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CAMBRIDGE TRACTS IN MATHEMATICS General Editors B. BOLLOBAS, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK 143 Analysis ...

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CAMBRIDGE TRACTS IN MATHEMATICS General Editors

B. BOLLOBAS, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK

143

Analysis on Fractals

Jun Kigami Kyoto University

Analysis on Fractals

Hf CAMBRIDGE '^fpF UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, United Kingdom 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, VIC 3166, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org ©

Jun Kigami 2001

This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2001

A catalogue record for this book is available from the British Library ISBN 0 521 79321 1 hardback Transferred to digital printing 2003

To my parents, Yasuko and Masao

Contents

Introduction 1 Geometry of Self-Similar Sets 1.1 Construction of self-similar sets 1.2 Shift space and self-similar sets 1.3 Self-similar structure 1.4 Self-similar measure 1.5 Dimension of self-similar sets 1.6 Connectivity of self-similar sets Notes and references Exercises 2 Analysis on Limits of Networks 2.1 Dirichlet forms and Laplacians on a finite set 2.2 Sequence of discrete Laplacians 2.3 Resistance form and resistance metric 2.4 Dirichlet forms and Laplacians on limits of networks Notes and references Exercises 3 Construction of Laplacians on P. C. F. Self-Similar Structures 3.1 Harmonic structures 3.2 Harmonic functions 3.3 Topology given by effective resistance 3.4 Dirichlet forms on p. c. f. self-similar sets 3.5 Green's function 3.6 Green's operator 3.7 Laplacians 3.8 Nested fractals Notes and references Vll

1 8 8 12 17 25 28 33 38 39 41 41 51 55 63 66 66 68 69 73 83 88 94 102 107 115 127

viii

Contents

Exercises Eigenvalues and Eigenfunctions of Laplacians Eigenvalues and eigenfunctions Relation between dimensions Localized eigenfunctions Existence of localized eigenfunctions Estimate of eigenfunctions Notes and references 5 Heat Kernels Construction of heat kernels 5.1 5.2 Parabolic maximum principle Asymptotic behavior of the heat kernels 5.3 Notes and references Appendix A Additional Facts A.I Second eigenvalue of Ai A.2 General boundary conditions A.3 Probabilistic approach Mathematical Background B B.I Self-adjoint operators and quadratic forms B.2 Semigroups B.3 Dirichlet forms and the Nash inequality B.4 The renewal theorem Bibliography Index of Notation Index 4 4.1 4.2 4.3 4.4 4.5

128 131 132 137 141 146 152 155 157 158 164 111 178 180 180 180 185 193 196 196 199 202 207 212 221 222

Introduction

What is "Analysis on Fractals"? Why is it interesting? To answer those questions, we need to go back to the history of fractals. Many examples of fractals, like the Sierpinski gasket, the Koch curve and the Cantor set, were already known to mathematicians early in the twentieth century. Those sets were originally pathological (or exceptional) counterexamples. For instance, the Koch curve (see Figure 0.1) is an example of a compact curve with infinite length and the Cantor set is an example of an uncountable perfect set with zero Lebesgue measure. Consequently, they were thought of as purely mathematical objects. In fact, they attracted much interest in harmonic analysis in connection with Fourier transform, and in geometric measure theory. There were extensive works started in the early twentieth century by Wiener, Winter, Erdos, Hausdorff, Besicovich and so on. See [181], [32] and [124]. These sets, however, had never been associated with any objects in nature. This situation had not changed until Mandelbrot proposed the notion of fractals in the 1970s. In [122, 123] he claimed that many objects in nature are not collections of smooth components. As evidence, using the experiments by Richardson, he showed that some coast lines were not smooth curves but curves which have infinite length like the Koch curve. Choosing

Fig. 0.1. Koch curve 1

2

Introduction

Fig. 0.2. Sierpinski gasket words more carefully and accurately, we need to say that some coast lines should be modeled by curves with infinite length rather than (compositions of) smooth curves. Mandelbrot coined this revolutionary idea and introduced the notion of fractals as a new class of mathematical objects which represent nature. The importance of his proposal was soon recognized in many areas of science, for example, physics, chemistry and biology. In mathematics, a new area called fractal geometry developed quickly on the foundation of geometric measure theory, harmonic analysis, dynamical systems and ergodic theory. Fractal geometry treats the properties of (fractal) sets and measures on them, like the Hausdorff dimension and the Hausdorff measure. From the viewpoint of applications, it concerns the static aspects of the objects in nature. How about the dynamical aspects? There occur (physical) phenomena on those objects modeled by fractals. How can we describe them? More precisely, how does heat diffuse on fractals and how does a material with a fractal structure vibrate? To give an answer to these questions, we need a theory of "analysis on fractals". For example, on a domain in E n , diffusion of heat is described by the heat (or diffusion) equation, du where u = u(t,x), t is time, x is position and A is the Laplacian defined by ]Cr=i TP1 If o u r domain is a fractal, we need to know what the

Introduction

3

Pi

P2

Go

P3

Gi

G2

Fig. 0.3. Approximation of the Sierpinski gasket by graphs Gm "Laplacian" on it is. This problem contains somewhat contradictory factors. Since fractals like the Sierpinski gasket and the Koch curve do not have any smooth structures, to define differential operators like the Laplacian is not possible from the classical viewpoint of analysis. To overcome such a difficulty is a new challenge in mathematics and this is what analysis on fractals is about. During the 1970s and 1980s, physicists tried to describe phenomena on fractals. They succeeded in calculating some of the physical characteristics of fractals, for example, the spectral exponent, which should describe the distribution of the eigenstates. (See, for example, [118] and [75] for reviews of studies in physics.) However they did not know how to define "Laplacians" on fractals. See Note and References of Chapter 4 for details. Motivated by studies in physics, Kusuoka [106] and Goldstein [51] independently took the first step in the mathematical development. They constructed "Brownian motion" on the Sierpinski gasket. Their method of construction is now called the probabilistic approach. First they considered a sequence of random walks on the graphs which approximate the Sierpinski gasket and showed that by taking a certain scaling factor, those random walks converged to a diffusion process on the Sierpinski gasket. To be more precise, let us define the Sierpinski gasket. Let {pi,£>2,P3} be a set of vertices of an equilateral triangle in C. Define fi(z) = (z—pi)/2+pi for i = 1,2,3. Then The Sierpinski gasket K is the unique non-empty compact subset K of R that satisfies K=

f1(K)Uf2(K)Ufs(K).

See Figure 0.2. Let VQ = {pi,P2,P3}- Define a sequence of finite sets {Vm}m>o inductively by Vm+1 = fi(Vm)Uf2(Vm)Ufs(Vm). Then we have the natural graph Gm whose set of vertices is V^. (See Figure 0.3.)

4

Introduction

For p G Vm, let Vm^p be the collection of the direct neighbors of p in Vm. Observe that #(F m , p j = 4 if p £ Vo and #(F m , p ) = 2 if p G Vb, where #(A) is the number of elements in a set A. Let X^ be the simple random walk on Gm. (This means that if a particle is at p at time £, it will move to one of the direct neighbors with the probability #(V^ljP)~1 at time t + 1.) What Kusuoka and Goldstein proved was that as m —» oo, where X* was a diffusion process, called Brownian motion, on the Sierpinski gasket. In this probabilistic approach, a Laplacian is the infinitesimal generator of the semigroup which is associated with the diffusion process. Barlow and Perkins [20] followed the probabilistic approach and obtained an Aronson-type estimate of the heat kernel associated with Brownian motion on the Sierpinski gasket. See Notes and References of Chapter 5. Then, in [116], Lindstr0m extended this construction of Brownian motion to nested fractals, which is a class of finitely ramified self-similar sets with strong symmetry. See 3.8 for the definition of nested fractals. (Roughly speaking, finitely ramified self-similar sets are the self-similar sets which become disconnected if one removes a finite number of points. See 1.3 for details.) On the other hand, in [82], a direct definition of the Laplacian on the Sierpinski gasket was proposed. Under this direct definition, one could describe the structures of harmonic functions, Green's function and solutions of Poisson's equations. This alternative approach is called the analytical approach. Instead of the sequence of random walks, one considered a sequence of discrete Laplacians on the graphs and then proved that by choosing a proper scaling, those discrete Laplacians would converge to a good operator, called the Laplacian on the Sierpinski gasket. More precisely, let £(Vm) = {/ : / maps Vm to R}. Then define a linear operator Lm : e(Vm) - i(Vm) by

(Lmu){p)= Y^ (u(q)-u(p)) qeVmjP

for any u G £(Vm) and any p G Vm. This operator L m is the natural discrete Laplacian on the graph G m . Then the Laplacian on the Sierpinski gasket, denoted by A, is defined by 5m(Lmu)(p) - (A«)(p) as m —> oo. This A is now called the standard Laplacian on the Sierpinski

Introduction

5

gasket. (Of course, it needs to be shown that A is a meaningful operator in some sense with a non-trivial domain, as we will show in the course of this book. Also we will explain why 5 m is the proper scaling. See 3.7, in particular, Example 3.7.3.) This analytical approach was followed by Kusuoka [107] and Kigami [83], where they extended the construction of the Laplacians to more general class of finitely ramified fractals. Since those early studies, many people have studied analysis on fractals and obtained numerous results using both approaches. Naturally the two approaches are complementary to each other and share the same goal. In this book, we will basically follow the analytical approach and study Dirichlet forms, Laplacians, eigenvalues of Laplacians and heat kernels on post critically finite self-similar sets. (Post critically finite self-similar sets are the mathematical formulation of finite ramified self-similar sets. See 1.3.) The advantage of the analytical approach is that one can get concrete and direct description of harmonic functions, Green's functions, Dirichlet forms and Laplacians. On the other hand, however, studying the detailed structure of the heat kernels, like the Aronson-type estimates, we need to employ the probabilistic approach. (Barlow's lecture note [6] is a self-contained and well-organized exposition in this direction. See also Kumagai [104] for a review of recent results.) Moreover, the probabilistic approach can be applied to infinitely ramified self-similar sets, for example, the Sierpinski carpet (Figure 0.4) as well. In the series of papers, [7, 8, 9, 10, 11, 12], Barlow and Bass constructed Brownian motions on the (higher dimensional) Sierpinski carpets and obtained the Aronson-type estimate of the associated heat kernels by using the probabilistic approach. Except for Kusuoka and Zhou [109], so far, the analytical approach has not succeeded in studying analysis on infinitely ramified fractals. One may ask "why do you only study self-similar sets???. Indeed, selfsimilar sets are a special class of fractals and there are no objects in nature which have the exact structures of self-similar sets. The reason is that self-similar sets are perhaps the simplest and the most basic structures in the theory of fractals. They should give us much information on what would happen in the general case of fractals. Although there have been many studies on analysis on fractals, we are still near the beginning in the exploration of this field. We hope that this volume will contribute to fruitful developments in the future. The organization of this book is as follows. In the first chapter, we will explain the basics of the geometry of self-similar sets. We will give the definition of self-similar sets, study topological structures of self-similar sets and introduce self-similar measures on them. The key notion is the self-

6

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similar structure which is a purely topological description of a self-similar set. See 1.3. Also, we will define post critically finite self-similar structure in 1.3, which will be our main stage of analysis on fractals. In Chapter 2, we will study analysis onfinitesets, namely, Dirichlet forms and Laplacians. The important fact is that those notions are closely related to electrical networks and that the effective resistance associated with them gives a distance on the finite set. Getting much help from this analogy with electrical networks, we will study the convergence of Dirichlet forms on a sequence of finite sets. This convergence theory will play an essential role in constructing Dirichlet forms and Laplacians on post critically finite selfsimilar sets in the next chapter. Chapter 3 is the heart of this book, where we will explain how to construct Dirichlet forms, harmonic functions, Green's functions and Laplacians on post critically finite self-similar sets. The key notion here is the "harmonic structure" introduced in 3.1. In this chapter, we will spend many pages to argue how to deal with the case when a harmonic structure is not regular and also when K\Vo is not connected, where K is the self-similar set and Vo corresponds to the boundary of K. These cases are of interest and sometimes really make a difference. However one would still get most of the essence of the theory by assuming that the harmonic structure is regular and that K\VQ is connected. So the reader may do so to avoid too many proofs. In Chapter 4, we will study eigenvalues and eigenfunctions of Laplacians

Introduction

7

on post critically finite self-similar sets. We will obtain a Weyl-type estimate of the distribution of eigenvalues in 4.1 and show the existence of localized eigenfunctions in 4.4. In the final chapter, we will study (Dirichlet or Neumann) heat kernels associated with Laplacians (or Dirichlet forms). In 5.2, it will be shown that the parabolic maximum principle holds for solutions of the heat equations. In 5.3, we will get on-diagonal estimates of heat kernels as time goes to zero. This book is based on my graduate course at Cornell University in the fall semester, 1997. I would like to thank the Department of Mathematics, Cornell University for their hospitality. In particular, I would like to express my sincere gratitude to Professor R. S. Strichartz, who suggested that I wrote these lecture notes, and gave me many fruitful comments on the manuscript. I also thank Dr. C. Blum and Dr. A. Teplyaev who attended my lecture and gave me many useful suggestions. I am also grateful to the Isaac Newton Institute of Mathematical Science, University of Cambridge, where a considerable part of the manuscript was written during my stay. I would express my special thanks to Professors M. T. Barlow and R. F. Bass who carefully read the whole manuscript and helped me to improve my written English. I would also like to thank all the people who gave me valuable comments on the material; among them are Professors M. L. Lapidus, B. M. Hambly, V. Metz, T. Kumagai, Mr. T. Shimono and Mr. K. Kuwada. Finally I would like thank the late Professor Masaya Yamaguti, who was my thesis adviser and introduced me to the study of analysis on fractals.

1 Geometry of Self-Similar Sets

In this chapter, we will review some basics on the geometry of self-similar sets which will be needed later. Specifically, we will explain what a selfsimilar set is (in 1.1), how to understand the structure of a self-similar set (in 1.2 and 1.3) and how to calculate the Hausdorff dimension of a self-similar set (in 1.5). The key notion is that of a "self-similar structure" introduced in 1.3, which is a description of a self-similar set from a purely topological viewpoint. As we will explain in 1.3, the topological structure of a self-similar set is essential in constructing analytical structures like Laplacians and Dirichlet forms. More precisely, if two self-similar sets are topologically the same (i.e., homeomorphic), then analytical structure on one self-similar set can be transferred to another self-similar set through the homeomorphism. In particular, we will introduce the notion of post critically finite selfsimilar structures, on which we will construct the analytical structures like Laplacians and Dirichlet forms in Chapter 3. 1.1 Construction of self-similar sets In this section, we will define self-similar sets on a metric space and show an existence and uniqueness theorem for self-similar sets. First we will introduce the notion of contractions on a metric space. Notation. Let (X, d) be a metric space. For x £ X and r > 0 , Br(x) = {y:yeX,d(x,y)

< r}

Definition 1.1.1. Let (X, dx) and (Y, dy) be metric spaces. A map / : 8

1.1 Construction of self-similar sets

9

X —> Y is said to be (uniformly) Lipschitz continuous on X with respect to dx,d>Y if

L=

sup

w(*)./0/))
x,y€X,x^y

dX{X,y)

The above constant L is called the Lipschitz constant of / and is denoted byL = Lip(/). Obviously, by the above definition, a Lipschitz continuous map is continuous. Definition 1.1.2 (Contraction). Let (X, d) be a metric space. If / : X —+ X is Lipschitz continuous on X with respect to d and Lip(/) < 1, then / is called a contraction with respect to the metric d with contraction ratio Lip(/). In particular, a contraction / with contraction ratio r is called a similitude if d(f(x), f(y)) = rd(x, y) for all x, y G X. Remark. If / is a similitude on (Rn, d#), where d# is the Euclidean distance on R n , then there exist a G R n , U G 0{n) and r > 0 such that f(x) = rUx + a for all x G W1. (Exercise 1.1) The following theorem is called the "contraction principle". Theorem 1.1.3 (Contraction principle). Let (X,d) be a complete metric space and let f : X —• X be a contraction with respect to the metric d. Then there exists a unique fixed point of f', in other words, there exists a unique solution to the equation f(x) = x. Moreover ifx* is the fixed point of f, then {fn{a)}n>o converges to x* for all a £ X where fn is the n-th iteration of f. Proof. If r is the ratio of contraction of / , then for m> n, d(fn(a),

fm(a))

< d(fn(a),

<{rn

+

f^1

(a)) + ••• + d(fm-\a),

... + r m-i) d (o,/(a)) <

fm(a))

-^^dia,/(a)).

Hence {/n(a)}n>o *s a Cauchy sequence. As (X,d) is complete, there exists x* G X such that fn(a) —• x* as n —• oo. Using the fact that /(n+i)( a ) = /(/^(a)), we can easily deduce that z* = f(x*). Now, if f(x) = x and f(y) = y, then d(x,y) = d(f(x),f(y)) < rd(x,y). Therefore d(x, y) = 0 and x = y. So we have uniqueness offixedpoints. • Remark. In general, for a mapping / from a set to itself, a solution of f(x) = x is called a fixed point or an equilibrium point of / .

10

Geometry of Self-Similar Sets

We now state the main theorem of this section, which ensures uniqueness and existence of self-similar sets. Theorem 1.1.4. Let (X,d) be a complete metric space. If fa : X —» X is a contraction with respect to the metric d for i = 1,2,... ,N, then there exists a unique non-empty compact subset K of X that satisfies K =

f1(K)U---UfN(K).

K is called the self-similar set with respect to {/i, / 2 , . . . , /jv}-

Remark. In other literature, the name "self-similar set" is used in a more restricted sense. For example, Hutchinson [76] uses the name "self-similar set" only if all the contractions are similitudes. Also, in the case that all the contractions are affine functions on R n , the associated set may be called a self-affine set. The contraction principle is a special case of Theorem 1.1.4 where N = 1. In the rest of this section, we will give a proof of Theorem 1.1.4. Define

F(A)=

(J

fj(A)

!<3
for A C X. The main idea is to show existence of a fixed point of F. In order to do so, first we choose a good domain for F, defined by C(X) = {A : A is a non-empty compact subset of X}. Obviously F is a mapping from C(X) to itself. Next we define a metric 6 onC(X), which is called the Hausdorff metric on C(X). Proposition 1.1.5. For A,B G C{X), define 6(A, B) = inf{r > 0 : Ur(A) D B and Ur(B) D A}, where Ur(A) = {x G X : d(x,y) < r for some y € A } = UyeABr(y). Then 6 is a metric on C(X). Moreover if (X^d) is complete, then (C(X),6) is also complete. Before giving a proof of the above proposition, we recall some standard definitions in general topology. Definition 1.1.6. Let (X,d) be a metric space and let K be a subset of X. (1) A finite set A C K is called an r-net of K for r > 0 if and only if UxeABr(x) 2 K. (2) K is said to be totally bounded if and only if there exists an r-net of K for any r > 0 .

1.1 Construction of self-similar sets

11

It is well-known that a metric space is compact if and only if it is complete and totally bounded. Proof of Proposition 1.1.5. Obviously, we see that 8(A,B) = 8(B,A) > 0 and 6(A, A) = 0. 8(A, B) = 0 => A = B: For any n, U1/n(B) D A. Therefore for any x e A, we can choose xn G B such that d(x,xn) < 1/n. As B is closed, x G B. Hence we have A C B. One can obtain B C A in exactly the same way. Triangle inequality: If r > 5(A,£) and 5 > 6(B,C), then £/r+8(i4) 2 C and £/r+e(C) D A. Hence r + s > £(A, C). This implies 5(A, B)+S(B, C) > 8(A,C). Next we prove that (C(X),8) is complete if (X,d) is complete. For a Cauchy sequence {An}n>i in (C(-X"),5), define Bn = Ufc>n-A/c- First we will show that £ n is compact. As Bn is a monotonically decreasing sequence of closed sets, it is enough to show that B\ is compact. For any r > 0, we can choose m so that Ur/2(Am) 3 A^ for all k > m. As A m is compact, there exists a r/2-net P of A m . We can immediately verify that U xG pi? r (:r) 2 #r/2CAm) 3 Uk>mAk. As U x e p5 r (x) is closed, it is easy to see that P is an r-net of Bm. Adding r-nets of A\,A*i, • • • , Am-i to P, we can obtain an r-net of B\. Hence B\ is totally bounded. Also, Bγ is complete because it is a closed subset of the complete metric space X. Thus it follows that Bn is compact. Now as {Bn} is a monotonically decreasing sequence of non-empty compact sets, A = nn>iBn is compact and non-empty. For any r > 0, we can choose m so that Ur(Am) 2 A^ for all k> m. Then Ur(Am) 2 Bm D A. On the other hand, Ur(A) 2 -E?™ 2 >lm for sufficiently large m. Thus we have 8(A, Am) < r for sufficiently large m. Hence Am —> ^4 as m —>• CXD in the Hausdorff metric. So (C(X),<5) is complete. • Theorem 1.1.4 can be stated in the following way using the Hausdorff metric (C(X),8). Theorem 1.1.7. Let (X, d) be a complete metric space and let fa : X —• X be a contraction for i = 1,2,... , n. Define F : C(X) -> C{X) by F(A) = ^i oo with respect to the Hausdorff metric. Lemma 1.1.8. For Au A2,BUB2

G C(X),

8(Ai U A2, Bγ U B2) < max{5(Ai, Si), 8(A2, B2)}

12

Geometry of Self-Similar Sets

Proof. If r> max{6(AuB1),6(A2,B2)}, then Ur{A{) 2 # i and Ur(A2) 2 B2. Hence Ur{A\ U A2) 2 B\ U I?2- A similar argument implies Ur(B\ U J32) 2 -Ai U A2. Hence r > <5(Ai U A2,Bi U B 2 ). This completes the proof. D Lemma 1.1.9. If f is a contraction with contraction ratio r, then, for any A, Be C(X), 6(f(A), f(B)) < r6(A, B). Proof If Us(A) D B and US(B) D A, Usr(f(A)) D f(Us(A)) D f(B). Also the same argument implies U8r(f(B)) 2 f(A). Therefore, 6(f(A),f(B)) < rs and this completes the proof. • Proof of Theorem 1.1.7. Using Lemma 1.1.8 repeatedly, we obtain 8(F(A),F(B)) = SiUi&KjffjiA^Ui&KsfjiB)) < ^ ^ ( / . ( A ) , / , ( £ ) ) . By Lemma 1.1.9, 6(fi(A),fi(B)) < riS(A,B), where r^ is the contraction ratio of fc. If r = maxi 1, we define W£ = { 1 , 2 , . . . ,N}m = {wlW2 .. .wm : Wi € { 1 , 2 , . . . ,N}}. w G W^ is called a word of length m with symbols { 1 , 2 , . . . , N}. Also, for m = 0, we define Wjf = {0} and call 0 the empty word. Moreover, set W* = U m >oW^ and denote the length of w G W? by \w\. (2) The collection of one-sided infinite sequences of symbols { 1 , 2 , . . . , N} is denoted by E ^ , which is called the shift space with iV-symbols. More precisely, EN = { 1 , 2 , . . . ,iV} N = {0JHJ2U3 . . . : Wi e { 1 , . . . , N} for i G N}. For k G { 1 , 2 , . . . , N}, define a map a^ : Y,N —> T,N by ak(uJiuJ2^3 • • •) =

1.2 Shift space and self-similar sets kui(jJ2^3

13

•)— ^2^3^4 Also define a : T,N —> T,N by a(uJiUJ2^s • •

o"

is called the shift m a p . Remark.

