynamical Systems of 127
Algebraic Origin 129
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The Moduli Space of Curves DUIStERMAAT. Fourier Integral Operators
131
GINDIKIN/LEPOWSKY/WILSON (eds).
Functional Analysis on the Eve of the 21st Century. In Honor of the Eightieth
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Functional Analysis on the Eve of the 21st Century. In Honor of the Eightieth
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Sub-Riemannian Geometry 145 ALBERTBROUZET/DUFOUR (eds).
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Gabriel P. Paternain Geodesic I lows
Geodesic lions are of considerable current interest since they are, perhaps, the most remarkable class of conser,ative dynamical systems. They provide a unified arena in which one can explore numerous interplays among several fields, including smooth ergodic theory, symplectic and Riemannian geontetr%. and algebraic topology.
The work begins with a concise introduction to the geodesic flow of a
complete Riemannian manifold. emphasizing its symplectic properties and culminating with carious applications, such as the non-existence of continuous invariant Lagrangian subbundles for manifolds with conjugate points. Subsequent chapters develop the relationship between the exponential growth rate of the aserage number of geodesic arcs between
two points in the manifold and the topological entropy of the geodesic (lows. :\ complete proof of \lanc's formula relating these two quantities is
presented. A final chapter explores the link between the topological entropy of the geodesic How and the homology of the loop space of a manifold.
This self-contained monograph ss ill be of interest to graduate students and researchers of ds nautical systems and differential geometr%. Numerous exercises and examples as swell as a comprehensive index and make this work an esceIIent self-studs resource or test for a one-semester course or seminar. I
IS" O-SIT's-41"-9
Rirkhdiiser ISBN 0-8176-4144-11
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