T h e two sided infinite sequence of { 1 , 2 , . . . , N},

{1,2,... ,N}Z = {...uj-2u-iuouiu2...

:ui e {1,2,... ,N} for i E Z}

may also be called the shift space with TV-symbols. If one wants to distinguish the two, the above E ^ should be called the one-sided shift space with iV-symbols. In this book, however, we will not treat the two-sided symbol space. For ease of notation, we write Wm, W* and E instead of W£[, W^ and Obviously, o> is a branch of the inverse of a for any k E { 1 , 2 , . . . , TV}. If we choose an appropriate distance, it turns out that o> is a contraction and the shift space E is the self-similar set with respect to {CTI, O"2, •.. , GN}Theorem 1.2.2. Foruj,r E E withu ^ r andO < r < 1, define8r(u,r) — S(U),T) = min{ra : u;m ^ r m } — 1. {i.e., n = 5(0;, r ) if and rs(uj,T)^ wjiere only if uoi = r» for 1 < i < n and o; n+ i ^ r n + i . ) yl/50 de/ine 5 r (a;,r) = 0 if (jj = T. 6r is a metric on E and (E,<5r) is a compact metric space. Furthermore, o> is a similitude with Lip (o>) = r and E is the self-similar

set with respect to {<Ji, 02, • • • ^/v}, Proof. It is obvious by the definition that 8r(u,r) > 0 and ^ r (cj,r) = 0 implies uj = r. As min{5(a;, r),s(r, K)} < S(LJ,K) for a;,r, K E E, we can see that <5r(u;, r) 4- 6r (r,«) > 5r(o;, K). Now for every z/; = w\W2 -.. wm E W*, we define Tiyj = {u = UJ\U)2W3 . . . E E : UJ\U)2 • • ^m • — W1W2 . . . Wm}'

Let {wn}n>i be a sequence in E. Using induction on m, we can choose r E E so that { n > 1 : (u>n)j = TJ for j = 1,2,... , 771} is an infinite set for any m > 1 . So there exists a subsequence of {u; n} that converges to r as n —> 00. Hence (E,<5r) is compact. Finally, it is obvious that o> is a similitude with Lip (o^) = r. Also we can easily see that E = o"i(S) U • • U •crjv(E). This implies that E is the self-similar set with respect to {o"i, o"2,... , O~N}. • E is called the (topological) Cantor set with iV-symbols. See Example 1.2.6. For the rest of this section, we assume that (X, d) is a complete metric space, fi : X —> X is a contraction with respect to (X, d) for every i E { 1 , 2 , . . . ,7V} and that K is the self-similar set with respect to

14

Geometry of Self-Similar Sets

{/l, /2, • • •/iv}, Also, for A C I , the diameter of A, diam(A), is defined by diam(A) = s u p ^ ^ d(x, y). The following theorem shows that every self-similar set is a quotient space of a shift space by a certain equivalence relation. fw1ofw2o---ofwrri Theorem 1.2.3. Forw = wiw2 . ..u> m G W*, set fw = andKw = fw(K). Then for any LJ = 00^2^3 . . . e E, nm>i-KWlfa,2...u,m contains only one point. If we define n : £ —* K by {TT(UJ)} = r\m>iKCJlUj2.m.UJm, then 7T is a continuous surjective map. Moreover, for any i G { 1 , 2 , . . . , N}, ft O O~i =

fiOTT.

Proof. Note that

As ATa;ia;2...a;m is compact, n m >il£ a ; lU ; 2 ... a , m is a non-empty compact set. Set R — maxil). Hence diam(i^u;iU;2...u;m) < Rmdia,m(K). So diam(D m>ii ;Ca;ia;2 ...a;m) = 0. Therefore nrn>iKUJlu>2--.uJm should contain only one point. If 8r((j,T) < r m , then 7r(o;),7r(r) G KUJl(Jj2,,mUJrn = Kr1T2...rrn- Therefore d(7r(o;),7r(r)) < i?m diam(i ; C). This immediately implies that TT is continuous. Using {n(<Ti(v))} = nTn>iKiuj1uJ2...uJrn = n m >i/t(if a ; ia , 2 ... Wm ) = {/i(7r(u))}, we can easily verify that TT O ai = /^ o TT. Finally we must show that TT is surjective. Note that 7T(E)

-

7T(<7I(E) U

• • U•o-iv(E))

- TT^xCE)) U • • • ^U( ^ ( E ) ) =

/I(TT(E)) U

- • -U

/^(TT(E)).

As 7T(E) is a non-empty compact set, the uniqueness of self-similar sets (Theorem 1.1.4) implies that TT(E) = K. • Proposition 1.2.4. Define w = www... 7r(ii;) is ^/le unique fixed point of fw.

if w e W* and w ^ 0. T/ien

Proof. As fw is a contraction, it has a unique fixed point. By Theorem 1.2.3, 7T(W) = 7T(WW) = fw{K{w)). Hence 7T(W) is the fixed point

of/w.

•

Using the above proposition, we see that TT(VIV2 ... Vkw) = fv(Pw) where w G W*, w ^ 0, v = i;i^2.. .Vk € W* and p ^ is the fixed point of Z^,.

1.2 Shift space and self-similar sets

15

This relation helps us to understand n in many examples. Moreover, since periodic sequences are dense in E, we have K = {pw :weW*,w^®}. In fact, 7T determines a topological structure on self-similar sets. Proposition 1.2.5. Suppose fi is infective for every i G { 1 , 2 , . . . , AT}. Then, TT(UJ) = TT(T) for uj ^ r G E if and only z/7r(crmu;) = 7r(arnr), where

m = s(c<;,r). Proof If w = uJiU)2 — -Mm = T1T2 • • T•m , t h e n 7T(U;) = fw(ft{o~'mLu)) a n d m 7T(T) = /iy('7r(cr T)). As /^ is injective, we have TT(O;) = TT(T) for CJ,T G E

if and only if 7r(crmo;) = 7r(crmr). The other direction is obvious.

•

Note that if TT(LJ) = TT(T), then 7r(am(cc;)) = 7r(crm(r)) G C x , where m = S(LJ,T) and C^ = Ui K is injective. By Theorem 1.2.3, ?r is also surjective and hence it is a homeomorphism between E and K. In particular, if a = b = 1/3, X is called the Cantor ternary set or the Cantor middle third set. Example 1.2.7 (Koch curve). Let X = C. Suppose that a G {z : z G C, |z|2 + |1 - z\2 < 1}. Set h(z) = az and / 2 (z) = (1 - a)(z - 1) + 1. Let D be a triangle domain with vertices {0, a, 1}, including the boundary. Then it follows that fx{D) U f2(D) C D and /i(L>) n f2(D) = {a}. See Figure 1.1. Hence K(a) C D, where i(T(a) is the self-similar set with respect to {/i,/ 2 }. Also note that /i(0) = 0, / 2 (1) = 1 and / 2 (0) = /i(l) = a. Denoting n = ?ra, we obtain that 7ra(i) = 0,7ra(2) = 1 and 7Ta(12) = 7Ta(2i) = a. Moreover, CK = Ki H K2 = {a}. Hence we can deduce that 7ra(uj) = 7Ta(r) and u ^ r if and only if there exists w G W* such that {o;,r} = {it;12,^2i}. In particular, if (1/2) = [0,1] and if (a) is called the Koch curve if a = | + ^^\/—T- See Figure 0.1. Note that fta ° TTi/2"1 is a homeomorphism between [0,1] and if (a). Example 1.2.8 (Sierpinski gasket). Let X = C and let {pi,P2,Ps} be a set of vertices of an equilateral triangle. Define fj(z) = (z — Pj)/2 + pj for j = 1,2,3. The self-similar set with respect to {/i, / 2 , fy} is called the Sierpinski gasket. It is easy to see that n(j) — Pj for j = 1,2,3. Let T be

16

Geometry of Self-Similar Sets a = 7T(12) = 7r(2i)

H2

0 = 7T(i)

|l-a|2

1 = TT(2)

Fig. 1.1. /i(D) U f2(D) (a = 0.4 H- 0 . 3 ^ ) Pl=7T(i)

/

/i(T)

\

/

/ P2 =

7T(2)

93

«h

/

\

/2(r)

\

qi= Hi

/

\

/ 91

Q2

\

/s(T)

7T(23) = 7T(32)

q2 = 7T(13) = 7T(32)

93 = 7T(12) = 7T(2i)

\ P3 =

7T(3)

Fig. 1.2. Topological structure of the Sierpinski gasket the equilateral triangle with vertices {pi,P2iP3}, including the boundary. Then /i(T) U f2(T) U fs(T) C T. Hence iiT C T. Also /i(i^) n f2(K) = /i(T)n/2(T) and this set contains only one point, which is denoted by {#3}. Then T T " 1 ^ ) = {2i, 12}. In the same way, if f2(K) n f3(K) = {qi} and h(K)nfi(K) = {q2} then T T " 1 ^ ) = {23,32} and T T " 1 ^ ) = {3i, 13}. See Figure 1.2. By those facts, if TT(UJ) = TT(T) and u) ^ r, there exists it; G VF* such that {o;,r} = {^12,w;2i} or {w23, w32} or {if;3i,^13}. Example 1.2.9 (Hata's tree-like set). Let X = C. Set fr(z) = cz, f2(z) = (1 - \c\2)z 4- |c| 2 , where |c|, |1 - c\ G (0,1). The self-similar set K with respect to {fi,f2} is called Hata's tree-like set. Let A = {t :

1.3 Self-similar structure »

S^\c\2=ir(112) ^— 0 = 7T(i)

17

C = 7T(12)

= w(2i)

*—

• X.

j2(A) *

1 = 7T(2)

7T(212)

Fig. 1.3. Hata's tree-like set; /i(A) U f2(A)

Fig. 1.4. Hata's tree-like set(c = 0.4 + O.Sv^-T)

0 < t < 1} U {tc,0 < t < 1}. Then it follows that /i(A) U f2(A) D A. Hence if Am = Uw^WmfwiA), then {Am}m>o is an increasing sequence and K — Um>oAm. Also we can easily observe that fi(K) n /2(^) = {|c|2}, A(0) = 0,/ 2 (l) = 1 and /i(/i(l)) = / 2 (0) = |c| 2 . Hence TT" 1 ^) = {i}, TT-^l) = {2}, TT-^c) = {12} and TT- 1 (|C| 2 ) = {112,21}. See Figure 1.3. Moreover, if TT(UJ) = TT(T) and u / r, there exists w G W* such that {U;,T} = {^112,^21}.

1.3 Self-similar structure From the viewpoint of analysis, only the topological structure of a selfsimilar set is important. For example, suppose you want to study analysis on the Koch curve. Recalling Example 1.2.7, there exists a natural homeomorphism between [0,1] and the Koch curve. Through this homeomorphism, any kind of analytical structure on [0,1] can be translated to its

18

Geometry of Self-Similar Sets

counterpart on the Koch curve. So it is easy to study analysis on the Koch curve. The notion of self-similar structure has been introduced to give a topological description of self-similar sets. Definition 1.3.1. Let K be a compact metrizable topological space and let 5 be a finite set. Also, let F{ be a continuous injection from K to itself for any i G S. Then, (K, 5, {Fi}ie$) is called a self-similar structure if there exists a continuous surjection TT : E —> K such that Fi o n = n o Oi for every Z G 5 , where E = SN is the one-sided shift space and c^ : E —> E is defined by Gi{w\W2W^ ...) = iw\W2W^ ... for each W1W2W3 ... G E. E is called the shift space with symbols 5. We will define Wm = S m , W# = Um>oWm, a : E —• E and so on in exactly the same way as in 1.2. Also the topology of E is given in exactly the same way as in 1.2. If we need to specify the symbols S, we use E(5), Wm(S) and W*(S) in place of E, Wm and W* respectively. In many cases, we think of S = {1,2,... , N}. Obviously if K is the self-similar set with respect to injective contractions {/l j /2J • • • /jv}j j then (if, {1,2,... , iV}, {/i}^) is a self-similar structure. It is possible that two different self-similar sets have the same topological structure. For example, the self-similar structures corresponding to the self-similar sets K(a) in Example 1.2.7 are all essentially the same. More precisely, they are isomorphic in the following sense. Definition 1.3.2. Let Cj = (Kj^Sj^F^j^Sj) be self-similar structures for j = 1,2. Also let TTJ : E(5^) —> Kj be the continuous surjection associated with Ci for j = 1,2. We say that C\ and £2 are isomorphic if there exists a bijective map p : S\ —> S2 such that TT2 O LP O TTI"1 is a well-defined homeomorphism between K2 and K\, where ip is the natural bijective map induced by 7, i.e., L{UJ\UJ2 • • •) =p(uJi)p(u>2) We say that two self-similar structures are the same if they are isomorphic. Proposition 1.3.3. If(K, 5, {Fi}ies) is a self-similar structure, then n is unique. In fact, {7T(U;)}=

0

FUlUa...Um(K)

m>0

for any u = w\u)2 . . . G E. Proof. By the above definition, we have i ^ o TT = TT O U^ for any w G W*. Hence, ir((j) G n m > 0 F a ; i a ; 2 ... a ; m (X). For x G n m > 0 Fa;ia; 2 ...a; m (^), there

1.3 Self-similar structure

19

exists xm G E ^ ^ . . . ^ such that 7r(xm) = x. Note that TT is continuous. Since xm —> u as m —> oo, it follows that x = 7r(xm) —• TT(U;) as m —> oo. Hence x = TT(U;). D Definition 1.3.4. Let £ = (if, 5, {.Fi}ies) be a self-similar structure. We n F K define CC,K = UiJesMj(Fi(K) A )), Cc = ^~\CC,K) and Vc = Un>i<Jn(Cc). Cc is called the critical set of C and Vc is called the post critical set of C. Also we define Vo(C) = TT(VC)For ease of notation, we use C, V and Vo instead of Cc, Vc and Vo(C) as long as it can not cause any confusion. The critical set and the post critical set play an important role in determining the topological structure of a self-similar set. For example, if C = 0, (and hence Vy Vo are all empty sets), then K is homeomorphic to the (topological) Cantor set E. Also Vo is thought of as a "boundary" of K. For example, define F\(x) = \x and F2{x) = \x + \ and recall Example 1.2.7. Then we find that C = {12,2i} and V = {i,2}. Hence Vo = {0,1}. See also Exercise 1.3 for another example. Proposition 1.3.5. Let C = (K,S,{Fi}ies) be a self-similar structure. Then (1) 7T-i(Vo)=V. (2) / / E w n E v = ®forw,ve W*, thenKwr\Kv = Fw(V0)nFv(V0), where Kw = FwyK). (3) C = 0 if and only if n is infective. Proof. (1) If 7T(U) G Vb, then there exist r e C and m > 1 such that 7r(amr) = TT(LJ). Set u/ = T\T2 . . . Tma;. Then ir(u') = F n ^... Tm (7r(o;)) = ^T 1 T 2 ...T rr >(a r V)) = TT(T) G CC,KHence W ' G C and LJ eV. (2) It is obvious that Fw(Vo)nFv(Vo) C ^ n ^ . For a: G X ^ H ^ , we can choose a;, r G E so that x = TT^U;) = TT(VT). AS E-^ PI E V = 0, there exists k < min{|it;|, \v\} such that WiW2 .. .Wk — v\v
20

Geometry of Self-Similar Sets

Proposition 1.3.6. Let C = (K,S,{Fi}ies) For any x G K and any m>0, define

be a self-similar structure.

weWm'xeKw

Then {^m^jm^o is a fundamental system of neighborhoods of x. Proof. Let d be a metric on K which is compatible with the original topology. First we show that max^v^m diam(Kw) —• 0 as m —> oo. If not, there exists {w(m)}m>o such that w(m) G Wm for any ra, and inf m >odiam(K iy ( m )) > 0. Choose uj(m) G E^m) for any m > 0 . Then since E is compact, there exists a subsequence {u(rrii)}i>i which converges to some LJ G E as i —> ex). Note t h a t diam(KWlW2._Wm)

<

diam(KWlW2_.Wri)

if m > n. It follows that liminfm_^oodiam(Xu;ia;2...u,m) > 0. This contradicts Proposition 1.3.3. Secondly, we show that ifm?a; is a neighborhood of x. Let {x m } m >i be a sequence in K which converges to x as m —• oo. Choose LJm G 7r~1(xm) for any m>1. Then there exists a subsequence {u;mi}i>i that converges to some u ; G E a s i - > o o . Since TT is continuous, TT(UJ) = x. Hence xrrii G Km,x for sufficiently large i. Therefore KmjX is a neighborhood of x. Combining the two facts, we conclude that {ifm,x}m>o is a fundamental system of neighborhoods of x. • A self-similar structure (K, S, {Fi}ie$) m&y contain an unnecessary symbol. For example, let K = [0,1] and define S = {1,2,3}, Fi(x) = \x, F2(x) = \x + \ and F3(x) = \x+\. Then obviously K = F^K) U F2{K) and we don't need F 3 to describe K. This example may be a little artificial but there are more natural examples. To explain such examples, we need to introduce some notation. Let C — (K,S,{Fi}ies) be a self-similar structure. Let W be a finite subset of W*\Wo. Then E(W) = WN can be identified as a subset of E(5) = SN in a natural manner. Set K(W) = TT(E(W)). Then (K(W),W, {Fw}wew) is a new self-similar structure. We denote this selfsimilar structure by C(W). Using this notation, we can rephrase the above example as K({1,2}) = K(S). The following is a more natural example. Example 1.3.7. Let K = [0,1] and define S = {1,2}, Fi(jc) = \x and F2(x) = \x + \. Then £ = (if, 5, {Fi, F2}) is a self-similar structure. Set W = {11,22}. Then K(W) = if because K = Fn(K) U ^ ( i f ) . This means that to describe K, we don't need the words {12,21}.

1.3 Self-similar structure

21

You may notice that this kind of unnecessary symbol (or word) occurs when the overlap set CC,K (or equivalently Cc ) is "large". The following theorem justifies such an intuition. Theorem 1.3.8. Let C = (K,S,{Fi}i^s) be a self-similar structure. The following conditions are equivalent. If C satisfies one of the following conditions, we say that C is minimal. (Mil) Ifn(A) = K for a closed set ACJ2, then A = E. (Mi2) For any w G W*, Kw is not contained in Uvewm\{w}Kv, where m = \w\. (Mi3) IfKiW) = K forW C Wm, then W = Wm. (Mi4) Kw is not contained in CC,K for any w G W*. (Mi5) int(C£) = 0. (Mi6) int(P £ ) = 0. (Mi6*) Vc ^ S. (Mi7) int(Vb) = 0. (Mi7*) Vo ^ K. As we can see from (Mi3), a minimal self-similar structure does not have any unnecessary symbol (or word). It is easy to see that the self-similar structures corresponding to the self-similar sets in 1.2 are all minimal. Proof. (Mil) => (Mi4) Assume that C D Kw for some w G VF*. Let k G S be the first symbol of w. Then for any x G Kw, there exists some j T^ k such that x G Kj. If m = \w\ and A — UvGvym\{u;}^t;5 then A is closed and TT(A) = K. (Mi4) =» (Mi5) Assume that int(C) ^ 0. Then C D Ew for some u> G W*. Hence C D X^. m (Mi5) =^ (Mi6*) Assume that P = E. Then as P = U m >ia C, Baire's m category argument shows that int(a C) ^ 0 for some m. (See, for example, [186] about Baire's category argument.) Hence, o~mC 2 S w for some w G W*. Therefore afcC = E for k = m+\w\. Now akC = Uvewko-k(T,vnC). k Again using Baire's category argument, it follows that a (T,v DC) 2 E u for some v G W& and it G W*. Therefore C D T,vu. (Mi6*)=> (Mi6) Assume that int(P) 7^ 0. Then P D S W for some w G W*. Since cr m P C P for m = |tu|, we have E = V. (Mi6) =^> (Mi7) As T T " 1 ^ ) = T^, we have Tr-^int^o)) C int(P). (Mi7) => (Mi7*) =4> (Mi6*) This is obvious by the fact that T Γ " 1 ^ ) = V. (Mi6*) => (Mi2) Assume that iiT^ C Uvewm\{w}Kv for some m and w G Wm. Then for any u G E, there exist v G Wm\{ii;} and r G E such that 7r(u>o;) = TT(I;T). AS W ^ v, we can choose k < m so that W1W2 ... Wfc-i = vi^2 . • - Vk-i and Wk ^ Vk- Since Fiu,llt;2...tl;fc_1 is injective, we see that ak(wu) G C. Therefore ueV. So V = E.

22

Geometry of Self-Similar Sets

(Mi2) => (Mil) If there exists a closed subset A c S with n(A) = if, then Ac is a non-empty open set and so it should contain T,w for some w G W*. Since A D Uvewm\{w}^v, where m = \w\, we have Kw G Uv€wm\{w}Kv. (Mi2) => (Mi3) Let VF be a proper subset of Wm and assume K(W) = if. Then for w G Wm\W, Kw C K = UveWKv. Hence (Mi2) does not hold. (Mi3) => (Mi2) If Kw C UveWrn\{wyKv, where m = \w\, then if = \JvewFv(K), where W = W m \{w}. Hence, for any x G if, there exists a; G E(W) such that ?r(cc;) - x. Therefore K(W) = K. D Remark. It seems quite possible that the condition int(Cc,K) = 0 is also equivalent to those conditions in Theorem 1.3.8. Unfortunately this is not true. In fact, there is an example where int(Cc,K) = 0 but int(Cc) ¥" 0See Exercise 1.5. Definition 1.3.9. Let S be a finite set. We say that a finite subset A C W*(S) is a partition of E(5) if E^ n T,v = 0 for any w ^ v G A and E = U^^AE-U;. A partition Ai is said to be a refinement of a partition A2 if and only if either E^ C E v or E^ D E v = 0 for any (w, v) G Ai x A2. Wm is a partition for any m > 0 and Wn is a refinement of Wm if (and only if) n> m. Lemma 1.3.10. Let C = (if, 5, {Fi}ies) be a self-similar structure. Define V(A,C) = UweAFw(V0) if A is a partition of" E. T/ien V(AUC) 2 y(A2,>C) i/Ai Z5 a refinement 0/A2. Proo/. Assume that Ai is a refinement of A2. Set x = TT(WU) for i<; G A2 and uj e V. Then there exists v G Ai such that v = wui...Uk. As Uk+iWk+2 . . . G P , we can see that x = n{wui) G V(Ai, C). D Lemma 1.3.11. Let C = (K,S,{Fi}i^s) be a self-similar structure. Define Vm(C) = V(Wm,C). Then Vm(C) C Vm+1(C) and Vm+1(£) =

UiesFi(Vm(C)).

Furthermore, set V*(C) = Um>oVm(C). IfVo ^ 0, then V^(C) is dense in K. Proof. The proof of the first statement is immediate from Lemma 1.3.10. If x = 7T(UJ) G if, then for r G P, xn = 7T(U>I . . . unr) converges to a: as n —> 00. Hence V^ (C) is dense in K. • We write V(A), F m and K instead of V(A, £), F m (£) and K(£) respectively if no confusion can occur.

1.3 Self-similar structure

23

Let A be a partition of D(5). If C = (K,S,{Fi}ies) is a self-similar structure and A ^ Wo, then we can define a self-similar structure £(A) = (if (A), A, {F^l^eA) as before. (See the definition of C(W) between Proposition 1.3.6 and Example 1.3.7.) Immediately, by Definition 1.3.9, it follows that K(A) = K and E(5) = E(A). (More precisely, E(A) can be identified with E(5) in a natural manner.) Of course, the topological structures of K and K(A) are expected to be the same since they are virtually the same self-similar structures. Proposition 1.3.12. Let C = (K, 5, {i7i}i€SF) be a self-similar structure and let A be a partition of E(5). Then Vc 2 Vc{h), where we identify E(5) and E(A) through the natural mapping. Furthermore, if A = VFm(5) for m>1, then Vc = ^£(A)Proof. Let a = OL\OL2 • • • £ ^C(A)> where c^ G A. Then there exists ft ft.. -An e W*(A)\Wo(A) and 7 = 7172... e E(A) such that TT(/3) = 7T(7) and ft ^ 71, where /? = Pifo.-.pm<x e E(A). Hence, if ft = W1W2 •..w m G Wm(S) and 71 = v\v2 ...vn G Wn(5), we can find k so that w\w2 ... Wk = v\V2 ..-vie and Wk+i ^ Vk+\. Therefore, as elements in E(5), 7r(akp) = 7r(ak-f) and hence a^/? G CC- This implies that a G P £ . Next let A = Wm for m > 1. For uj = LJIL>2 • - • ^ ^£ 5 there exists ^ G W*(S)\Wo(S) and r G E(5) such that TT(WLJ) = TT(T) and w1 / n . Now we can choose v G W*(S) so that vio = ft ft • • • /?? and vr = 7172 ... with ft, 7i G A and ft ^ 71. If a; = aia2 . . . , where a^ G A, then it follows a that ft ft ... /3jCKi«2 • • - G C/:(A). Therefore a; = aia 2 ... G Vc(A)Even if A ^ H^(5), P £ (A) often coincides with Vc- In general, however, this is not true. See Exercise 1.6 and 1.7 for examples. Finally, we will give the definition of post critically finite (p. c. f. for short) self-similar structure, which is one of the key notions in this book. Definition 1.3.13. Let C = (K, 5, {Fi}ieS) be a self-similar structure. C is said to be post critically finite or p. c. f. for short if and only if the post critical set Vc ls a finite set. If C = (if, 5, {Fi}ies) is post critically finite, Vm is a finite set for all m. In particular, Kw f) Kv is a finite set for any w ^ v G Wm. Such a self-similar set is often called a finitely ramified self-similar set. Obviously, a p. c. f. self-similar set is finitely ramified. The converse is, however, not true. Later, in Chapter 3, we will mainly study analysis on post critically finite self-similar sets.

24

Geometry of Self-Similar Sets

be post critically finite and let Lemma 1.3.14. Let C = (K,S,{Fi}ies) p G K. If Fw(p) = p for some w G W* and w^0, then 7r~1(p) = {w}. Proof Obviously, w G 7r~1(p). First we consider the case when w = k G W\. Assume that there exists r = T\T2 . . . G £ such that TT(T) = p and r T^ k. Without loss of generality, we may suppose that T\ ^ k. Let Tn = (o~k)nT for any n > 1 . Then rn G ft~1(p) and hence 7r~1(p) is an infinite set. On the other hand, since p G K^ H KTl, 7r~1(p) is contained in the critical set. As TT~1(P) is an infinite set, this contradicts the fact that C is post critically finite. Now for general case, let w G Wm. Then by Proposition 1.3.12, Cm = (K, Wrn,{Fv}V£Wm) ls a l s o P o s t critically finite. So applying the above argument to £ m , we see that 7r~1(p) contains only one element. Hence TT'1^ = {w}. • Example 1.3.15 (Sierpinski gasket). Let K be the Sierpinski gasket defined in Example 1.2.8. Then C = (K,S,{fi}ieS), where 5 = {1,2,3} and the fo are the same maps as in Example 1.2.8, is a post critically finite self-similar structure. In fact, CC,K = {^1,^2,93}, Cc = {12,2i,23,32,3i, 13} and Vc = {1,2,3}. Also Vo = {pi,P2,Ps} and V1=V0U {fc,ft,g 3 }See Figure 1.2. Example 1.3.16 (Hata's tree-like set). Let / i , f2 and K be the same as in Example 1.2.9. Then C = (/T,{l,2},{/i,/ 2 }) is a p. c. f.self-similar structure. In fact, C ^ = {|c2|}, C£ = {112,2i} and Vc = {12,2, i}. See Figure 1.3. Hence Fo = {c,0,l}. Also l^ = {c,0,1, |c| 2 ,/ 2 (c)}. Note that self-similar structures are isomorphic for all c with |c|, |1 — c\ G (0,1). Of course there are numerous examples of non-p. c. f. self-similar structures. One easy example is the unit square. (See Exercise 1.3.) Another famous example is the Sierpinski carpet, which may be thought of as the simplest non-trivial non-p. c. f. self-similar structure. Example 1.3.17 (Sierpinski carpet). Let pi = 0, p2 — 1/2, ps — 1, P4 = 1 + v/=T/2, ps = 1 + V-i, PG = 1/2 + v=1, pi = V=i and p8 = v/ z T/2. Set fi(z) = (z - pi)/S + pi for i = 1,2,... , 8. The self-similar set K with respect to {/i}i=i,2,...,8 is called the Sierpinski carpet. See Figure 0.4. Let C be the corresponding self-similar structure. The C is not post critically finite. In fact, CC,K-> CC and Pc are infinite sets. In particular, VQ equals the boundary of the unit square [0,1] x [0,1].

1.4 Self-similar measure

25

1.4 Self-similar measure In this section, we will introduce an important class of measures on a selfsimilar structure, that is, self-similar measures. First we will recall some of the fundamental definitions in measure theory. (X, M) is called a measurable space if X is a set and M is a α-algebra whose elements are subsets of X. A measure /x on a measurable space (X, M) is a non-negative α-additive function defined on M. Definition 1.4.1. Let (X, d) be a metric space and let ^ be a measure on a measurable space (X, M). (1) The Borel <j-algebra, S(X, d), is the minimal α-algebra which contains all open subsets of X. An element of #(X, d) is called a Borel set. If no confusion can occur, we write B(X) instead of S(X, d). (2) μ is called a Borel measure if M contains B(X). (3) μ is called a Borel regular measure if it is a Borel measure and, for any A G M, there exists B G B(X) such that μ{A) = /i(£) and ACB. (4) We say that μ is complete if any subset of a null set is measurable, i.e., B G M if B C A G M and μ(A) = 0. (5) μ is called a probability measure if and only if μ{X) = 1. The following proposition is one of the most important facts about Borel regular measures. See, for example, [124] and [158]. Proposition 1.4.2. Let (X, d) be a metric space and let μ be a Borel regular measure on (X,M). Assume that μ{X) < oo. Then, for any A G M, μ(A) = inf^(U) :U is a open set that contains A} = sup{/i(F) : F is a closed set that is contained in A} Proposition 1.4.3 (Bernoulli measure). Let S be a finite set. If p = {Pi)ieS satisfies ^2iesPi == 1 an^ 0 < Pi < 1 for any i G S, then there exists a unique complete Borel regular measure yIP on (E,.M P ), where E = 5 N , p or an w that satisfies μ {Tlw) = pWlpW2 .. -Pwm f V — ^1^2 • • • w m G W*. p This measure μ is called the Bernoulli measure on E with weight p. Remark. In this book, all the measures we will encounter are supposed to be complete unless otherwise stated. Also the Bernoulli measure with weight p is characterized as the unique Borel regular probability measure on E that satisfies

μ(A) = Y^piμ(ar\A)) ies

for any Borel set A C E.

26

Geometry of Self-Similar Sets

Proposition 1.4.4 (Self-similar measures). Let C =

(K,S,{Fi}i(zs)

be a self-similar structure and let n be the natural map from E to K associated with C. Ifp = {pi)ies £ ^S satisfies Y^ieSPi = 1 and 0 < Pi < 1 for p p p for A £ M = {A : any i £ S, then we define v by v {A) = μ^t (^(A)) 1 p p A C KJ7T~ (A) £ M }. Then, v is a Borel regular measure on {K,NP). p v is called the self-similar measure on K with weight p. It is known that vp is the unique Borel regular probability measure on K that satisfies

u(A) = ^2Piu(F-\A)) ies

for any Borel set Ac fact. By definition,

K. See [34, Chapter 2] and [76] for the proof of this

vp(Kw)

> pWlpW2..

(1-4.1)

.pWm

for any w = w\W2 .. • wm £ W*. Intuitively, it seems that equality holds in (1.4.1) rather than inequality if the overlapping set Cc is small enough. More precisely, we have the following theorem. Theorem 1.4.5. Let C = (K, 5, {Fi}ies) be a self-similar structure and let 7T be the natural map from E to K associated with C. Also letp = {pi)ies satisfy Y^ieSPi = 1 and 0 < pi < 1 for any i e S. Then vp(Kw)=pWlpW2...pWrn for any w = w\W2 • •. wm £ W* if and only if μp^Xoo) = 0, where loo

1

= W e E : #(7T- (7TM)) =

Remark. We will show that l^

+oo}.

p

£ M.

Lemma 1.4.6. For any A G MP, define m

Ao = {uj G E : a uj G A for infinitely many m £ N.}. p

Then Ao £ M

p

and μ (Ao)

p

> μ (A).

In particular, if A £ S(E) then

ioGB(S). Proof. Define aw = o~Wl o • • • oaWrn for w = w\W2 . . . wm £ W*. Set Am = p ^weWmO'w(A). Then Ao = limsup m _, o o Am. Hence Ao £ M and by Fatou's lemma, we have μp(Ao) > limsup m _, oo /x p (^4 m ). (Note that μp is a finite measure.) On the other hand, μp(Arn) = ^2weWrn Vp(o~w(A)) = p Y^weWmPw^i-A) = A* (^)> where pw = pWlpW2.. . p ^ m . Hence it follows

that μ^o)

> μp{A).

D

1.4 Self-similar measure

27

Lemma 1.4.7. Define I = {u G S : ^ T r " 1 ^ ) ) ) > 1}. Then 1 G B(S), loo e Mp, l o C I ^ C I and /iP(2o) = ^{I^) = /jf>(T). Proof. Set Im = U^^vGV^ml^ni^v). Then it follows that J m is closed and J = Um>i7T~1(/m). Hence I G 6(D). Now if u G lo, by using induction, we can choose {rrik}k>i, {n>k}k>i and {u^}k>1, {T^}k>1 C £ so that 1 < mi < rt\ < rri2 < n^ < • • • < m^ < n^ < rrik+i < • • • , <7mfeu; G I,α^u j= r(fc),7r(amfeu;) = TT(T^), U{h)

= LJKJ2 . . . UmkT{k\

UJi(jJ2 • • • ^

^

= 0; ( f c ) i0; ( / c ) 2 . . . o / ^ n ^

and (jjnfe T^ (jj(k^nk. This implies that 7r(u/fc)) = TT(O;) and hence u G X^. Thus we have shown that J o C J ^ C J . By Lemma 1.4.6, μp(I) < μ p (Jo), we can see that μp(I) = //P(XO). As μp is complete, l^ G A

0 implies /xp(T) > 0. Hence by Theorem 1.3.8, we have the following corollary. p

Corollary 1.4.9. If v (Kw)

= pw for any w G W*, then C is minimal.

Although the next theorem does not directly relate to self-similar measures, it tells us a useful fact: two Borel regular measures on a self-similar set are comparable if they are comparable on Kw for all w G W*. Theorem 1.4.10. Let C = (K, 5, {Fi}ies) be a self-similar structure. Let μ and v he Borel regular measures on (K,M^)) and (K,M(v)) respectively. Assume that v(K) < oo and v(I) = 0. // there exists c > 0 such that μ^yj) < cv(Kw) for any w G W*, then μ(A) < cis(A) for any A G M^μ) n M{y). In particular, //(I) = 0. Proof. Let U be an open subset of K. Set W(U) = {w G W* : Kw C U}. For w,v G W(U), we define w > v if and only if Hw D Ev. Then > is a partial order on W(U). If W~*~(U) is the collection of maximal elements in

28

Geometry of Self-Similar Sets

W(U) with respect to this order, then U = Uwew+(U)Kw and KWC\KV c X for w ^ v e W+(U). Therefore wew+(u)

wew+(U)

Now, by Proposition 1.4.2, for any A G A4(/x) D M(v), there exists a decreasing sequence of open sets {Ok}k>1 such that A C Ok for any k, μ(^k>lOk) = μ{A) and u(nk>iOk) = v(A). As ^(Ofc) < cv{Ok), we have /x(A) < ci/(A). D

1.5 Dimension of self-similar sets In this section, we will introduce the notion of the Hausdorff dimension of metric spaces and show how to calculate the Hausdorff dimension of self-similar sets. Definition 1.5.1. Let (X, d) be a metric space. For any bounded set A C X, we define H86(A) = i n f l ^ d i a m ^ ) 5 : A C U ^ i ^ d i a m ^ ) < 6}, i>1

where diam(i£) is the diameter of E defined in 1.2. Also, we define HS(A) = s limsup6ion 6(A). It is well-known that Hs is a complete Borel regular measure for any s s>0. See Rogers [157] and Falconer [32] for example. H is called the 5-dimensional Hausdorff measure of (X, d). Lemma 1.5.2. Let (X, d) be a metric space. For 0 < s
HUE) < S^HKE) for any E C X. Proof. If E C Ui>iEi and diam(Ei) < 6 for any i, then

Y^di&miEiY < ^diam^'MiamCEi) 5 < ^ " 5 ^ d i a m ( ^ ) s i>1

i>1

i>1

• By Lemma 1.5.2, we can obtain the following proposition. Proposition 1.5.3. For any E C X, sup{s : ns(E)

= oo} = inf{s : HS(E) = 0}.

(1.5.1)

1.5 Dimension of self-similar sets

29

S

Proof. By Lemma 1.5.2, if s < t, then H (E) < oo implies 7^(E) = 0 and S • also H\E) > 0 implies H (E) = oo. Now it is easy to see (1.5.1). Definition 1.5.4 (Hausdorff dimension). The quantity given by the equality (1.5.1) is called the Hausdorff dimension of E, which is denoted by dimuE. Remark. The Hausdorff measure and the Hausdorff dimension depend on a metric d. In this sense, if we need to specify which metric we are looking at, we will use the notation of dimn(^, d) instead of dimnE. The following lemma is often useful to calculate the Hausdorff dimension of a metric space. It is often called "Frostman's lemma". See, for example, Mattila [124]. It is also called the "mass distribution principle" in Falconer [33]. a Lemma 1.5.5. Let (K,d) be a compact metric space. IfH (K) < oo and there exist positive constants c, lo and a probability measure μ on Ksuch that

μ^iix))

< cl<*

for all x G K and any I G (0, IQ), then μ{A) < cHa(A) for any Borel set Ac K. In particular, 0 < Ha(K) < oo. Remark. According to the discussion of Moran [134], the converse of the above lemma is true: if 0 < 7ia(K) < oo, then there exists a probability measure μ on K such that, for some c > 0 , μ^xix))

for all x € K and / > 0. Moran proved this fact if K was a compact subset of Euclidean space. His argument, however, can be easily extended to this case. Proof. For U C K and x G 17, note that U C -Bdiam(i7)(x)A C \JiUi, then c^diam^r,

μ(A) < Y^M(Bdiam(tf,)te)) i

Hence, if

i

where x» G £/». Therefore μ{A) < cHf(A). Letting / —> 0, it follows that μ{A)
30

Geometry of Self-Similar Sets

Now let (K, { 1 , 2 , . . . , iV}, {i^}i
rVk for v = v\v2—

e W^,rWlW2,,,Wm_1 > a> rw}, Vk € W^.

Remark. A(r, a) is a partition of E. The following is our main theorem. This theorem was introduced in [88]. The ideas are, however, essentially the same as in Moran [134]. T h e o r e m 1.5.7. Assume that there exist r = ( r i , r 2 , . . . ,r/v) where 0 < 7*i < 1 and positive constants c\, c2, c* and M such that d i a m ^ ) < cxrw

(1.5.2)

#{w : w G A(r, a),d(x, Kw) < c2a) < M

(1.5.3)

for all w e W* and

31

1.5 Dimension of self-similar sets

for any x G K and any a G (0, c*), where d(x, Kw) = miy£Kw d(x, y). Then there exist constants C3, C4 > 0 such that for all A G B(K, d), czv(A) < na{A)

(1.5.4)

< c4v(A),

where v is a self-similar measure on K with weight (ria)i
and a is

e

f > « = l.

(1.5.5)

i=1

In particular, 0 < 7ia(K) < 00 and d i n i n g , d) = a. 1

Remark. Under the assumption (1.5.3), it is easy to see that #(7r~ (a:)) < M for any x G K. Hence, by Corollary 1.4.8, rwa

=

v(Kw) for any w G W*. Also u(I) = 0.

a

a

Proof We write A a = A(r,a). First we show that H (Kw) < (ci) v(Kw) for all w G W*. For w = W\W2 .. .wm G Ty*, we define Aa(w) = {v = v\V2 . •. Vk • wv G Att}^ where wv = W\W2 . . . wrnv\V2 . . . Vfc- Then we can see that A a (w) is a partition for sufficiently small a. Hence

rwa=

rwva.

^2

(1.5.6)

veAa(w)

By (1.5.2), it follows that diam(Kwv) < C\rwv < c\a for v G Aa(w). note that X^, = UveAa{w)Kwv Then

W?^^) < d

e

a

2

rwv

Also

a

= {Clrw) .

veAa(w)

Letting a —> 0, we obtain W Q ( ^ ) < (ci)Qr-wQ = ( d ) 0 * / ^ ) . a

a

Next we show that i/(KTO) < Mc2~ H (Kw). Let μ be the Bernoulli meaa sure on S with weight (ri )i
ir-l(BC2a{x))c

(J E™,

where Aa>a; = {t<;: w G A a , d(x, X^) < C2a}. Hence it follows that

v{BC2a{x))<

Y^ ^ € A a) X

M(S W ).

32

Geometry of Self-Similar Sets

Since μ ^ j = r£ < aa and #(A O , X ) < Af by (1.5.3), we have a

*(£<*«(*)) <

a

Mc2- (c2a) .

Lemma 1.5.5 implies that u(A) <

Mc2-ocna(A).

for any A G B(K, d). Hence there exist C3, C4 > 0 such that c 3 ^ C M < Ha(Kw)

<

c4v(Kw).

By Theorem 1.4.10, we can verify (1.5.4).

•

In the rest of this section, we show that the open set condition implies (1.5.2) and (1.5.3) of Theorem 1.5.7. Proposition 1.5.8. Suppose K is a subset ofRk, d is the Euclidean metric k k k ofR and Fi : R —> R is an ri-similitude for i = 1,2,... , N with respect to d. If the open set condition holds, i.e., there exists a bounded non-empty open set O C Rk such that N

\jFi(O)cO

and F^0) n Fd(0) = 0

fori^j,

i=1

then there exist constants c\,c2,M > 0 such that diam(Kw) < c\rw for all w G W* and #{w : w e A(r,a),d(x,Kw)

< c2a} < M

for all 0 < a < 1 and x G K. Proof. We can see that Kw C Ow for any w G W*, where O^ = F^(0). (By Exercise 1.2, it follows that O 2 -K\) Without loss of generality, we may assume that diam(O) < 1. Then, for all w G W*, dian^if^) < diam(0ly) < rw. Let m be thefc-dimensionalLebesgue measure and let < a}. Then U ^ A ^ O U ; C 52a(^)Aa,x = {w : w e A(r,a),d(x,Kw) Since the Ow are mutually disjoint, we have Y^weA m{Ow) < m(B2a(x)). Hence it follows that #(AaiX)rwkm(0) < 2kCak, where C = m(unit ball). Since rw > aR where R = min{ri,r2,... ,rN}, we see that #(Aa>iC) < k k 2 CR- m(0)-K D Corollary 1.5.9 (Moran's theorem). If K satisfies the open set condition, then dining, d) = a, where a is given by (1.5.5) with r^ = Lip(Ff).

1.6 Connectivity of self-similar sets

33

1.6 Connectivity of self-similar sets Let (K, 5, {Fi}i€s) be a self-similar structure. In this section we will give a simple condition for connectivity of K and also show that K is connected if and only if it is arcwise connected. As a reminder, the definition of connectivity is as follows. Definition 1.6.1. Let (X, d) be a metric space. (1) (X, d) is said to be connected if and only if any closed and open subset of X is X or the empty set. Also a subset A of X is said to be connected if and only if the metric space (A, <1\A) is connected. (2) A subset A of X is said to be arcwise connected if and only if there exists a path between x and y for any #, y G A: there exists a continuous map 7 : [0,1] —> A such that 7(0) = x and 7(1) = y. (3) (X, d) is said to be locally connected if and only if, for any x E X and any neighborhood U of x, there exists a connected neighborhood V of x with V C U. Of course, arcwise connectivity implies connectivity, but the converse is not true in general. Now we come to the main theorem of this section. Theorem 1.6.2. The following are equivalent. (1) For any i,j € S, there exists {ik}k=o,i,... ,n £ S such that IQ = i, in — j and Kik n Kik+1 ^ 0 for any k = 0 , 1 , . . . , n — 1. (2) K is arcwise connected. (3) K is connected. Proof. Obviously (2) =» (3). So let us show (3) => (1). Choose i € S and define A C S by A = {j E S : there exists {ik}k=o,i,... ,n Q S such that io = h *n= j and Kik D Kik+1 ± 0 for any k = 0 , 1 , . . . , n - 1}. If U = Uf e A #j and F = U^lf,-, then C/ H V = 0 and Cf U V = X. Also both J7 and V are closed sets because Ki is closed and A is a finite set. Hence U is an open and closed set. Hence U = K or U = 0. Obviously KiQU and hence [/ = X. Therefore V = 0 and hence A = 5. To prove (1) => (2), we need the following lemma. Lemma 1.6.3. For a map u : [0,1] —• K and for t G [0,1], we define D(u,t) = s\xp{\ims\xpd(u(tn),u(sn)) n—•<»

: lim tn = lim sn = t}. n—»-oo

n—>oo

7f / n : [0,1] —• K is uniformly convergent to f : [0,1] —* K as n —> 00 and lim<«_.~> D( f^.s) = 0. £/ien f is continuous at s.

34

Geometry of Self-Similar Sets

Proof of Lemma 1.6.3. Let d be a metric on K that is compatible with the original topology of K. litn-+s and sn —> s as n —> oo, then <*(/(*»), /(«»)) < d(/(*n), /m(tn)) + d(/m(*»), /„,(«„)) + d(f m(sn),

f(sn)).

Set r m = sup{d(/ m (£),/(£)) : 0 < t < 1}. Then the above inequality implies D(f,s) < 2rm + D(/ m ,s). Now letting ra —* oo, we can see that D(f, s) = 0. Hence / is continuous at s. D Now we return to the proof of (1) => (2). Define P = {/ : X 2 x [0,1] -> X : /(p,g,0) = p and /(p, g, 1) = g for any (p, g) € K2}. Also for f,g £ P, set dp(/,0) = sup{d(/(p,(7, t),(/(p, (/,*)) : (p,g,t) G X 2 x [0,1]}. Then (P, dp) is a complete metric space. By (1), for any (p, q) e K2, we can choose n(p,g)-, {ik(p,q)}o /* as m -> oo in P. Also set £>(/) = sup{£>(/(Pi9),t) : (p,q,t) G K2 x [0,1]} for f e P, where / ( M ) ( * ) = f(p,q,t). Then £>(Gm/) < r m Z)(/). Hence by Lemma 1.6.3, /*(p, <7, t) is continuous with respect to £. As /*(p, q, i) is a continuous path between p and o is a system of fundamental neighborhoods of x. If K is connected then Kw is connected for any w G W*. Hence Km,x is connected. • For p. c. f. self-similar structure Theorem 1.6.2 can be Written as follows. Corollary 1.6.5. If (K,S,{Fi}ies) is post critically finite, then K is connected if and only if, for any p,q G V\, there exist {pi}o
1.6 Connectivity of self-similar sets

35

In the rest of this section, we study connectivity of K\Vo for a connected p. c. f. self-similar structure. Proposition 1.6.6. Let (K, 5, {Fi}ies) be a connected post critically finite self-similar structure. Let C be a connected component of K\VQ. Then C is arcwise connected. Moreover, for any x G C and any p G C, there exists a path between x and p in C U {p}. Proof Let x G C. Note that Km,x is arcwise connected. (See Proposition 1.6.4 and its proof.) Since C is open, Km^x is contained in C for sufficiently large m. Hence if O = {y : y G C, there exists a path between x and y in C}, then O is open. Therefore, O = C. So C is arcwise connected. Let p G C. If p G C, the statement is obvious because C is arcwise connected. So assume that p G Vo- Choose m so that KmyP n Vb = {p}. Let y G ifm,p n C. Since lfm,p is (arcwise) connected, there exists a path between y and p. This path is included in C U {p}. Also there exists a path between x and y which is included in C. Joining those two paths, we obtain the desired path between x and p. • Proposition 1.6.7. Let (K,S,{Fi}i^s) be a connected post critically finite self-similar structure. Let p be the fixed point of F{. If C is a connected component of K\{p}, then Cfl Vb ^ 0. In particular, the number of connected components of K\{p} is finite. Moreover, let {Cj}j=i^,...}m be the collection of all connected components of K\{p}. Then there exists a permutation of {1,2,... , m}, p, such that Fi(Ck) = Cp^ n Ki. Proof. Suppose that C/i,... ,17/ are connected components of K\{p}. Then we may choose n so that Uj is not contained in Fin(K) for all j = 1,2,... , /. By Proposition 1.3.5-(2), UJOF^CVQ) ^ 0 for any j = 1,2,... , I. Therefore I #(Vb)- This implies that the number of connected components of K\{p} is finite. Now, let Ci,... ,C m be the collection of all connected components of K\{p}. Note that Fi(Cj) is connected and Uf=1Fi(Cj) = Ki\{p}. Therefore, there exists p(j) such that Fi(Cj) C Cp(jy Lemma 1.3.14 implies that KijP = Ki. Hence Ck H K{ ^ 0 for any k and there exists j such that Cj n F^~1(Cfc) T^ 0. This implies that p is a permutation of {1,2,... , m}. As X A W = UJLiCKi n Ci), we see that i ^ n C p(j) = F^Cj) for any j . Next choose n so that Cj is not contained in Fin(K) for any j . Then, it follows that Cpn(k) n i^71 (Vb) ^ 0 for any k. Since Cpn(fc) n Fi n (F 0 ) = Fin(V0 n Ck), we obtain Cfc n Vb ^ 0 for any jfe. • Proposition 1.6.8. Let C = (K, 5, {Fi}ies) be a connected post critically

36

Geometry of Self-Similar Sets

finite self-similar structure. (1) For any p G Vo, let J(p) be the collection of all connected components of K\{p}. Then J(p) is a finite set. (2) The number of connected components of K\Vo is finite. (3) Forpe Vo, define J(p, Vo) = {C : C is a connected component of K\Vo,p G C}. Set m(p, Vo) = #(«/(p, Vo)) and m(p) = #(J{j>)). Ifp eV0 is a fixed point of Fw for some w G W*\Wo, then m(p, Vo) = m(p). Proof. (1) Suppose that J(p) is an infinite set. Let {Ci}i>\ be a collection of connected components of K\{p}. Assume that Cj ^ Cj if i ^ j . Claim 1. # { j : diam(Cj) > e} < oo for any e > 0 . Suppose that diam(Cjfc) > e for any k>0. Then for any k, there exists Xk G Cjh such that d(p,Xk) > e/2. Since K is compact, we may choose a subsequence {xkn}n>i so that Xkn —>• x as n —» oo for some x G K. Let C be the connected component of K\{p} with x G C. Since C is a neighborhood of x, £fcn belongs to C for sufficiently large n. This contradicts the fact that each Xkn belongs to a different Cj. Hence we have the claim. Now since C is p. c. f., we may write n~1(p) = (u;(l),... ,u;(ra)}. Then there exists k such that, for any j = 1,2,... , m, o;(j) = w(j)v(j) for some w(j) G Wfc and some v(j) G VF*. Let pj = 7r(v(j)) for jf = 1,2,... ,ra. Note that p^ G Vo and Fw^(pj) = p for all j . Also, by Proposition 1.6.7, the number of connected components of K\{pj} is finite. (We may change the self-similar structure to Cn = (K, W n , {Fw}wewn)i where n = \v(j)\. Then apply Proposition 1.6.7 to pj, which is a fixed point of Fv^y) Since Kkfp = \JjLiFw(j)(K), it follows that the number of connected components of Kk,p\{p} is finite. (See Proposition 1.3.6 for the definition of Kk,p.) By the way, since K^^ is a neighborhood of p, Claim 1 implies that K^^ contains infinitely many connected components of K\{p}. This contradiction shows that the number of connected components of K\{p} is finite. (2) Suppose that the number of connected components of K\Vo is infinite. Then there exists p G Vo such that J(p,Vo) is an infinite set. By the same discussion as in the proof of (1), it follows that, for any e > 0 , #{C|C G J(p,Vb), diain(C) < e} = oo. On the other hand, let e = ming€y0\{p} d(p, q)/2 and assume that C G J(p, Vo) and diam(C) < e. Then C is a connected component of K\Vo- By (1), {C\C G J(p,Vo), diam(C) < e} is a finite set. (3) Let k = m(p,Vo) and let U = (UCeJ(P,v0)C) u W - T n e n u i s a

1.6 Connectivity of self-similar sets neighborhood of p. Hence, if w(n) = w...w,

37

then Kw(n) C U for suffi-

n times

ciently large n. Therefore if k' is the number of connected components of Kw{n)\{p}i then k' > k. Since m(p) = k\ we see that m(p) >ra(p,Vb)On the other hand, a connected component of K\{p} contains at least one C G J(p, Vb). Hence m(p) < m(p, Vb). • The next proposition also concerns a connected p. c. f. self-similar structure. It gives a sufficient condition for K\VQ to be connected. be a connected post critically fiProposition 1.6.9. Let (K,S,{Fi}ies) nite self-similar structure. Assume that, for any p,q G Vb, there exists a homeomorphism g : K —* K such that g(Vo) = Vb and g(p) = q. Then K\Vo is connected. If a connected p. c. f. self-similar structure satisfies the assumption of the above proposition, we say that the self-similar structure is weakly symmetric. To prove the above proposition, we need the following lemmas. Lemma 1.6.10. Assume the conditions in Proposition 1.6.9. Let C be a connected component of K\VQ. Then #(C C\ Vo) > 2. Proof. Suppose that there exists a connected component of K\Vb satisfying #(C Pi Vo) = 1. Let po be the unique po G Vo with p0 G C. For any p G Vb, there exists a homeomorphism g : K —• K such that g(po) = PLetting Cp = ^(C), we see that Cp is a connected component of K\Vo and CpCiVo = {p}. Then it follows that Cp is a connected component of K\{p}. Now, since C is post critically finite, there exists p G Vo such that p is a fixed point of Fw for some w G W*\Wo. Let m = \w\. By exchanging the self-similar structure C with Cm = (K, Wmi {Fv}vewm), we can use Proposition 1.6.7 and obtain that Cp D Vb ^ 0. This contradicts the fact that CpnV0 = {p}. • Lemma 1.6.11. Assume the conditions in Proposition 1.6.9. Thenjor all pe Vo, m(p,Vo) =m(p). Proof. Since C is weakly symmetric, m(p) and m(p1 Vo) are independent of the choice of p G Vb- Hence, as in the proof of the last lemma, we may assume that Fw(p) = p for some w G W*\WQ. Then the statement is immediate by Proposition 1.6.8-(3). • Proof of Proposition 1.6.9. Let J be the collection of connected components of K\VQ. Define V = Vo U J and E = {(p, C):peV0,C G J(p, Vo)}.

38

Geometry of Self-Similar Sets

Let G = (V, E) be a undirected graph, where V is the set of vertices and E is the set of edges. Note that G is connected. First we show that this graph G contains no loop. Suppose that there exists a loop in G : there exist {pi}i=i,2y...,n C VQ and {Ci}i=it2,...,n C J such that (pi,d), {pi+i,Ci) G J51 for z = 1,2,... , n, wherep n +i = plm Then Ci and Ci-i-i are connected components of K\VQ whose closures contain p^. Let A = \Jj=1(Cj U {pj}). Then for any i, A\{pi} is connected and is contained in one connected component of K\{pi}. Hence, C% and C^+i are contained in the same connected component of K\{pi}. This contradicts Lemma 1.6.11. Since G contain no loop, G is a tree : for any x,y G V, there exists a unique sequence of edges from x to y. Hence G has an end point, i.e., there exists x G V such that #{?/ G V : (#,2/) € E or (y,x) E E} = 1. By Lemma 1.6.10, we see that x G Vfo. Hence m(x,Vo) = 1. Therefore, m (p, Vo) = ^i(x, Vo) = 1 for any p G Vb. On the other hand, if # J > 2, then there exists p G Vb such that m(p, Vo) > 2 because G is connected. Hence # J = 1. •

Notes and references The general references to this chapter are Falconer[32, 33] and Yamaguti et al.[lS4\. 1.1 The notion of self-similar sets could be traced back to the Cantor set, which is the first mathematical example of what are now called "fractals". In [134], Moran considered a class of sets which are extension of the Cantor set and calculated the Hausdorff dimensions of them. His class includes, for example, the Koch curve and the Sierpinski gasket. After Mandelbrot proposed the notion of fractals in [122, 123], Hutchinson[76] formulated the mathematical definition of self-similar sets, which is more restrictive than what we call "self-similar sets" in this book. See the remark after Theorem 1.1.4. Theorem 1.1.4 was essentially obtained in [76]. See also Hata[64] for an extension of Theorem 1.1.4. 1.2 The shift space E and the shift map a are important concepts in dynamical systems. For example, they play an essential role in the study of interval maps. See [133] for example. Theorem 1.2.3 was essentially obtained in [76]. 1.3 Partly motivated by Kameyama[80], the notion of the self-similar structure was introduced in [83]. 1.4 Hutchinson has given the definition of self-similar measures in [76].

Exercises

39

1.5 See Rogers[157] for details on the Hausdorff measures. See also [124] for results from the view point of geometric measure theory. 1.6 Theorem 1.6.2 was obtained in [64]. The results after Proposition 1.6.6 are new.

Exercises n

Exercise 1.1. Let / : R —* Rn be a similitude with Lipschitz constant r. Show that there exist a G Rn and U G O(n) such that f(x) = rUx + a for all x G R n . (Hint: if g(x) = (/(z) - / ( 0 ) ) / r , one can see that \g(x) — g(y)\ = \x-y\. This implies that the natural inner product of Rn is invariant under g. Also one should show that g is a linear map.) Exercise 1.2. Let (X,d) be a complete metric space and let fo : X —» X be a contraction for i = 1,2,... ,7V. For A C X, define F(A) = Ui if (resp. E(5) -• if7) is denoted by n (resp. TT'). Note that both TT and TT' are homeomorphisms. S e t / 3 = /iO7T/o7r-1.

(1) Show that fs is a contraction on if. (2) Let C - (if,{l,2,3},{/i,/ 2 ,/ 3 }). int(C£,K) = 0.

Show that int(C£) ^ 0 and

40

Geometry of Self-Similar Sets

Exercise 1.6. Prove that VC(A) = ^c for any partition A for the selfsimilar structures corresponding to Example 1.2.7, 1.2.8, 1.2.9 in the last section. Exercise 1.7. Let S = {1,2,3}. Set u ~ r if and only if {LJ,T} C {wl212, w3i} for some w e W*(S) or u> = r. (1) Let K = Y,(5)l~ with the quotient topology. Also define Fi : K —> K by Fi(x) = ^ ( ^ ( T T - ^ X ) ) ) for x 6 l Then prove that C = (K, 5, {i7!}^^) is a self-similar structure. (2) Let A = {1,21,22,23,3}. Prove that VC(A) is a proper subset of VcExercise 1.8. Let C = (K,S,{Fi}ies) be a self-similar structure and let A be a partition of E(5). Show that C is post critically finite if and only if £(A) is post critically finite. Exercise 1.9. Evaluate the Hausdorff dimensions of the self-similar sets introduced in Examples 1.2.6-1.2.9 under the Euclidean metric.

2 Analysis on Limits of Networks

In this chapter, we will discuss limits of discrete Laplacians (or equivalently Dirichlet forms) on a increasing sequence of finite sets. The results in this chapter will play a fundamental role in constructing a Laplacian (or equivalently a Dirichlet form) on certain self-similar sets in the next chapter, where we will approximate a self-similar set by an increasing sequence of finite sets and then construct a Laplacian on the self-similar set by taking a limit of the Laplacians on the finite sets. More precisely, we will define a Dirichlet form and a Laplacian on a finite set in 2.1. The key idea is that every Dirichlet form on a finite set can be associated with an electrical network consisting of resistors. Prom such a point of view, we will introduce the important notion of effective resistance. In 2.2, we will study a limit of a "compatible" sequence of Dirichlet forms on increasing finite sets. Roughly speaking, the word "compatible" means that the Dirichlet forms appearing in the sequence induce the same effective resistance on the union of the increasing finite sets. In 2.3 and 2.4, we will present further properties of limits of compatible sequences of Dirichlet forms.

2.1 Dirichlet forms and Laplacians on a finite set In this section, we present some fundamental notions of analysis on a finite set, namely, Dirichlet forms, Laplacians and effective resistance. Notation. For a set V, we define £(V) = {/ : / maps V into R}. If V is a finite set, £(V) is considered to be equipped with the standard inner product (•,•)defined by (u,v) = J2Pev u(p)v(p) f° r a n y v>, v e ^00First we give a definition of Dirichlet forms on a finite set V. In B.3, one 41

42

Analysis on Limits of Networks

can find a definition of Dirichlet forms for general locally compact metric spaces. Definition 2.1.1 (Dirichlet forms). Let V be a finite set. A symmetric bilinear form on £(V), 8 is called a Dirichlet form on V if it satisfies ( D F l ) £(u, u)>0

for any u G t(V),

(DF2) £(u, u) = 0 if and only if u is constant on V and (DF3) For any u G £(V), £(u,u) > £(u,u), where u is defined by

{

1

ifu(p)>l,

u(p)

ifO
0

if w(p) < 0.

We use VT(y) to denote the collection of Dirichlet forms on V. Also V!F{y) is the collection of all symmetric bilinear forms on £(V) with (DFl) and (DF2). Obviously VT{V) C VF(V). Condition (DF3) is called the Markov property. This definition is a special case of Definition B.3.2 when X is a finite set V and the measure μ is the discrete measure on V. Notation. Let V be a finite set. The characteristic function subset U C V is defined by v

.

XU(Q)

fi

=
Xu °f

a

if 9 e^,

., . otherwise.

If no confusion can occur, we write xu instead of \u- If U = {p} for a point p G V, we write x P instead of X{p}- H -^ : ^(^) ~^ ^(V) is a linear map, then we set Hpq = (Hxq){p) for p,q G V. For / G ^(V),

(Hf)(p) = Y,qeVHpgf(q).

Definition 2.1.2 (Laplacians). A symmetric linear operator H : £(V) —> ^(F) is called a Laplacian on F if it satisfies (LI) iJ is non-positive definite, (L2) Hu — 0 if and only if u is a constant on V, and (L3) flp, > 0 for all p ^ q G F. ^_ We use CA(V) to denote the collection of Laplacians on V. Also CA(V) is the collection of symmetric linear operators from £{V) to itself with (LI) and (L2).

2.1 Dirichlet forms and Laplacians on a finite set

43

Obviously CA(V) C CA(V). There is a natural correspondence between V!F(y) and CA(V). For a symmetric linear operator H : £(V) —> £(V), we can define a symmetric quadratic form £H(*, •) on £(V) by £H(U,V) = —(u,Hv) for u, v G £(V). If we write TT(H) = £H, it is easy to see that TT is a bijective mapping between symmetric linear operators and symmetric quadratic forms. This correspondence between Dirichlet forms and non-negative symmetric operators is a special case of the correspondence described in Theorem B.3.4. Proposition 2.1.3. n is a bijective mapping between CA(V) Moreover, ir(CA(V)) = VT{V).

andVTiV).

Proof. It is routine to show IT(CA(V)) = VF{V). To show n(CA(V)) = VTiy\ first note that £H{u,u) = ^EP,qev Hpq(u(p) - u(q))2. By this expression, it is easy to see that 7T(CA(V)) C VT{V). NOW suppose H e CA(V)\CA(Y). So there exist p ^ q e V with Hpq < 0. We can assume that Hpq = — 1 without loss of generality. Set u(p) = x, u(q) = y and u(a) = z for all a G V\{p, q}. Then we have £H(U, U) = a(x — z)2 + P(y — z)2 — (x — y)2. As £H is non-negative definite, a and β should be non-negative. If x = 1, z = 0 and y < 0, then £H{u, u) = a - 1 + 2y + (/3 - l)y 2 and ^H(^> ti) = a — 1. If |y| is small, we have £#(i6, w) < £i/(u, ix). Hence £H £ VTiy). This shows that n(H) G PJ^(F) if and only if H G CA(V). D Example 2.1.4. Let V be a set with three elements, say, pi,P2»P3- Set / - ( l + c) 1 e \ H= 1 -2 1 . Then5 f f («,u) = (;E-2/)2 + ( y - ^ 2 + V e 1 -(1 + e)/ e(a; — 2) 2 , where a; = u(pi),y = u(p2) and 2; = u(jpz). Letting X = x — y and y = y — z, we have £H(u, u) = X2 + Y2 + e(X + Y)2 = (1 + 2e)(X2 + y 2 ) - e(X - Y)2 So it is clear that if e > - | , then H € ^4(1^) and if c > 0, then H £ CA(V). If V is a finite set and H is a Laplacian on V, the pair (V, # ) is called a resistance network (an r-network, for short). In fact, we can relate an rnetwork to an actual electrical network as follows. For an r-network (V, H), we will attach a resistor of resistance rpq = Hpq~1 to the terminals p and q for p, q G V. Also the plus-side of a battery is connected to every terminal p while its minus-side is grounded so that we can put any electrical potential

44

Analysis on Limits of Networks

on each terminal. For a given electric potential v G £(V), the current ipq between p and q is given by ipq = Hpq(v(p) — v(q)). So the total current i(p) from a terminal p to the ground is obtained by i(p) — (Hv)(p). Let (V, if) be an r-network and let U be a proper subset of V. We next discuss the proper way of restricting if onto U from the analytical point of view. Lemma 2.1.5. Let V be a finite set and let U be a proper subset of V. For H G CA(V), we define Tv : £{U) -> £(U),JV : £(U) -» £(V\U) and Xu : £(V\U) - £(V\U) by H

-\Ju

Xv)>

where iJ\j is the transpose matrix of J\j. (When no confusion can occur, we use T, J and X instead ofTjj, Jjj and Xjj.) Then, X — Xu is negative definite and, for any u G £(V), £H(U,U) = £x(ui + X~1Juo,ui

+ X"1JUQ)

+£T-tJX-1j(uo,uo),

(2.1.1) where UQ = u\jj and u\ = u\y\u. Proof. For v € £(V\U), define v by v\jj = 0 and v\v\u = v. Then £x(y,v) = £H(V,V) > 0. By (L2), we see that if £x(v,v) = 0 then v should be a constant on V. This implies v = 0. Hence £x is positive definite. By the definition of £x, X is negative definite. Now (2.1.1) can be obtained by an easy calculation. • Theorem 2.1.6. Assume the same situation as in Lemma 2.1.5. For u e £(U), define h(u) e £(V) by h(u)\v = u and h(u)\V\u = -X~xJu. £H(V, V). Also Then h{u) is the unique element that attains mmve^v^v\u=u 1 define PV,u(H) =T- VX" J. Then, Pv,u : CA(V) -+ CA(U) and £pv,u{H){u,u) = £H(h(u),h(u)) =

min ve£(V),v\u=u

£H{v,v).

(2.1.2)

Moreover, if H G CA(V), then PV,u(H) G CA(U). Proof By (2.1.1), minve^V)}V\u=u £H(VJ V) is attained if and only iiv\v\u+ X~xJu = 0. Hence we have the first part of the theorem. Next we show that PV,u(H) = T- VX" 1 J G £A{V). By (2.1.1), we can verify (2.1.2). Hence, £pvu(H) is non-negative definite. By (2.1.2), £pv,u{H)(v">u) — 0 implies that h(u) is a constant on V and therefore u is a constant on U. Thus we can show that Py^u{H) G CA(U).

2.1 Dirichlet forms and Laplacians on a finite set

45

Finally, if H G CA(V), then £pv,u(H)(u,u) = £H(h(u),h(u)) > £H(h(u),h(u)). As h(u)\u = u, we obtain £H(h(u),h(u)) > £pV!U(H)(u,u). Hence £Pvu^H>} has the Markov property. By Proposition 2.1.3, PV^(H) G CA{U). • The linear operator Pγ^u (H) may be thought of as the proper restriction of H onto U from the viewpoint of electrical circuits. In fact, H and Py,u(H) give exactly the same effective resistance (which will be defined in Definition 2.1.9) on U. When no confusion can occur, we write [H]jj in place of Pv,u(H). Remark. In general, Pvu is not injective. For example, set V = {pi,P2^Ps} / - ( l + e) 1 e\ and U = {pi,p2}- If He = 1 - 1 0 for e > 0, then He €

V

CA(V) and [Ht]v = ("j

1

€

0

-e)

1

J ^.

Note that /i(w) is the unique solution of (Hv)\v\u = 0 a n ( l vlc/ = uTherefore, if we think of U as a boundary of V, /i(tt) is called the harmonic function with boundary value u G -£({/). For a Laplacian if on V, we obtain the following maximum principle for harmonic functions. Proposition 2.1.7 (Maximum principle). Let V be a finite set and let H e CA(V). Also let U be a subset ofV. For p <E V\U, set Up = {q eU : There exist pi,P2,... ,p m € V\U with pi = p such that HPiPi+1 > 0 for i = 1,2,... ,ra— 1 and ifPmQ > 0}. Then if (Hu)\v\u — 0? *ften, /or an?/ p G V\[/, min w(g) < u(p) < maxu(q).

qeUp

qeUp

Moreover, u(p) = maxg€[/p u(q) (or u(p) = mmqejjp u(q)) if and only if u is constant on Up. Proof. For p G V, set Np = {q : Hpq > 0}. Also define Wp = {q G V\17 : there exist pi,P2, • •,Pm • G V\J7 with p\=p and p m = # such that HPiPi+1 > 0 for i = 1,2,... ,ra— 1} and Vp = Wp U C/p. First assume u(p*) = maxqevpu(q) for p* G Wp. Then ^ C Vp and (ffu)(p#) = E gG iv p , Hp^q(u(q) - u(p*)) = 0. Since

46

Analysis on Limits of Networks

Hp*q > 0 and u(q) — u(p*) < 0 for any q G NPm, we have u(q) — u(p*)

for all q G Np#. Iterating this argument, we see that u is constant on Vp. Using the same argument, it follows that if there exists p* G Wp such that u (p*) — mmgeVp U(Q)I then u is constant on Vp. Hence, min u(q) — min u(q) < u(p) < m&xu(q) = maxii(^). The rest of the statement is now obvious.

•

The following corollary of the maximum principle is called the Harnack inequality. Corollary 2.1.8 (Harnack inequality). Let V be a finite set and let H G CA(V). Also let U be a subset of V'. Assume that A C V\U and that Vp = Vq for any p,q G A. Then there exists a positive constant c such that maxu(p) < c min u(p) peA

p£A

for any non-negative u G £(V) with (Hu)\y\u = 0. Proof Let V = Vp for some p G A. (Note that V is independent of the choice of p G A.) Set A = {u : (Hu)V\u = 0,minpev u(p) > 0,maxpGv" u(p) — !}• By Proposition 2.1.7, m.inpeAu(p) > 0 for u £ A. If Ao = {u\v : u G .4.}, then Ao is a compact subset of £(V). Therefore, c = inf{miiip^A u{p) '• u ^ A$} > 0. By the definition of Ao, it follows that c = inf{miupeA u(p) : u G A}. This immediately implies the Harnack inequality. • Next, we define effective resistances associated with a Laplacian or, equivalently, a Dirichlet form. Prom the viewpoint of electrical circuits, the effective resistance between two terminals is the actual resistance considering all the resistors in the circuit. Definition 2.1.9 (Effective resistance). Let V be a finite set and let H G CA(V). For p^qeV, we define RH(p,q) = (mm{£H(u,u) : u G l(V\u{p) = l,u(q) = O})"1.

(2.1.3)

Also we define RH(P,P) = 0 for all p G V. Rn(p,q) is called the effective resistance between p and q with respect to H. By Theorem 2.1.6, if U = {p, q}, then it follows that

2.1 Dirichlet forms and Laplacians on a finite set

47

Definition 2.1.10. Let V{ be a finite set and let H{ e llA(Vi) for i = 1,2. We write (Vi,#i) < (V2,H2) if and only if Vx C y 2 and Pv^Vx(H2) = Hx. The next proposition is obvious by the above definitions. Proposition 2.1.11. Let V* be a finite set and let Hi e CA(Vi) for i = 1,2. //(Vi,ffi) < (V2,H2), then RHlip,q) = RH (P,Q) for anyp,qe V^ 2

In fact, the converse of the above proposition is also true if both H\ and H2 satisfy (L3). This fact is a corollary of the following theorem, which says that a Laplacian is completely determined by its associated effective resistances. Theorem 2.1.12. Let V be a finite set. Suppose HuH2e CA(V). Then Hi = H2 if and only if RHl(p,q) = RH2(P,q) for anyp,qeV. Proof. We need to show the "if" part. We use an induction on #(V). When #(V) = 2, the theorem follows immediately by (2.1.4). Now suppose that the statement holds if # ( V ) < n. Let V = {pi,P2>--- >Pn}« We write hij = (H!)PiPj and Hi5 = (H2)PiPj. Also let V{ = V\{pi} and let

Di^Ti-tjtixir'ji for k = 1,2, where T^ : ^(^) -> ^(VJ), Jj : ^(^) -^ ^(fe» and Xj : ^({pj) -^ t({Pi}) a r e defined by tJl

ft=f^

)

Since (Vi,D{) < (V,Hk), Rpifaq) = RDi(p,q) for all p,q e V^. By the induction hypothesis, D\ = D\. Now define D% = D\ = D\ and dlkl = (D%)PkPl. Calculating directly and then using the fact that h^ — hki w e and Hik = Hku obtain tfsi — hki — hikhii/hii = Hki — HikHii/Hii. In particular, 2

dik = hkk - h ik/hu = Hkk - H?k/HH. 1

(2.1.5)

Exchanging k and i, we see that d^/d ^ = ha/hkk = Ha/Hkk- Therefore, there exists t > 0 such that Hα = thu for i = 1,2,... , N. Again, by (2.1.5), we have (hik) = hkkhii — dlkkhii

48

Analysis on Limits of Networks

and (Hik)2 = HkkHii - d\kHii = t2hkkhii - tdkkhu. As -hkk = J2i:i^k hik

and

~thkk = -Hkk = J2i:i^k H*k, we obtain

-hkk = 22 yhkkhii - d\khii = 22 yhkkhii - dlkkhii/t. iii^k

i-.i^k

As a function of t, the right-hand side of the above equation is monotonically increasing. Hence the above equality holds only for t = 1. Therefore we obtain H\ = H^ • • Corollary 2.1.13. Let Vi be a finite set and let Hi e CA(Vi) for i = 1,2. Then (Vi,#i) < (V2,H2) if and only if RHAP^Q) = R 0. Unfortunately, we don't know whether such an extension is true or not. One reason why effective resistance is important is that it is a metric on V if H e CA(V). This metric, called the effective resistance metric, will play a crucial role in the theory of Laplacians and Dirichlet forms on (post critically finite) self-similar sets. Theorem 2.1.14. Let V be a finite set and let H e CA(V). Then RH(-, •) is a metric on V. This metric RH is called the effective resistance metric on V associated with H. Remark. Not every metric on a finite set V corresponds to an effective resistance metric with respect to a Laplacian H e CA(V). See Exercise 2.1 and Exercise 2.2. We need the following well-known formula about electrical networks to show Theorem 2.1.14. Lemma 2.1.15 (A—Y transform). Let U = {pi,P2,P3} o,nd let V = {Po} U U. Set Rij = H~}p. for H G CA(U), HPiPj > 0. Define

R\2 "I" -^23 H" -R3I

R\2 H~ -R23 H~ R3I

where we assume that

R\2 "I" ^ 2 3 + -^31

A

2.1 Dirichlet forms and Laplacians on a finite set Pi

P2

Pi

Ri T>

K-1/^

PS

R2S

49

^ ^ \

Po

T>

\it3

P2

A-circuit

PS

(upside-down) Y-circuit

Fig. 2.1. A-Y transform

IfH'eCA{V)

is defined by H,

1

=(R-

PiPj

\o

if i = o, othewise,

for i < j , then [Hf]u = H. A direct calculation shows this formula. As we mentioned before, we can associate an actual electrical circuit with a Laplacian. In the above lemma, the circuit associated with H e CA(U) has three terminals {pi,P2,P3} and the terminals pi and pj are connected by a resistor of resistance Rij. Let us call this circuit a A-circuit, which reflects the triangular shape of the circuit. At the same time, the circuit associated with Hf e CA(V) consists of four terminals {po,Pi,P2,P3} and each terminal pi is connected only to po by a resistor of resistance Ri for i = 1,2,3. po is a focal point of the circuit. Let us call this circuit a Ycircuit because of its "upside-down Y" shape. The A-Y transform says that the A-circuit and the Y-circuit are equivalent to each other as electrical networks. See Figure 2.1. Proof of Theorem 2.1.14. Definition 2.1.9 and (2.1.4) imply that #jf(p,g) > 0 and that RH(P> Q) = 0 if and only if p = q. Next we must show the triangle inequality. We may assume that # ( ^ ) > 3. For U = {pi,P2,P3} C V, let H' = [H]u- By Proposition 2.1.11, we have RwipuPj) = RH(PUPJ)> First assume that H'PmPn > 0 for any m ^ n. Then the A-Y transform

50

Analysis on Limits of Networks

shows

1

where Rmn = (-H"pmPn}~ and {z,j, &} = {1,2,3}. Hence we easily see that RH(PUP2)

+ RH(P2,P3)

RH(PUPS)'

f

Next, suppose H piV3 = 0 for instance. Then RH'(PI,P2) = R23 and RH'(PI,P3)

= R12 + ^23, where Rij

= R12, 1

= {H^.)' .

verify the triangle inequality.

RH'(P2,P3) So we

can

•

RH('i •) is not a metric on V for general H e CA(V). In fact, if #(V) > 3, there exists H £ CA(V) such that RH{', ) is not a metric on V. (See Exercise 2.4 and Exercise 2.5.) As we will see, however, ^RH{-, •) is always a metric on F for all if e ^ 4 ( F ) . The following is an alternative expression for the effective resistance. Proposition 2.1.16. Let V be a finite set and let H e *CA(V). Then, for any p,q eV,

RHfaq) = m a x { K ^ 7 ^ ) | 2 : u G l(y),£H(u,u) ± 0)}.

(2.1.7)

Proof. Note that \u(p) — u(q)\2/£H(U,U) = \v(p) — v(q)\2/£H(V,V) if v = OLU^-β for any a ^ e R with a ^ 0 . For given u € £(V) with u(p) ^ u(q), there exist a and /? such that v(p) = 1 and v(q) = 0, where ^ = au + /?. Hence the right-hand side of (2.1.7) equals max{

*

: i; G ^(V),t;(p) = l, V (g) = 0}.

Now (2.1.3) immediately implies (2.1.7).

•

Applying (2.1.7), we can obtain an inequality between \u(p) — u(q)\, Rit(p,q) and £H(U,U). Corollary 2.1.17. Let V be a finite set and let H G CA(V). p,q€V and any u G £(V), \u(p)-u(q)\2

< RH(p,q)£H(u,u).

For any (2.1.8)

This estimate will play an important role when we discuss the limit of a sequence of r-networks in the following sections. As another application of Proposition 2.1.16, we can show that \/RH('I •) is a metric on V.

2.2 Sequence of discrete Laplacians

51

Theorem 2.1.18. Let V be a finite set and let H G CA(V). Set ## / 2 (p, q) = \JRH{PI Q)- Then R^ (•, •) is a metric on V. Proof. We only need to show the triangle inequality. By (2.1.7),

R]i2(p,q) = m a x { K g - ^ ) l

:u

e t(V),eH(u,u) ± 0}.

This immediately implies t h e triangle inequality for R^

(•,•)•

^

2.2 Sequence of discrete Laplacians In this section, we will discuss the limit of r-networks on an increasing sequence of finite sets that satisfy a certain compatible condition, namely: Definition 2.2.1. Let Vm be a finite set and let F m G lCA(V) for each m>0. {(Vm, #m)}m>o is called a compatible sequence if (V^n,i/m) < ( y m + i , i J m + i ) for all m > 0 . Let S = {(V r m ,i7 m )} m > 0 be a compatible sequence. Set V* = Um^oV^ and define JT(S) = {u : u e £(V*), lim eHm{u\VrnJu\vJ

< +oo},

(2.2.1)

m—>oo

£s(u,v) = Jim^SHm(u\vm,v\vJ,

(2.2.2)

for u, v £ J^(<S). Also, for p, q £V*, define the effective resistance associated with S by Rs(p,q) = RHm(P,q),

(2-2.3)

where m is chosen so that p,q G VmIn the next chapter, we will approximate a self-similar set by a sequence of increasing finite sets. Then we will construct Dirichlet forms and Laplacians on the self-similar set by taking a limit of a compatible sequence of r-networks. Throughout this section, S = {(V^n,ifm)}m>o is assumed to be a compatible sequence. Let us regard Vm as a boundary of F*. Then for any u G £{Vm), we consider a minimizing problem of £s(', *) under the fixed boundary value u as follows. Lemma 2.2.2. There exists a linear map hm : £(Vm) —> ^(S) such that, for any u G £(Vm), ^m(^)|vm = u and SHm(u,u)

= £s(hm(u),hm(u))

=

min

vEJr(S),v\vrn=u

£s(v,v).

(2.2.4)

52

Analysis on Limits of Networks

Moreover, if v £ J~(S) with v\ym = u attains the above minimum then v = hm(u). Proof As [fl"n]vm = Hm for n > m, we can apply Theorem 2.1.6 with V = Vn, U = Vm and H = Hn. Set hn,m = h where h is the linear map £(U) -> £(V) defined in Theorem 2.1.6. Then define hm(u)\Vn = ftn,m(w). For any n > ra, this definition is compatible and hm(u) G £(V*) is welldefined. By (2.1.2), we have £Hm K for all n > m. immediately.

w

) = £Hn

Therefore hm(u)

(hm(u)\vn,hm(u)\Vri) 6 ^ ( 5 ) . Also (2.1.2) implies (2.2.4) •

Let us fix m. Then hm(u) is also characterized as the unique solution of

{

(Hnvn)\Vn\Vrn v

\vm

=0

for all n> m,

= u,

where v G t(V*) and vn = v\vn. So hm(u) may be thought of as the harmonic function with boundary values u G Vm. If Hm G £^4(V^) for all m > 0, we can show the following maximum principle for harmonic functions. We will sometimes regard l(Vm) as a subset of T(S) by identifying £(Vm) with hm(£(Vm)) through the injective map hm. With this identification, one can write £um(% u) = £s(u, u) for any u G £(Vm) C ^"(5). Lemma 2.2.3 (Maximum principle). Assume ifm G £.4(1^) /or a// 7n > 0. If v £ £{y*) satisfies (Hnvn)\vn\vm ~ ^ for a^ n > m> where min v(flf) < v(p) < max v(a) w - v ^ ; - qeVrn K*J

qeVm

for any p G V*.

Proof This follows immediately by the maximum principle for harmonic functions on a finite set, Proposition 2.1.7. • Next we discuss the effective resistance lls( , *)• As in the case for finite sets, y/Rs( i') is a metric on V*. Proposition 2.2.4. If Kg (•, •) = y/Rs(m, •)> ^ e n -^5 ^5 a rne^c on V*. Moreover if Hm G £^4(1^) /or aZ/ m > 0 , t/ien i?,s is a metric on V*. Proof This is an easy corollary of Theorem 2.1.14 and Theorem 2.1.18 along with Proposition 2.1.11. •

2.2 Sequence of discrete Laplacians

53

The following lemma follows immediately from its counterpart, Proposition 2.1.16. Lemma 2.2.5. For any p, q G V*, Rsfaq) = (min{£s(u,u) : u G F(S),u(p) = l,u(q) = O})"1 = maxj

1 ^"^1 £s(u,u)

: u G F(S),£s(u,u)

> 0}. (2.2.5)

This lemma implies that \u(p) -u(q)\2 < Rs(p,q)£s(u,u) for any u G T(S)

and p,q e V*. By (2.2.6), f(S)

(2.2.6) C C{V*,R)!2).

For a

metric space (X, d), C(X, d) is the collection of real-valued functions on X that are uniformly continuous on (X, d) and bounded on every bounded subset of (X, d). Next we present important results on the limit of compatible sequences. In the following chapter, the results will be applied in constructing Dirichlet forms and Laplacians on a self-similar set. Theorem 2.2.6.

(1) f(5)cC(K,4 / 2 )-

(2) £s is a non-negative symmetric form onJ-(S). Moreover, £s(u,u) = 0 if and only if u is a constant on V*. (3) Define an equivalence relation ~ on F(S) by letting u ~ v if and only ifu — v is constant on V*. Then £s is a naturally defined positive definite symmetric form on F(S)/~ and (J-(S)/~,£s) is a Hilbert space. (4) Assume that iJ m G CA(V) for all m>0. If u is defined as in (DF3) for any u G F(S), then u G FiS) and £s{u,u) < £s(u,u). Proof. Every statement but (3) follows easily from the results and discussions in this and the previous section. We will use £ and T in place of £s and T{S) respectively. To show (3), first note that £(u,u) = £(v,v) if u rsj v. Hence £ is a well-defined positive definite symmetric form on F/~. Choose any p G V* and set Tv = {u : u G ,F,u(p) = 0}. Then (.77~,£) is naturally isomorphic to (!FP,£). Hence it suffices to show that (Tv, £) is a Hilbert space. Now let {vn}n>o be a Cauchy sequence in (.?>,£) and let v% = hm(vn\Vm). Then by Lemma 2.2.2 £(vF - < \ V ? - vD r

£(vk -vhvk-

vi).

Note that p G Vmfc> sufficiently large m. Hence £ is an inner product on Tv n £(Vm), where £(Vm) is identified with /im(^(Kn))« So there exists

54

Analysis on Limits of Networks

m

m+1

m

m

v G Tv Pi £(Vm) such that v™ -> v as n -> oo. As v | y m = v , there 171 exists v G ^(Vi) such that v\ym = v . On the other hand, let C = s u p n > 0 £(vn, vn). Then we have £(v™, vm) < sup n > m £ « \ O = C. Hence t; G > . Now, we fix e > 0. Then, we can choose n so that £(vn — Vk,vn — Vk) < e for all A: > n. Also, we can choose m so that 171 I ^ K - v,vn - v) - £(y™ - v ^™ - v^1 < 6.

Furthermore, we can choose k so that k > n and \£«

m - V™, < - V?) - £(V™ - V™,V™ - V )\ < 6.

As£(v™-v%l,v™-v%i) <£(vn-vk,vn-vk) < e, wehave £(vn-v,vn-v) < 3e. Thus we have completed the proof of (3). • Finally we show two examples. The first one is related to one of the most basic examples in probability theory. Example 2.2.7 (Simple random walk on Z). Let F m = {i e Z : \i\ < = 1 if \i-j\ = 1, ( # m ) ^ = m} and let Hm e CA(Vm) be defined by (H^ r 0 if \i — j\ > 1. Then S = {(V m ,i7 m )} m >i is a compatible sequence. We can easily see that V* = Z, Rs(i,j) = \i — j\ and £s(u, v) = Y^{u(i + 1) - u(i))(v(i + 1) - v(i)). Also we can see that (^,^"(5) PI L2(Z,//)) is a regular Dirichlet form on L 2 (Z, μ) for every Borel measure /i on Z that satisfies 0 < μ^i}) < oo for all i e Z . (See Definition B.3.2 for the definition of a regular Dirichlet form.) 2 Define a linear operator Aμ on L (Z,/x) by (Apu)(i) = M O '

1

^ + 1) + u{% - 1) - 2u{%)).

Then A^ is a non-positive self-adjoint operator on L 2 (Z, μ). Also Dom(AM) C J-(5)nL 2 (Z,/i)and £s(u,v) = - /

UA^μ

Jz for any u, v G Dom(AM). From this fact, AM is identified as the self-adjoint operator associated with the closed form (£s, F(S) DL2(Z, μ)) on L 2 (Z, /i). (See B.I for more information about closed forms and their associated selfadjoint operators.) Now if u(i) = 1 for all i G Z, then Au is the self-adjoint operator

2.3 Resistance form and resistance metric

55

associated with a simple random walk on Z in the following sense. Let u0 G Dom(AI/) and think about the following evolution equation with discrete time n = 0,1,2,...:

One can easily see that un = (I-\-Au/2)nuo for any n. For i G Z, if wo(^) = 1 and uo(k) = 0 for any k ^ z, then un(j) is the transition probability from i at time 0 to j at time n for the simple random walk on Z. The next example is an extreme case where Rs is the trivial metric on

Example 2.2.8 (Discrete topology). Let Vm = {1,2,... ,ra} and let tfm G £A(V m ) be defined by ( i f m ) ^ = 2/ra for i ^ j. Then 5 = {(y m , # m ) } m > 2 is a compatible sequence. We can easily verify that V* = N and Rs{i,j) = 1 for z ^ j. This metric ife induces the discrete topology on N. As Xi G F{S) for all i e N , (£5, ^ ( 5 ) ) is a regular Dirichlet form on L 2 (N, μ) for every Borel measure /i on N that satisfies 0 < /x({i}) < 00 for all i G N. In particular, if /i({z}) = 1 for all i G N, then L 2 (N,^) = ^ 2 (N). We see that ^ 2 (N) n ^ ( 5 ) = ^ 2 (N) and, for all u, v G ^ 2 (N),

£s(u,v) = 2

uvdμ. JN

2.3 Resistance form and resistance metric In the previous section, we constructed a quadratic form (£s,T(S)) and a metric Rs from a compatible sequence of r-networks <S = {(V^, #m)}m>oIn this section, we will give characterizations of the form (£5, .F(«S)) and the metric Rs and show that there is an one-to-one correspondence between such forms and metrics. First we give a characterization of quadratic forms.

Definition 2.3.1 (Resistance form). Let X be a set. A pair {£,F) is called a resistance form on X if it satisfies the following conditions (RFl) through (RF5). (RFl) J 7 is a linear subspace of £(X) containing constants and £ is a nonnegative symmetric quadratic form on T. £(u,u) = 0 if and only if u is constant on X. (RF2) Let rsj be an equivalent relation on T defined by u ~ v if and only if u — v is constant on X. Then (F/^, £) is a Hilbert space.

56

Analysis on Limits of Networks

(RF3) For any finite subset V C X and for any v G £(V), there exists u G T such that u\y = v. (RF4) For a n y p , g G X , lu{p)

snP{

Ui

- f:u^,S(u,u)>0}

C{U,U)

is finite. The above supremum is denoted by M(p,q). (RF5) If u G T, then u G T and £(iZ, u) < £(u,u), where u is defined in the same way as (DF3) in Definition 2.1.1. We use TZ!F{X) to denote the collection of resistance forms on X. Also we define UT(X) = {(£,?) : {£,T) satisfies the condtions (RFl) through (RF4)}. The condition (RF5) is called the Markov property. Let V be afinitejet. Then (£,£(V)) G nF(V) (or (£,£{V)) G nT(V)) if and only if £ G VT(V) (or £ G VT(V) respectively.) Also, immediately from Theorem 2.2.6, (£s,.F(<S)) belongs to KFiy*) for any compatible sequence S = {(Vm, i^m)}m>o- Moreover, if Hm G CA(Vm) for all m, then {£s,F(S)) is a resistance form on V*. Next we consider a characterization of metrics. Definition 2.3.2 (Resistance metric). Let X be a set. A function R : X x X -^ [0, +oo) is called a resistance metric on X if and only if, for any finite subset V C X, there exists Hγ G £*A(F) such that R\vxv = RHV, where RHV is the effective resistance with respect to Hγ. The collection of resistance metrics on X is denoted by IZM(X). Also we define UM{X) = {R : X x X -> M+ : for any finite subset V C X, there exists ffv € £A(V) with R\VxV

= RHv and fT^ = [HV2]Vl if VI C V2}.

Remark. Recall that [i^v2]vi = ^2,^1(^1^) by definition. Notice that by Corollary 2.1.13, the condition Hy1 = [Hy2]y1 is satisfied for a resistance metric R. If we could extend Theorem 2.1.12 to CA(V), which is quite likely, then we could remove the assumption HY1 = [Hy2]y1 from the definition of UM(X). Since RHV is a metric on V, a resistance metric R is a distance on X. Also, for R G IZM(X), y/R(-,-) is a distance on X. Let V be a finite set.JThen R G UM{V) (or i? G 72^4(V)) if and only if R = RH for some H G £A(V) (or iJ G £«A(V) respectively). Furthermore, it is natural to expect that Rs is a resistance metric for a compatible sequence S. More precisely, we have the following proposition.

2.3 Resistance form and resistance metric

57

Proposition 2.3.3. If S = {(V^,il m )} m >o is a compatible sequence, then Rs G TIM(V*). In particular, if Hm G CA(V) for all m, then Rs is a resistance metric. Proof Let V be a finite subset of V*. Then V C Vm for sufficiently large m. If iJy = [iirm]y, then RHV — -R«s|vxv- The rest of the conditions are obvious. • There is a natural one-to-one correspondence between resistance forms and resistance metrics. First we will construct a resistance metric from a resistance form. Theorem 2.3.4. // (£,.F) G TIT{X), then min{£(u,u) : u G J-,u(p) = l,u(q) = 0} exists for any p,q G X withjpjfi- q. If we define R(p, q)~x to be equal to the minimum value, then R G 71A4(X) and R(p,q) = m a x { | u ( p j ~ " ( g ) | 2 : u € T,£{u,u) > 0}. c(u, u)

(2.3.1)

Moreover, if (S.J7) G TIT{X), then R G TIM(X). To prove the above theorem, we need the following lemma. Lemma 2.3.5. // (£,T) G 11T{X) and V is a finite subset of X, then there exists a linear map hy : £(V) —> T such that hy{u)\y = u and £(hv(u),hv(u))=

min

E(v,v).

(2.3.2)

Furthermore, hy(u) is the unique element that attains thejibove minimum. Also set £v(u1,u2) = £(hv(ui),hv(u2)). Then £v G VTiy). Moreover, 7 v if (£, J ) is a resistance form, then £ G VTiV). Proof For p G V, let Tv = {u : u G T,u\v\{p} = 0}. Then by (RF3), Tv is not trivial. By (RF2), {Tv\S) is a Hilbert space. Define $ p : Tv -^ R by *p(w) = u(p). Then by (RF4) we have, for q G V\{p}, \$p(u)\2 < M(p,q)£(u,u) for all u G Tv. Hence <J>P is a continuous linear functional on (TV,E). Therefore there exists gp G Tv such that for all u G Tv\ £(gp,u) = $p(u) = u(p). Noting that £(gp,gp) = gp(p) > 0, we define ^p

=9p/9p(p)'

Now for any u G £(V), define hy(u) = J2peV u(p)ip^. If v G T with

58

Analysis on Limits of Networks

v\y — u, set v — v — hy(u). Then £ (v, v) = £ (v + hy(u), v + hy (u)) = S(v, v) + 2£(6, ftv(w)) + £(hy(u),

hv(u)).

As £(£, /iv(ii)) = ^Zpey u(p)v(p)/gp(p) = 0, we have £(v,v) = £(v,v) + £(hy(u),hy(u)) > £(hy (u), hy (u)). Equality holds only when v is constant on X and so v = 0 on X. It is easy to see that £v G VF(V). Also the Markov property (RF5) of [£,T) implies the Markov property (DF3) of £v. Thus we have completed the proof. • Proof of Theorem 2.34. By Lemma 2.3.5, min{£(u,u) : u G T, u(p) = l,u(q) = 0} exists for any p,q £ X with p ^ q. Now define Hy G CA(V) by £v = £HV . IfViCF 2 ,then 2

x

ve£(v2)Mv1=u

=

min

v'eJr,v'\v1=u

f^, i;;) = £H 1 (u, u).

Hence [i^y2]vi = Hyx. This fact also implies that RHV — R\vxv- Therefore we see that R G KM(X). If (f,^ 7 ) G HT{X)\ then # y G Pjr(y) and hence R G 7JA<(X). The same argument as in the proof of Proposition 2.1.16 implies (2.3.1). • Theorem 2.3.4 says that each (£, T) G 1Z,T(X) is associated with R G UM(X). So we can define a map F M X : 11T{X) -> 1ZM(X), which is called the "form to metric" map, by R = FMx{{£,F)). This form to metric map is, in fact, bijective: we can construct the inverse of FMx> Theorem 2.3.6. For R G IZM(X), there exists a unique (£.J7) G KJr(X) that satisfies (2.3.1). Moreover, if R G IZM(X), then [£,T) G 11T{X). Assuming the above theorem, we can define the "metric to form" map MFx : UM{X) -> nT(X). It is easy to see that MFX is the inverse of FMX. We will only present the proof of a special case of Theorem 2.3.6, namely Theorem 2.3.7. Theorem 2.3.6 can be proven by using routine and tedious arguments about limiting procedures similar to the special case. If R G nM(X), R1/2(-r) = VR(-r) is a metric on X. Assume that

2.3 Resistance form and resistance metric

59

the metric space (X, R1/2) is separable. Equivalently, there exists a family of finite subsets {Vm}m>o of X that satisfy Vm C Vm+i for m > 0 and K = Um>oVrm is dense in X. Set # m = HVrn. Then iJ m G CA(Vm) and [ffm+ilvm = #m by definition. Hence (Vm,Hm) < (y m + i,iJ m + i) and so S = {(Vm,iJm)}m>o is a compatible sequence. We know that (£s,f(S)) G T£F(V*). Also it is obvious that R = Rs on V*. Now as .F(«S) G C(V*,RJ ) , U E F(S) has a natural extension to a function in C(X, R1/2). We will think of F(S) as a subset of C(X, Rl/2) in this way. Then it is easy to see that (£5,.F(S)) satisfies (RFl) and (RF2). This (£s-,Jr(S)) is the candidate for (£,F) in Theorem 2.3.6. The problem is to show (RF3) and (2.3.1) for any p,q G X. (We already know that (2.3.1) holds for p, q G V* by Lemma 2.2.5.) We do this in the next theorem. Theorem 2.3.7. For R G UM(X), assume that (X,R1/2) is separable. Let {Vm}m>o be a family of finite subsets of X such that Vm C Kn+i for any m>0 and that V* = Um>oKn is dense in X. Set S = {(Vm, iJ m )} m > 0 where Hm = HVm. Then (£S,F(S)) G 11T{X) and R(p,q) = m a x j

1

^"^

1

: u G F(S),£s{u,u)

> 0}

(2.3.3)

for all p, q G X. Moreover, (£s,F(S)) is independent of the choice of {Vm}m>*. Also if Re 1ZM{X), then (£S,F(S)) G UT^X). Before proving the theorem, we need two lemmas. L e m m a 2.3.8. Let (£^) G TZJ-(X) and let {Vm} be a sequence of finite subsets of X such that Vm C Vm+i for m>0 and that V* = Dm>oVm is dense in (X.R1/2), where R - FM X ((£,.F)). If S = { ( F m , ^ m ) } m > 0 where Hm = HVrn, then ( ^ , ^ ( 5 ) ) = (£,T).

Remark. In this lemma, again we regard F(S) as a subset of C(X, R1/2) because R - Rs on V* and ^(5) C C(V*, R)/2). Proof. First we show that ^"(5) C T and £s(u,u) = £(u,u) for u G T(S). Let u G F(S). Set um = hvm(u\vm)i w n e r e ^ym is defined in Lemma 2.3.5. As £Hm(u\vmju\vm) ~ £(um->um), w e obtain £(um,um) < £(um+i,Um+i) < £s(u,u). Now without loss of generality we may assume that u(p) = 0 for some p eVo. (We can just replace u by u — u(p).) Note that (um — un)\vn = 0 for m > n. Then, recalling the definition of hy in the proof of Lemma 2.3.5, it follows that £(um - un,un) = ^2 (um(p) - un(p))u(p)I'gp(p) = 0 pevn

60

Analysis on Limits of Networks

for m> n. Hence £(um — ^n^m - un) = £(urn,urn) - £(un,un) —•0 as ra, n —> oo. Therefore {wm}m>o is a Cauchy sequence in {Tv, £). As (J^, £) is complete by (RF2), there exists u* G Tv such that £(u*—u m,w* — wm) —> 0 as m -^ oo. So £(u*,u*) = limrn^oo£(urn,urn) = £s(u,u). For g G V*, we have I^*(?) -Um(q)\2

R(p,q)£(u* -um,u*

-um).

Letting m —> oo, we obtain u\y^ = it*|v;. As it and w* are continuous on X with respect to i? 1 / 2 , it follows that u — u*. Thus we have shown that u e T and £(u,u) = £s(u,u). Secondly, for u G f , define um exactly the same as before. Then £{um, um) =

min

£(v, v) < £(u, u).

V$LJ~\V Y =U\V

Hence u G ^"(5). Now using the argument of the latter half of this proof, we see that £s('u>, u) = £(u,u). • Lemma 2.3.9. Let (£,T) G UT(Y) and let ( F , ^ 1 / 2 ) be the completion of(Y,R1^2), where R = F M y ( ( f , / ) ) . Then, for anyp,qe Y, R(p,q) = m a x { K p j " ^ ) | 2

: u G F,e(u,u)

> 0}.

Proof First we will show that R(p,q) = s u p { | t x ( ^ " t A ( g ) | 2 : u G .F,£(ti,u) > 0}. (2.3.4) £(u,u) We will denote the right-hand side of (2.3.4) by M(p, q). Choose {pn}, {qn} C Y so that Pn^ P and qn -^ q as n —> oo. Note that, i^,^)

=m

a x

{ M ^ M

: Me

jr,f (M,w) > 0}.

(2.3.5)

Hence, we have \u(pn) — u{qn)\2 < R(pn, qn)£(u,u) for any u G T. Letting n —± oo, we obtain \u(p)—u(q)\ 2 < R(p, q)£(u, u). Hence M(p, q) < R(p, q). Suppose M(p, q) < R(p, q). Then we can choose e so that for all u G T", |w(p) - ^(g)I < (yR(P>q) -

5e)y/£(u,u).

On the other hand, since R(pn,qn) —> R(p,q) as n —> oo, using (2.3.5), there exists {un} such that £[un, un) = 1 and \un(pn) - un(qn)\2 —> i?(p, (yJ~R(p,q) - e)

2.3 Resistance form and resistance metric

61

and R(pn,Pm),R(qn,qm) < e2 for all m > n. Furthermore, we can choose m so that m> n and un(pm) - un{p)\ < e and

\un{qm) - un(q)\ < e.

Now we have

\un(Pn) - Un(qn)\ < \un(pn) - Un(pm)\ + \un(pm) - Un{p)\ + \un(p) - un(q)\ + \un(q) - un(qm)\ + \un(qm) - un(qn)\ < \Un{Pn) - Un(Pm)\ + |^n(<7n) ~ Un(qm)\ + (yR(p,q) - 3c). Hence \un(pn) - un(pm)\ > e or |wn(^n) - un(qm)\ > e. This contradicts the fact that R(pniprn),R(qniqrn) < e2. Therefore we have shown (2.3.4). Now using the same argument as in the proof of Lemma 2.3.5, it follows that there exists \j) £ T such that ip{p) = 1, ip(q) = 0 and ip attains the supremum in (2.3.4). • Proof of Theorem 2.3.7. As we mentioned before, (£s, ^F(S)) (recall that we think of JF(S) as a subset of C ^ i ? 1 / 2 ) ) satisfies (RFl) and (RF2). To show (RF3), set V^ = Vm U V for a finite set V C X. Let H'm = Ev> and let S' = {(V^H'm)}. Then for any u G £(V), there exists v G F(S') such that v\v = u. As (Vm,Hm) < (V^H'J, £Hm{v\vmMvm) £ H^(v\v^,v\v^) < £s'{V)V). Hence ^nim^QoEH^v^Mvm) £s'(v,v). Therefore v G ^(5). This shows (RF3). Next, applying Lemma 2.3.9 to the case that Y = V*, we obtain (2.3.3) because X C F . This implies (RF4). Thus we have shown (£s,JF(<S)) G nF(X). Furthermore, (2.3.3) also implies R = FMx((£s,f ($)))• Let {Um} be a sequence of finite subsets of X that satisfies the same condition as {V^} and let Si be the compatible sequence associated with Um. Then, applying Lemma 2.3.8, we can see that (£s,.F(<S)) = (£s1,Jr(Si)). Hence (Ss^iS)) is independent of the choice of {V^}. Finally, if R G KM(X), then HVm G CA(Vm). Hence ( ^ , ^ ( 5 ) ) has the Markov property. • Using the discussions in this section, we can show another important fact about resistance forms and resistance metrics. If S = {(Fm,i]rm)}rn>o is a compatible sequence, then (£5,.F(S)) G T£F(V*) and Rs G 11M(V*). The space V* is merely a countable set. So if we constructed analytical objects like Laplacians or Dirichlet forms from (^5,^"(«S)), we would end up with analysis on a countable set. That is hardly what we want! One way of overcoming this difficulty is to consider the completion of

62

Analysis on Limits of Networks

V* with respect to the metric Rj . Let (Qs^Rg ) be the completion of (F*,i?£ ). Then (Qs^Rj ) might be an interesting uncountably infinite set. As we mentioned before, F(S) can be naturally regarded as a ). Hence (£s,3~(S)) can be considered as a quadratic subset of C(QS,RJ form on (Qs, RJ ). There is, however, a delicate question about this completion procedure. Is the extended R$ in 1ZAi(Qs)7 Equivalently, do we have (£s,.F(<S)) G 7^^-*(X)? This is not a trivial problem. In fact, this is not true in general. See Exercise 2.7 for a counterexample. Fortunately, if we assume the Markov property, i.e., Hm G CA(Vm) for all m > 0, then it follows that RlJ2 G nM(tts) and (£S,F(S)) G ilF^s) by virtue of the next theorem.

Theorem 2.3.10. Let (S.J7) G HT(X). If (X,R) is the completion of (X, R), where R = FMX((£, J 7 )), then (£, T) G TLT{X) and R G UM(X). Proof (RFl), (RF2) and (RF5) follow immediately. Also (RF4) is an obvious consequence of Lemma 2.3.9. Instead of (RF3), we will show the following (RF3*). (RF3*) For each finite subset V C X and for each p G V, there exists u G T such that u\y — Xp, where Xp i s the characteristic function of the one point set {p}. We use an induction on #(V) to prove (RF3*). If #(V) = 2, say V = {p, q}, then by Lemma 2.3.9, there exists u G T such that u(p) ^ u(q). If we set / = (u- u(q))/(u(p) - u(q)), we have f\v = xPNext suppose (RF3*) holds for #(V) < n. Let V = {pi,P2,-- ,Pn}Then by the induction hypothesis, there exists u G T such that u(pi) = 1 and u(pi) = 0 for i > 3. Case 1. If u(p2) < u(pi), then for some a,/? E M, v = aw + /? satisfies v{p1) = 1 and v(pj) < 0 for j > 2. Define I; as in (DF3) of Definition 2.1.1. Then by the Markov property (RF5), we have v G T. Obviously v\y — xPl • Case 2. If ufa) = w(pi), then choose / G T that satisfies /(pi) > /(P2) and |/(pi)| < 1/2 for alH = 1,2,... , n. For some a, /? G R, v = a(w+/)-h^ has the same properties as v in Case 1. Case 3. If u(pi) < u(p2), then using the same argument as in Case 1, we can find v G T that satisfies v\y = x P2 . Thus if / = u — u{p2)v, then f\v = XpiThus we have shown that (RF3*) holds for #(V) = n.

D

63

2.4 Dirichlet forms and Laplacians on limits of networks 2.4 Dirichlet forms and Laplacians on limits of networks

In the last section, we studied relations among a compatible sequence of r-networks, a resistance form and a resistance metric. In this section, we will take a first step to establish an "analysis" on limits of networks. In particular, we are interested in constructing a counterpart of the Laplacian defined as a differential operator in classical calculus. By the results in the last section, it is reasonable to start from a compatible sequence of r-networks S = {(V m ,# m )}. (We are not concerned with how to obtain a compatible sequence of r-networks in this section.) Then we obtain a resistance form (£,«7r) and a resistance metric R by taking a limit of S. Naturally, the resistance form (£, T) and the resistance metric it! are important elements in our "analysis". However, those are not enough. We need to introduce integration, that is, to introduce a measure on the space. The following is a general result concerning a resistance form, a resistance metric and a measure. Theorem 2.4.1. Let R G UM(X) and suppose that (X.R1^2) is separable. Set (£, J7) — MFx(R). Also let μ be a α-finite Borel measure on (X,^ 1 / 2 ). Define £$u, v) = £(u,v) + / u{x)v{x)μ(dx) Jx for u, v G L2(X, μ) n T. Then (L2(X, μ) fl T, £$ is a Hilbert space. Moreover, if μ(X) < oo and Jx R(p,p*)μ(dp) < oo for some p* G X, then the 2 2 2 identity map from L (X, μ)C\J• with £\-norm to L (X, μ) with L -norm is a compact operator. Proof. Let {un}n>o be a Cauchy sequence in (L2(X, μ) D T,£\) and let vn = un — un(p) for p G X. Then by (RF2), there exists v G Tv such that £(vn — v, vn — v) —> 0 as n —> oo. By (RF4), we have \vn(q) - v(q)\2 < R(p,q)£(vn -v,vn-

v).

(2.4.1)

Since μ is cr-finite, there exists {if m } m >o such that Km C X is bounded, 0 < //(ifm) < oo and Um>oifm = X. By (2.4.1), we see that vn —> v as h —> oo in L 2 (i^ m ,^|x m )- Also {wn|Km}n>o is a Cauchy sequence in L2(Km^\Krn). As un(p) = (un - vn)\Krn, there exists c G R such that u n(p) - ^ c a s n - > oo. If we let u = v + c, then £ (u — un, u — un) —> 0 as n —> oo. Also, Unlxm -* ^\'Km as n ^ oo in L2(JFCm,//|i<:m). On the other hand, {^n}n>o is a Cauchy sequence in L2(X^) and so there exists w* G L2 (X, μ) such that un —• u* as n —> oo in L2 (X, μ). As

64

Analysis on Limits of Networks u

n

2

2

u*\Km = \Km i L (Krn^\Krn), u = u* in L (X, μ). Hence « n -> u as 2 7 n - > o o in (L (X, μ) n J , £\). 2 Now suppose μ(X) < oo. Let W be a bounded subset of (L (X, μ) D T,£i). If C = s u p u G W £i(u, u), then 2

|u(p)-u((Z)|
(2.4.2)

for all u G £/ and all p, g G X. Let V be a countable dense subset of 1S bounded for any p G V by (2.4.2). Hence, X. Note that {^(p)}new by standard diagonal construction, we can find v G ^(V) and {vn} C ZY satisfying vn(p) —>• v(p) as n —> oc for all p G V. Using (2.4.2), we see that v satisfies (2.4.2) for p , g G 7 . Therefore v extends naturally to a function 1 2 and it satisfies (2.4.2) for all p,q G X as well. For any v G C(X,R / ) p e l , choose {pn} C V so that pn —•> p as n —> ex). Then M P ) - V(P)| < \Vk(p) -Vk(Pn)\ + |Vfc(Pn) - V(pn)\ + |v(pn) ~ v(p)| + |Vfc(pn) - V(pn)\. < 2y/CR(p,pn) Hence vn(p) —> v(p) a s n - ^ o o for all p G X. By (RF4), K^/CCP) - vi(p)) - (vk(p*) - ^ ( p * ) ) | 2 < 5(vfc - ^z,^fc -

vi)R(p,p*).

As f ( u n , t ; n ) < C, the above inequality implies \vk(p) - vi(p)\ < ^/4CR(p,p*) -f |^(p*) - vt(p*)\. Letting / —> oo, we have |v/c(p) — v(p)\2 4Ci?(p,p*) 4-1 for large /c. Since 2 $xR(p,p*)lJL{dp) < oo, it follows that Vk —> v in L (X, μ) as A: —>• oo by Lebesgue's dominated convergence theorem. • Now we have collected enough facts to use some abstract theory from functional analysis. In fact, by Theorem 2.4.1, we can apply the welldeveloped theory of closed forms and self-adjoint operators, which is introduced in Appendix B.I. Theorem 2.4.2. Let R G 11M(X) and suppose that (X.R1/2) is separable. Set (£,F) = MFx(R)' Also let μ be a α-finite Borel measure on (X,R}/ 2 ). Also assume that L2(X^)nJ: is dense in L 2 (X,μ) with respect 2 to the L -norm. Then there exists a non-negative self-adjoint operator H 2 such that Bom(H^2) = T and £{u,v) = (H^u.H^v) for on L (K^) all u,v G T. Moreover, if ^(X) < oo and jx R(p,p*)μ(dp) < oo for some p* G X, then H has compact resolvent. Proof. Set n = L 2 ( X ^ ) , Q(-,-) = £ and Dom(Q) = T. Then Theorem 2.4.1 along with Theorem B.1.6 immediately implies the required results. •

2.4 Dirichlet forms and Laplacians on limits of networks

65

Assume that R G 7ZM(X). Also, in addition to the assumptions of Theorem 2.4.2, let us assume that X is a locally compact metric space. Then (£, T n L 2 (X, μ)) is a Dirichlet form on L2(X^). Moreover, if 2 2 JFnL (X, μ^coix] is dense in C 0 (X), then (£, JF n L (X, //)) is a regular Dirichlet form. (See B.2 for the definition of (regular) Dirichlet forms and Co(X).) In fact, if (S^J7) comes from a regular harmonic structure, which is defined in 3.1, we can verify all the conditions above and get a Dirichlet form and a Laplacian immediately from the theorems in this section. See the next chapter for details. From an abstract point of view, the self-adjoint operator —H should be our Laplacian. However, this abstract construction is too general to study detailed information about our Laplacian. For example, it is quite difficult to get concrete expressions for harmonic functions and Green's function from this abstract definition. So, we also need to construct a Laplacian on a self-similar set in a classical way, specifically as a direct limit of discrete Laplacians. See Chapter 3, in particular 3.7. Example 2.4.3. Let K be any closed subset of R. We can always find an increasing sequence of finite sets {T4n}m>o that satisfy Vm C Vm+i and Um>oKi = K- If Vm = {Pm,i}7=i a n d Pm,i < Pm,t+i for all i, then we define Hm G CA{Vm) by 1

,TT N

j \Pm,i - Prnjl'

(Mm)pm,iPm,j -

i

n

[0

if |i - j | = 1, ^U

•

otherwise,

for i ^ j . Then {(Vm>#m)}m>o is a compatible sequence. Also if i^ is the effective resistance defined on Um>oV^, then R coincides with the restriction of the Euclidean metric. Let (S,^) be the corresponding resistance form and let /x be a α-finite Borel regular measure on K. First note that f\x belongs to T for any piecewise linear function / on R with supp(/) compact. (supp(/) denotes the support of /.) By this fact, it follows that T"Pi I?{K,\L) is dense in I?{K^μ) with respect to the L2-norm. Set T\ = T n I?{K,\i). Then {Z,T\) is a local regular Dirichlet form on This example contains many interesting cases. The most obvious one is 1 the case where K = R. In this case, T\ coincides with i? (R), which is the completion of {u G C 1 (R) : / v!(x)2dx < oo,supp(^) is compact.}

66

Analysis on Limits of Networks 1

with respect to the iJ -norm || • ||i defined by f

2

u {x) )dx. Also £(u,v) — fRuf(x)v'(x)dx. If μ is the Lebesgue measure on R, then the non-negative self-adjoint operator H coincides with the standard — A = 2 2 -d /dx . One of the other interesting cases is the Cantor set. Let K be the Cantor set defined in Example 1.2.6. Let //bea self-similar measure on K. (See 1.4 for the definition of self-similar measures.) Then (£,T) is a local regular 2 Dirichlet form on L (if,/x). By Theorem 2.4.2, we obtain a non-negative self-adjoint operator H. Set Aμ = —H. Then A^ may be thought of as a Laplacian on the Cantor set K. In [39, 40], Fujita studied the spectrum of Aμ and the asymptotic behavior of the associated (generalized) diffusion process on the Cantor set.

Notes and references Most of the contents in this chapter are taken from [87]. New aspects are the introduction of 7XF(-), &A(-), UM(-) and T£F(-). Indeed, in [87], condition (DF3) was forgotten in the definition of the Dirichlet forms. (The author thanks V. Metz for having pointed this out.) This motivated the author to study what would happen without (DF3). The relation between electrical networks and Dirichlet forms (or Laplacians) on graphs is a classical subject. See, for example, Doyle & Snell [31]. 2.1 The operator Pjj,v appearing in Theorem 2.1.6 is known as the trace of Dirichlet forms. In 2.1, we only treat the special case of Dirichlet forms on finite sets. See [43, §6.2] for the definition of the trace of Dirichlet forms for general cases. This operation has been known in various areas. For example, it is called the shorted operator in electrical network theory and is an example of a Shur complement. See Metz [127] and Barlow [6, Remark 4.26] for details.

Exercises Exercise 2.1. Show that every metric on V coincides with an effective resistance metric associated with a Laplacian H £ CA(V) if #(V) = 3.

Exercises

67

Exercise 2.2. Let V = {pi,P2,P3,P4} and let d be a metric on V defined by

{

x if (i,i) = (1,2), 0 if» = i, for some x with 0 < x < 2. Show that there exists a Laplacian H £ CA(V) such that Rfj = d if and only if x < 3/2. Exercise 2.3. Verify that the A-Y transform remains true even if H G Exercise CA(V). 2.4. Show that if #(V) = 3, H G CA(V) if and only if RH(-, •) is a metric on V. Exercise 2.5. Let V = {pi,P2,P3,P4}- For i ^ j , set PxVo

\-e

if (ij) = (1,4),

where e > 0 . Show that if e is sufficiently small, H G CA(V) and it^j(•, •) becomes a metric on V. Exercise 2.6. Let V = {^1,^2,^3}- Define

(

—(1 + m)

l + 2m

—m

1 + 2m -2(1 + 2m) 1 + 2m -m l + 2m - ( 1 + m), Show that RHm(pi,Pj) converges a s m - > o o . Also show that there exists no H G CA(V) such that R = RH, where R(pi,Pj) = limm_+oo RHrn(pi,pj). Exercise 2.7. Let X = {a, 6} U {pm}m>i- Define #(a,&) = 2,iJ(a,p m ) = R(b,Pm) = (l + m)/(l + 2m) and R{pj,pk) = \k - j\/(l + 2fc)(l + 2j). (1) Show that i? G UM{X). (2) Let (X, R1/2) be the completion of (X, J?1/2). Show that R <£ UM(X). (Hint: see Exercise 2.6.)

Construction of Laplacians on P. C. F. Self-Similar Structures

In this chapter, we will construct the analysis associated with Laplacians on connected post critically finite self-similar structures. In this chapter, C = (K,S,{Fi}ies) is a post critically finite (p. c. f. for short) self-similar structure and K is assumed to be connected. (Also in this chapter, we always set £ = {1,2,... , N}.) Recall that a condition for K being connected was given in 1.6. The key idea of constructing a Laplacian (or a Dirichlet form) on K is finding a "self-similar" compatible sequence of r-networks on {V^n}m>o, where Vm = Vm(C) was defined in Lemma 1.3.11. Note that {Vm}m>$ is a monotone increasing sequence of finite sets. We will formulate such a self-similar compatible sequence in 3.1. Once we get such a sequence, we can use the general theory in the last chapter and construct a resistance form (£,F) and a resistance metric R on V*, where V* = Um>oV^. If the closure of V* with respect to the metric R were always identified with K, then we could apply Theorem 2.4.2 and see that (£,J-) is a regular local Dirichlet form on L2(K, μ) for any self-similar measure μ on K. Consequently, we could immediately obtain a Laplacian associated with 2 the Dirichlet form (£,.F) on L (if,//). Unfortunately, as we will see in 3.3, in general, the closure of V* with respect to R is merely a proper subset of K in certain cases. In spite of this difficulty, we will give a condition on the probability measure μ which is sufficient for (£, J-) to be a regular 2 local Dirichlet form on L (K^) in 3.4. As was mentioned in 2.4, there is an abstract way of constructing the Laplacian from a Dirichlet form (£,.?*) on L2(K^). (See B.I for details.) However, we will develop our analysis on a p. c. f. self-similar set K in a 2 2 classical and explicit way similar to that of the ordinary Laplacian d /dx on the unit interval. In 3.5, the Green's function will be given in a constructive manner. In the following sections, we will study some counterparts of 68

3.1 Harmonic structures

69

classical analysis on Euclidean spaces, for example, Green's operator in 3.6 and Gauss-Green's formula in Theorem 3.7.8. Finally, we will define a Laplacian as a scaling limit of discrete Laplacians on Vm in 3.7. Throughout this chapter, d is a metric on K which gives the original topology of K as a compact metric space. Also we write C(K) — C(K, d). Since (K, d) is compact, C(K) is the collection of all real-valued continuous functions on K. 3.1 Harmonic structures In this section, we start constructing Dirichlet forms and Laplacians on K. As was mentioned above, the basic idea is to find a "self-similar" compatible sequence of r-networks on {Vm}m>$ and to take a limit. (Recall that Vm C Vm+i by Lemma 1.3.11.) For any initial D G £A(Vo), we can construct a sequence of self-similar Laplacians i/ m G CA(Vm) as follows. Definition 3.1.1. If D G CA(V0) and r = (ri,r 2 ,... ,rN), where for i G 5, we define £^ G VTiy^) by

for w, v G £(ym),

w h e r e rw = rWl . . . rWrn for w = wiw2

Hm G CA(Ym) is characterized by S^

...wme

n>0

Wm. Also

= £Hm.

It is easy to see that N

u, v) - y

1 - £ ( m ) ( u o Fi, v o Fi)

(3.1.1)

for u,v e £(Vm). Also ffm = J2wewm ^RwDRw, where i ^ : e(ym) -> £(Vb) is defined by i ^ / = / o F^ for w G Wm. We write 5 m = S^ hereafter. Considering (3.1.1), we may regard (ym,Hm) as a self-similar sequence of r-networks. If it is also a compatible sequence, then it is possible to construct a Laplacian on K using the theory in the previous chapter. Definition 3.1.2 (Harmonic structures). (D,r) is called a harmonic structure if and only if {(K n ,iJ m )} m >o is a compatible sequence of rnetworks. Also, a harmonic structure (-D,r) is said to be regular if 0 < n < 1 for all i G S.

70

Construction of Laplacians on P. C. F. Self-Similar Structures

Once we get a harmonic structure, we can use the general framework in Chapter 2 (in particular, Theorem 2.2.6, Theorem 2.4.1 and Theorem 2.4.2) to construct a quadratic form {£, J7) on V* = Um>oV^ and an associated non-negative self-adjoint operator H on L2(17, μ), where ft is the completion of V* under the resistance metric associated with (5, J7) and ^ is a given cr-finite Borel regular measure. This H should be our Laplacian. It seems easy but there remains a "slight" problem: the topology on V* given by the resistance metric may be different from the original topology of K. In such a case, ft does not coincide with K. In fact, we will see in the next section that ft = K if and only if the harmonic structure is regular. Another important problem is whether there exists any harmonic structure on a p. c. f. self-similar structure. By virtue of the self-similar construction of Hm, we can simplify the condition for harmonic structures as follows. Proposition 3.1.3. (D, r) is a harmonic structure if and only if(V0, D) < Proof Assume that (ym-i,Hm-i) < (Vm,Hm). Then, for any u G we\mve£m-i(uoFi,uoFi) = min{£m(voFi,voF») : v G £(Vm+i), v\Vm u}. Hence, by (3.1.1), £m{u,u) = min{f m+ i(v,^) : v G i(Vm+1),v\Vm u). Therefore (V^,# m ) < ( V ^ + i ^ + i ) . So, by induction, if (V0,D) (Vijjffi), then (Vm,Hm) < (Vm+i,Hm+i) for any m > 0 . The converse obvious.

= = < is •

For given r = (n, r 2 ,... , rN), define Ur : CA(V0) -+ CA(V0) by

where Hi G CA(Vi) is given by Definition 3.1.1. 1Zr is called a renormalization operator on £A(Vo). By the above proposition, D is a harmonic structure if and only if D is a fixed point of 1Zr. Also, it is easy to see that 1Z\r(aD) — (\)~la1Zr(D) for any a, A > 0. Hence, if D is an eigenvector of 7£r, i.e.,7^(D) = AD, then D is a fixed point of TZ\r. So, the existence problem of harmonic structures is reduced to a fixed point problem (or eigenvalue problem) for the non-linear homogeneous map lZr. In general, this problem is not easy and we do not fully understand the situation yet. For example, it is not known whether there exists at least one harmonic structure on a p. c. f. self-similar set. The only general result on existence of a harmonic structure is the theory of nested fractals by Lindstr0m [116]. Nested fractals are highly symmetric self-similar structures. (See 3.8 for

3.1 Harmonic structures

71

the definition.) We will present a slightly extended version of Lindstr0m's result on the existence of a harmonic structure on nested fractals in 3.8. Example 3.1.4 (Interval). Set Fx(x) = x/2 and F2(x) = x/2 + 1/2. Then C = ([0,1], {1,2}, {i^, F2}) is a p. c. f. self-similar structure. We see that Vm = {i/2rn}i=o 1 2m. Let us define D e CA(V0) by

Then (D, r) is a harmonic structure on C if r = (7*1, r 2 ) satisfies r\ -f r 2 = 1 and 0 < r^ < 1 for i = 1, 2. Also, it is easy to see that those are all the harmonic structures on C Example 3.1.5 (Sierpinski gasket). Recall Examples 1.2.8 and 1.3.15. The Sierpinski gasket is a p. c. f. self-similar set with Vb = {pi,P2>P3}- Define D e CA(Vo) by D= Also set r = (3/5,3/5,3/5). Then we see that (D, r) is a harmonic structure on the Sierpinski gasket. (D, r) is called the standard harmonic structure on the Sierpinski gasket. There are other harmonic structures on the Sierpinski gasket if we weaken the symmetry. See Exercise 3.1. Example 3.1.6 (Hata's tree-like set). Let C be the self-similar structure associated with Hata's tree-like set appearing in Examples 1.2.9 and 1.3.16. Then Vb = {c, 0,1} as in the previous example. Define D € CA(Vo) by D= and define r = (r, 1 — r 2 ) for r G (0,1). If rh = 1, then (Z), r) is a regular harmonic structure on C. So far we have presented examples of regular harmonic structure. Of course, there are many examples of non-regular harmonic structures. Example 3.1.7. Set Fx(z) = z/2,F2(z) = z/2 + 1/2 and F3(z) = y/^ +1/2. Let K be the self-similar set with respect to {Fi,F 2 ,F 3 } and let C = (K, {1,2,3}, {Fi, F 2 , F3}). Then £ is a p. c. f. self-similar structure. In

72

Construction of Laplacians on P. C. F. Self-Similar

Structures

fact, Cc = { 1 2 , 2 i , 3 i } , P £ = {i,2} and Vo = {0,1}. If D = and r = (r, 1 — r, s) for r € (0,1) and s > 0 , then (D, r) is a harmonic structure on £. Obviously, (D,r) is not regular for s > 1. See Example 3.2.3 and Exercise 3.2 for a more natural example of nonregular harmonic structures. Proposition 3.1.8. For w G W*, let w denote the periodic sequence in S defined by w = www . . . . Let (D, r) be a harmonic structure and let w G P /or w G VF*. XTien r^ < 1. /n particular, there exists i £ S such that

n<1. Remark. If (if, 5, {jFi}ies) is a p. c. f. self-similar structure, then the post critical set P consists of eventually periodic points: for any uj e P , there exist w G W* and m > 0 such that o~mu) = w. Corollary 3.1.9. Let (D,r) be a harmonic structure. If r± = • ••= rjy, then Ti < 1 for any i G S. In particular, (D, r) is a regular harmonic structure. To prove Proposition 3.1.8, we need the following lemma. Lemma 3.1.10. Let V be a finite set and let H G CA(V). Suppose U C V and p G U. If there exists q* G V\U such that Hvq^ ^ 0 , then —h pp < -Hpp, where (hki)k,ieu = [H]uProof. H can be expressed as (

), where T : £(U) -> i(U),J

:

\J XJ £(U) -> e(V\U) and X : £(V\U) -+ i(V\U). Then [i7]t/ = T - VX" 1 J. Now let ^p = - X " 1 Jxjf, where Xp(x) = 1 if a; = p and x^(#) = 0 if x ^ p on [/. It follows that

qev\u As Hpq^ T^ 0, the maximum principle (Proposition 2.1.7) implies that

Vvte*) > 0. Therefore Eqev\u

Hpq^P{q) > 0.

D

Proof of Proposition 3.1.8. First we assume that #CPi(Vb) n Vo) < 1 for all i G 5.

As # i = E i l i £*/WW*i, we have (H^

(3.1.2)

= E^o^Vo^fa)^ ^ ^ « -

Set

p = 7r(ii;), where it; = WiW2 . . . w m G W* and w e V. By Lemma 1.3.14, TT-^TT^)) = {w}. Hence, {(^,i) : ^ G Vb,*i(tf) = P} = {(7T(CT^),K;I)}.

3.2 Harmonic functions

73

Therefore, (H\)pp = ^^Dqq, where q = 7r(aw). So, letting pi = 7r(al~1w) for i = 1,2,... ,ra+ 1 , then (Hi)PiPi = ^DPi+lPi+1, where we set wm+i = w\. Now by (3.1.2), we can apply Lemma 3.1.10 and obtain — DPi+lPi+1 < -{Hi)Pi+lPi+1. So we have II™ i ~ ( # i W i < r^~1112=1 -(Hi)Pl+lPi+1. Hence 7\y < 1. If (3.1.2) is not satisfied, we replace the original self-similar structure

C = (K,S,{Fi}ieS)

by Cm = (K, Wm, {Fw}weWJ.

Then, by Propo-

sition 1.3.12, Vc — T^cm' Also, it is easy to see that (Z},r m ), where r m = (rw)weWm, is a harmonic structure on £m. For sufficiently large m, Cm satisfies (3.1.2) and hence we can apply the above argument to the harmonic structure (JD, r m ) . Therefore ( r ^ ) m < 1. Thus we obtain that

rw < 1.

•

3.2 Harmonic functions Let (D, r) be a harmonic structure on a connected p. c. f. self-similar structure C = (K, S, {F,} i e 5 ), where S = {1,2,... , N}. Then {(Vm, ^ m )}m>o is a compatible sequence of r-networks. So we can construct (£,T) as in (2.2.1) and (2.2.2). By Theorem 2.2.6, (E,F) G nF(V*), where K = Um>oKn- In this section, we consider harmonic functions associated with (8,^). The arguments in the last chapter, in particular Lemma 2.2.2, imply the following result. Proposition 3.2.1. For any p G £(Vo), there exists a unique u G T such that u\y0 = p and £(u,u) = min{£(v,v) : v G .T7, v|y0 — p}. Furthermore, u is the unique solution of j(Hrnv)\Vm\Vo

=0

for all m>1,

<

{

v\v0

[6.Z.I)

= p.

The function u obtained in the above theorem is called a harmonic function with boundary value p. Corollary 3.2.2. Let u be a harmonic function. and any m>0, (Hmu)(p) = (Du)(p).

Then, for any p G Vo

Proof. By Lemma 2.2.2, £(u, u) — £Hrn (u, u) for any m > 0 . Hence, if v is a harmonic function, then £(u, v) — £um (uiv) f° r a n y m> 0- Now let v\y0 = Xp- Then £(u,v) = £Hrn(u,v) = - 52q£Vm v(q)(Hmu)(q) = (Hmu)(p). • Example 3.2.3. Let {pi,P2,P3}

De

the vertices of an equilateral triangle

74

Construction of Laplacians on P. C. F. Self-Similar Structures

in C and let

^ +)

^ +)

( +)

( +

Let 5 - {1,2,... ,7}. For z G S, let #(*) = A(z - p{) + ph where ft = ft = ft = ft ft = & = ft = 1 - 2/?, /?7 = 1 - W for 1/3 < /? < 1/2. If if is the self-similar set with respect to {Fi}ies, then the self-similar structure (if, S, {F^ j ^ 5 ) is independent of the value of P and is post critically finite. In fact 3

C= [J {Ik, (k + 3)k} k=1

U

(J

{Jfe/,raifc,raZ,/fc},

k
P = {i,2,3} and pk = n(k) for k = 1,2,3. Define qk = n(7k) = n((k + 3)fe) for jfe = 1,2,3. Also let
and r = (1,1,1,1,1,1,^) for £ > 0. Considering the symmetry of (D,r) and if, we see that 1Zr(D) = XD for some A. (Note that 1Zr preserves the symmetry of D.) So (D, Ar) is a harmonic structure. Now let us calculate the value of A by using Corollary 3.2.2. Let u be the harmonic function that satisfies u(p2) = ^(^3) = 0 and u(pi) = 1. (Note that the harmonic function u is independent of the value of A because it is the solution of (3.2.1).) Taking the symmetry into account, we deduce that u(qie) = u(q15),u(q2e) = u(q^),u(q2) = u(q^),u(q2^) = u(q34). By (3.2.1),

t+3 By Corollary 3.2.2, (Dw)(pi) = - 2 = A - ^ ^ i e J - l ) = (ffiu)(pi). Therefore, we obtain A = (3t + 7)/(7t + 15). Note that 0 < A < 1 for any t > 0. However, At > 1 if t > VE. So the harmonic structure (J9, Ar) is regular if and only if 0 < t < y/E. Note that (K,S,{Fi}ies) is an affine nested fractal. (See 3.8 for the definition of an affine nested fractal.)

3.2 Harmonic functions

75

Fig. 3.1. Pi,qi and ^ Let R be the resistance metric on V* associated with (S.J7). Then, by (2.2.6), u G C(K,-#). Note that V* is a countable subset of K and the topology of ("14, i?) may be different from that of V* with the relative topology from the original metric on K. We will see, however, that a harmonic function has a unique extension to a continuous function on K. Now recall that d is a metric on K which is compatible with the original topology of K. Theorem 3.2.4. Letu be a harmonic function. Then there exists a unique u e C(K) such that u\y^ = u\v+ Remark. As is shown in 3.3, the closure of (V*,R) equals K with the original topology if and only if (£), r) is regular harmonic structure. In such a case, the above theorem is obvious. Proof. Let u b e a harmonic function with boundary value p. Set H\ = T

J \ where T : i(V0) -> e(V0),J : £(V0) -> J X, ^(VA^o) -> ^(^A^o). Then it follows that

(uoFi)\Vo=Ri(u\Vl)

((V^VQ)

and X :

=R

As Fw is a bijective mapping between Vb and Fw(VQ) for to £ If,, we will

76

Construction of Laplacians on P. C. F. Self-Similar

Structures

identify £(V0) and £(FW(VO)) through Fw. Define a linear map A{ : £(V0) e(Fi(Vo)) = i(Vo) by

Then u\Fw(Vo)

= AWmAWm_1

. . . AWlp

(3.2.3)

for w = W\W2 •..Wm G W* and (^)pQ > 0 for any p, g G Vb and ^

:

=

:

Ww

.

(3.2.4)

First we prove the theorem assuming (3.1.2). Claim 1. Set v(f) = maxp,g€Vb |/(p) - f(q)\ for / G ^(F o ). Then v(AJ) < Proof of Claim 1: If fi"i^|vi\vb = 0 an< ^ fl'lvb — /> then Aif — g\Fi(y0)Applying the maximum principle (Proposition 2.1.7) and taking (3.1.2) into account, we can see that max geFi(Vb) 9(Q) ~ mmgGFi(Vb) 9(o) < v(f)Hence v(Aif)
/ e e(Yo).

Proof of Claim 2: Define Q : ^(F o) -> ^(^o) by

then v(/) = v(Q/) and v ( ^ / ) = v(A{Qf) ^

^

for any / G £(V0). Hence

: / G €(Vb),t;(/) ^ 0} = s u p { ^ ^ : / G £(Vb),t;(/) ^ 0}

f : / G <(%), J]) /(g) = 0,t;(/) = 1}. As {/ G ^(Vb) : Y^qzvo /(^) ~ ^' 'L'(^) = -^} ^s c o m P a c t ? t n e above supremum is less than 1. Now by Claim 2 and the maximum principle, vw(u) = sup{|/(p) - /((/)| :p,qeKwnV.}<

cmv(p)

for any it; G W*, where c = m a x ^ s Q . Hence, if {pi}i>i is a Cauchy sequence with respect to a metric on K which is compatible with the original

3.2 Harmonic functions

77

topology of K, then {u(pi)}i>i is convergent as i —• oo. Using this limit, we can extend u t o a continuous function u on K. Next if (3.1.2) is not satisfied, we can exchange the harmonic structure as in the proof of Proposition 3.1.8. Then again we can use the result under a new self-similar structure £ m and harmonic structure (D,r m ). Note that harmonic functions remain the same after we replace the harmonic structure. • Hereafter, we identify u with its extension u and think of a harmonic function as a continuous function on K. Immediately, by Lemma 2.2.3, we have the maximum principle for harmonic functions. Theorem 3.2.5 (Maximum principle: weak version). Let u be harmonic. Then, for any w G W*, min

u(p) < u(x) <

max

u(p)

for any x e Kw. This maximum principle is a little weak because we say nothing about the case when u(x) attains the maximum. We will give a stronger version of the maximum principle in Theorem 3.2.14. Example 3.2.6 (Sierpinski gasket). Let us calculate the stochastic matrices {Ai}ies for the standard harmonic structure on the Sierpinski gasket given in Example 3.1.5. Recall that Vb = {pi,P2,P3}- See Figure 1.2. It follows that Vi = {Pi,qi}ies, where S = {1,2,3}. Now let f(p1) = a, f(jp2) — b and f(ps) = c and solve the linear equation (Hif)(qi) = 0 for i e S. Then we get f(q1) = (2b + 2c + a)/5, f(q2) = (2c + 2a + 6)/5 and f(q3) = (2a + 26 + c)/5. By this result,

It is easy to see that the eigenvalues of A{ are 1,3/5,1/5. Note that the second eigenvalue 3/5 is equal to r^. In fact, this is not a coincidence. (Recall that r = (3/5,3/5,3/5).) In A.I, we will give a general result on the second eigenvalue of Ai. See Exercise 3.3 for more examples. The stochastic matrices {Ai}ies determine the harmonic functions through (3.2.3). The behavior of a harmonic function around a point TT(UJ) for u> £ ^(S) is given by the asymptotic behavior of (3.2.3) for m —> oo.

78

Construction of Laplacians on P. C. F. Self-Similar

Structures

This is the problem of random iterations of matrices and, in general, it is very difficult. Even in the above example, we do not know how to calculate the behavior of Au>rnAUJrn_1 . . . AUl as m —> oo unless the sequence u) is (eventually) periodic. Kusuoka used {Ai}ies to construct Dirichlet forms on finitely ramified self-similar sets in Kusuoka [107] and obtain some result about almost sure behavior of the random iteration of {Ai}iesAn important property of harmonic functions is the Harnack inequality, which follows from the discrete version, Corollary 2.1.8. Proposition 3.2.7 (Harnack inequality). If X is a compact subset of K that is contained in a connected component of K\VQ, then there exists a constant c>0 such that ma,xX£x u(x) < cminxex u(x) for any nonnegative harmonic function u on K. Proof Set Xm = U w e ^ m : ^ n x / 0 ^ - Then we can choose m so that Xm n Vo = 0. Now set V = Vm, U = Vo, H = Hm and A = Xm n Vm. The Harnack inequality (Corollary 2.1.8) implies that there exists c > 0 such that maxpGA u(jp) < cminpeA u(p) for any non-negative harmonic function u on K. By using the maximum principle, it follows that max x €x u(x) < maXpG,4 u(p) and min p €^ u(p) < min a ; e x^(^)- Hence we have shown the required inequality. • For p G Vo, let i^v be the harmonic function that satisfies ipp\v0 — X^0• It is easy to see that {ipp}Pev0 is a partition of unity on K: J2peVo ipP(x) = 1 for any x G K. Theorem 3.2.8. Let p G Vo and let x G K\V0. Then ipp{x) > 0 if and only if there exists C G J(p, Vo) which contains x. Recall that J(p,Vo) is the collection of all connected components of K\Vo whose closure contains p. Definition 3.2.9. (1) For p,q G Vm, {pi}f=i is called an iJ m -path between p and q if Pi G Vm\Vo for any i = 1,2,... ,n, (Hrn)PPl > 0, (Hm)Pnq > 0 and (Hm)PiPi+1 > 0 for any i = 1,2,... , n - 1. (2) For p, q G V^\Vo, we write p ~ q if and only if there exists an i7 m -path between p and q.

771

The notion ~ will not appear later in this section. It plays, however, an m

important role in 3.5, where we will show that p ~ q if and only if p and q m

belong to the same connected component of K\Vo by using the results in this section.

3.2 Harmonic functions

79

Note that (Hm)pq = T,W:P,qeFw(v0)(r™)~ lD(Fw)-Hp)(Fw)-Hq)Since D is a Laplacian on Vb, we may immediately verify the following lemma. Lemma 3.2.10. Let w G Wm and letp,q G FW(VO). (1) / / Fw(Vo) n Vo = 0, then there exists an H^-path between p and q contained in Fw(Vo). (2) If Fw(Vo) nVo = {q} and p ^ q, then there exists an H^-path between p and q contained in Fw(Vo). Proof of Theorem 3.2.8. First assume that x G C G J(p,V0). Then, by Proposition 1.6.6, there exists a path 7 : [0,1] —> C U {p} between x and p. Choose m so that sup diam(if m f 7 ( t ) )
i€[0,l]

where dist(A,JB) = mixeA,yeB d(x,y). Let { p i , . . . ,p n } = ^ m ri7([0,l]), where pi = p(U) and £1 < £2 < • •<•tn = 1. Note that p n = p. Then there exist w(0),... , i y ( n - 1 ) G W m such that x,p1 G ^ ( o ) and p»,Pi+i € ^ ( t ) for all i = 1,2,... ,n — 1. Note that Kw^ C\ Vo C {p} for any z. Let j = min{i : K^ ( i ) n Vo = {p}}. Suppose i^p{x) = 0. Then, by Theorem 3.2.5, there exists q G FW^(VQ) such that ipp(q) = 0. Let g = p$. Then, by Lemma 3.2.10, there exists an # m - p a t h between pi and pi+\ for any i = 0 , 1 , . . . , j — 1. Also, by Lemma 3.2.10, there exists an i7 m -path between pj and p. Joining those i7 m -paths, we may construct an if m -path between q and p. Let V — V m and let U = Vo. Then the maximum principle, Proposition 2.1.7, implies that i/;p(q) > 0. Therefore, if>p(x) > 0. Next let C be the connected component of K\Vo. Assume that p £ C. We show that ipp(q) = 0 for any q G C C\ V*. Let q G C C\ Vm- Define Um = U2,EVbnc(^™>2/ n Vrn)' Since ipp is continuous and i/>p(y) = 0 for any p G Vb n C, it follows that maxzGj7m i/)p(z) —> 0 as m —> CXD. Also g ^ C/m and p £ Um for sufficiently large m. Now apply the maximum principle, Proposition 2.1.7 with V = F m , t/ = Um U Vb and if = ffm. Since p £ C, any i7 m -path between p and g must intersect Um. Hence Uq C f/m, where f/q is defined in Proposition 2.1.7. So 0 < ipp(q) < max z6 [/ m i^p(z). Letting m —»• 00, we see that ^p(q) = 0. Since CflF* is dense in C, it follows that ^p\c = 0. • By the above theorem, we get a relation between the topological properties of K and the positivity of the elements of D, where (D,r) is the harmonic structure.

80

Construction of Laplacians on P. C. F. Self-Similar

Theorem 3.2.11. Let (D,r) be a harmonic structure. and only if J(p, Vo) D J(q, Vo) ^ 0.

Structures Then Dpq > 0 if

The condition that J(p, Vo) n J(q, Vo) ^ 0 is equivalent to the one that there exists a connected component of K\Vo whose closure contains both p and q. Proof: Parti. Let (£,T) be the resistance form associated with (D,r). Then it follows that Dpq = -£o(ipp\v0,ipq\v0) = -£m(^P\vm^q\vm)' Hence Dpq = (Hm^p)(q)

= (Hm^q)(p).

(3.2.5)

Assume that J(p,Vo) D J(q,Vo) — 0. Since Kmjq C Ucej(g,vo)Cf f° r sur E~ ciently large ra, Theorem 3.2.8 implies that ipp\Kmjq = 0. Hence, by (3.2.5), Dpq = 0. ' • To prove the rest of this theorem, we need the following lemmas. L e m m a 3.2.12. Let p and q belong to Vo. If Dpq > 0, then there exists a path 7 : [0,1] —> K between p and q that satisfies 7((0,1)) C K\VoProof. By the above proof of Theorem 3.2.11, if Dpq > 0, then J(p, Vo) H J(g 5 Vo) T^ 0- Let C G J(p, Vo)n J(q, Vo). Then, by Proposition 1.6.6, there exists a path 7 : [0,1] —> C U {p, g} with 7(0) = p and 7(1) = q. D Lemma 3.2.13. Let p and q belong to Vm. Let {pi}i=i,2,... ,n be an Hmpath between p and q. Then there exists a connected component of K\Vo, C, such that pi € C for all i = 1,2,... ,n, p e C and q e C. Proof. For each i G {1, 2 , . . . , n — 1}, there exists w G Wm such that Pi,Pi+i £ Kw and ^(Fu; )-1 (pi),(Ft/; )-1 (pi+ i) > 0- By Lemma 3.2.12, it follows that there exists a path 7^ between pi and Pi+i that satisfies 7i((0,1)) C i;f\T4n. Hence there exists a connected component of K\Vo which contains both pi and Pi+i. Inductively, we see that the connected component, say C, contains all pi. By the same argument as pi and p^+i, we may find a path 7 between p and pi that satisfies 7((0,1)) C K\Vm. So it follows that 7((0,1)) C C. Hence p G C. Exactly the same argument implies that qeC as well. • Part II of the proof of Theorem 3.2.11. Assume that J(p, Vo) H J(q, Vo) ^ 0. Let C G J(p,Vo) n J(q,Vo). Then, by Proposition 1.6.6, there exists a path 7 : [0,1] —> C U {p,^} with 7(0) = p and 7(1) = q. Now by the same argument as in the proof of Theorem 3.2.8, we may construct an iJ m -path between p and q from the continuous path 7. Let {pi}f:=1 be the iJ m -path between p and q. Then, by Lemma 3.2.13, pi belongs to C for

3.2 Harmonic functions

81

alH = 1,2,... ,ra. Since (Hm)Priq > 0 and ipp(pn) > 0, (3.2.5) implies that Dpq = (Hmil>p)(q) > (Hm)Pnq(^p(pn) - i/>p(q)) > 0. • By the above discussion, we also obtain a stronger version of the maximum principle.

Theorem 3.2.14 (Maximum principle: strong version). Let x G K\Vo and let C be the connected component of K\Vo with x G C. Then, for any harmonic function u, min_u(p) pevonc

< u(x) <

ma,x_u(p). pevonc

Moreover, if any of the equalities holds, then u is constant on C. Proof First assume that x G Vm for some m. Then the same arguments as in the proof of Theorem 3.2.8 imply that, for any p G C fl Vb, there exists an i/ m -path between x and p. Conversely, by Lemma 3.2.13, if there exists an i/ m -path between x and p G Vb, then p G C DVo- Hence, let V = Vm, U = Vo and H = Hm and apply Proposition 2.1.7. Then Ux = C n Vb. Therefore, the desired results are immediate from the conclusion of Proposition 2.1.7. Next if x £ V*, then choose w G W* so that x G Kw C C. Then, by the weak version of the maximum principle (Theorem 3.2.5),

min u(q) < u(x) < max u(q). F(V)

eF(V)

Also, if u(x) = max 9eFu; (y 0 ) u(q), then u(x) = u(q) for some q G FW(VO). Since FW(VQ) C C fl Vm, where m = \w\, the first part of this proof implies the desired results for this case as well. • Next we define the notion of piecewise harmonic functions. The following is an immediate corollary of Theorem 3.2.4. Corollary 3.2.15. For p G £(Vm), there exists a unique continuous function u on K such that ^|y m = p and £(u\v*,u\v+) = min{£'(v,^) : v G u in the above corollary is called an m-harmonic function with boundary value p. Another characterization of ra-harmonic functions is that u is an ra-harmonic function if and only if uoFw is a harmonic function for any w G Wm. For p G Vm, define 1$)™ to be the ra-harmonic function with boundary value xjfm. Then any ra-harmonic function u is a linear combination of {il>™}. In fact, u = £ p e Vm ™. For u G t(V+), we define P mu by

82

Construction of Laplacians on P. C. F. Self-Similar •If

Structures

We Write U

m = PrnU, t h e n £ m ( l i | Vm , u\ Vrn) =

In the rest of this section, we will give an expansion of u G £(V*) in a piecewise harmonic basis {V^lpGV*, where ipp = ip™ if p G V m \ y m _ i . Lemma 3.2.16. Let u be an m-harmonic function. Then £(u,f) — 0 for

feFiff\vm=0. Proof. For n > m, we have (Hnu)(p) = 0 if p € Vn\Vm and /(p) = 0 if peVm. Hence £n(«, / ) = - £ p e V n /(p)(ffn«)(p) = 0. • L e m m a 3.2.17. For u G J-, £(u — um,u — um) —>• 0 as m —>• oo? where Proof By Lemma 3.2.16, £(u — w m , i^m) = 0. Hence, £(iz — ixm, w — ixm) = S(u, u) - £{um, Um) = £{u, u) - £m{u, u). n Definition 3.2.18. Let u G £(V*). For p G Vrm\Vrm_i, define a p (i/) = Um(p) - um-i(p)' {pLp{u) = u(p) if p G Vo-) Also define a w (w) = ix m+ i o Fw - Um o F w for ^ G W m . Note that ap(u) — u(p) — um-i(p) definition,

for p G Vm\Vm-i-

By the above

where the infinite sum in the latter equality is pointwise convergent. (In fact, it is a finite sum for each p G V*.) This is the expansion of a function on V* in the piecewise basis {'0p}pev*Also, by Definition 3.2.18, aw(u) is a 1-harmonic function and aw{u)\yQ = 0. Using ap(u), aw{u) is given by a

Fw(p)(u)ipp.

(3.2.6)

Proposition 3.2.19. For u G £{V*), u G T if and only if £o(u,u) -f Y^ Moreover, ifuEJ7,

^2

—£i(a>w(u),a w(u))

< oc.

then the sum in (3.2.7) is equal to £(u,u).

Proof By Lemma 3.2.16, m-l

£m(u,u) = ^2 £k+i(uk+i -uk,uk+i k=0

-uk)

+Eo(u,u).

(3.2.7)

83

3.3 Topology given by effective resistance By the self-similarity of £k+i, we obtain

—£i(a w (u),a w (u)). weWk

Combining the above two equalities, we easily verify the statement of the proposition. •

3.3 Topology given by effective resistance As in the last section, we assume that (D,r) is a harmonic structure on C = (K, S, {Fi}ieS), where S = {1,2,... , N}. Let {(Vm, # m )}m>o be the compatible sequence induced by (D, r). Then, there exist (£, J7) G 1ZJ-(V*) and R G KM(V*) associated with { ( F m , # m ) } m > o . Moreover, if (il,R) is 7 the completion of (V*,#), the results in 2.3 and 2.4 imply that (S.J ) G In this section, we study the difference between Q and K, in other words, the difference between two topologies on V*. One is given by the resistance metric R and the other is given by the relative topology from the original metric space (K,d). Roughly speaking, it turns out that Q can be always identified with a subset of K and that Q, = K if and only if (Z), r) is regular. First, we observe an important property of (E^J7), that is, the selfsimilarity of (S^J7). 7

P r o p o s i t i o n 3 . 3 . 1 . For any u G f and i^S^uoFi^J . c< yU) VJ

==

Moreover,

Ls [U O JΓi) V O ±*i)

- y •—1 ^**

for any u,v G T. The above proposition is easily verified by (3.1.1). Moreover, if A is a partition of T, = 5 N , then an inductive argument on #(A) shows that —£(uoFw,voFw).

(3.3.1)

Now we start to study the difference between O and K. Proposition 3.3.2. There exists a continuous infective map 0 : Q, —> K which is the identity on V*. Lemma 3.3.3. If FW(V*) D FV(V*) = 0 for w,v G W*, then in£{R(p,q) :

84

Construction of Laplacians on P. C. F. Self-Similar

Structures

Proof. Choose m > 0 so that w, v G Wm. Define p G £(Vm) by {

0

otherwise.

Then p\Fv(y*)nVm ~ 0- Hence, if u is the ra-harmonic function with bound ary value p, then u\Fw{y^ = 1 and W|FW(V.) = 0- BY ( 2 - 2 -6), if P € and g G F , ( 1 4 ) , then

D Recall that d is a metric on K which is compatible with the original topology of if. Note that (if, d) is a compact metric space. Proof of Proposition 3.3.2. Step 1: Construction of 6 Let {pi} be a Cauchy sequence in (V*,R). By Lemma 3.3.3, defining 6m by min

(

inf

,Fw(v*)nFv(v*)=® \

then we have dm > 0. Choose n(m) so that R(pk,Pi) <$m for /c, / > n(m), then there exists w G W m such that pk G U vG ^ m :F u; (v,)nF.(y,)^0^( V *) f o r k > n(m). As max 1(;€ ^ m diam(K-u;,d) —>> 0 as m —• oo, {p^} becomes a Cauchy sequence in (if, d). (diam(-, d) means the diameter with respect to a metric d.) So let p be the limit point of {pi} in (V*, i^). Then we define 0(p) by the limit point of {pi} in (K,d). It is routine to show that 6 is a well-defined continuous map from ft to K and 0|y; is the identity. Step 2: # is injective. Assume that 0(p) = 6{q) for p,q G ft. If #(pi,p) - • 0 and iJ(^,g) -> 0 as i —> oo, then the discussion in Step 1 shows that d(pi,0(p)) —> 0 and d(qi,0(q)) —>• 0 as i —> oo. Let i? be an ?n-harmonic function on V*. Since v can be extended as a continuous function on both (if, d) and (ft, i2), we see that v(p) = limi_>oo v(Pi) = v(5(p)) = v(^(g)) = lim^oo vfe) = v(q). Now, for u G J", by (2.2.6)

where po G Vo- (Recall that um is an m-harmonic function defined by um = Y^xeVm u(x)^m-) Hence, using Lemma 3.2.17, we have limm-+oo Um(p) = u(p). In the same manner, we can also show that lim™-,^ Um{q) = ^((z)Since, wm is a m-harmonic function, it follows that um(p) = um(q)' Hence u(p) = u{q). Therefore, using (2.3.1), we can conclude that p — q. •

3.3 Topology given by effective resistance

85

By virtue of Proposition 3.3.2, we will identify ft with O(ft) and think of ft as a subset of K. Now we ask when ft is equal to K. The answer is given by the following theorem. Theorem 3.3.4. The following are equivalent. (1) ft = K. (2) (ft, R) is compact. (3) (ft, R) is bounded. (4) For any u G T, suppGfi \u(p)\ < oo. (5) (D, r) is regular. Moreover, if (D, r) is regular, then R is a metric on K which is compatible with the original topology. To prove the above theorem, we need the following lemmas. Lemma 3.3.5. For any w G W* and for any p,q G ft, rwR(p,q) > R(Fw(p),Fw(q)).

(3.3.2)

Proof. By Proposition 3.3.1, £(u,u) =

^

rw>~x£(u o Fwr,u o FW') > rw~xE(u o Fw,u o Fw)

for w G Wm. Therefore, we have 'W

\uoFw(p)-uoFw(q)\2 £(uo Fw,uo Fw)

\u(Fw(p))-u(Fw(q))\2 £(u,u)

>

Hence, by (2.3.1), we obtain (3.3.2).

•

Lemma 3.3.6. Let uo = u)\U)2 ... G S. Assume that limsup min 8(arnuj,uf) > 0,

(3.3.3)

where 6 is any of the metrics Sr on E defined in Theorem 1.2.2. If lim inf r m—>oo

m

^> 0

then 7T(UJ) £ ft. Proof Set if = J2peVl\Vo ipp and [0

otherwise,

for w G W*. By (3.3.3), there exist r G T\V and {m n } n >i C N such

86

Construction of Laplacians on P. C. F. Self-Similar

t h a t mn

<

ran+i

for any n > 1 and a^co

•(jOmn and define u = X ^ n > 1 n~1(Pwn>

Structures

—• r as n —• oo. Set wn = Then

M^) =-I 0 Note that liminfn^oo r w n > 0. Hence ^o(w,w)+^

^

otherwise.

—fi(a W ;(u),a 1( ;(^))

= ^o(^)^J + 2 ^ ~2 n>1

ti\fp,ip) < OO. ^

By Proposition 3.2.19, we see that £{u,u) < oo and u G T. Now we assume (3.1.2). Then, by Lemma 1.6.10, ip(x) > 0 for any x G K\VQ. Since r ^ P , we can choose m > 0 so that C = mhixeKT1T2...Trn (p(x) > 0. Without loss of generality, we may assume that o~mriuj G E riT2 ... Tm . So, if w(n) = wnT\T2 . . . Tm, then w(n) = o^i^ . . . a; m n + m and min

99wn (x) =

min

^(^) = C.

Hence inf{w(x) : x G Kw^ D V*} > X]fc=i ^ / ^ "^ °° a s n ~^ °°- Suppose that 7T(U) G fi. Then there exists {pi}i>i C V* such that pi —>• TT(LO) as i -^ oo in (fi, ,R). Since 0 is continuous, it follows that pi —> 7T(O;) as 2 —> oo in (K,d) as well. Therefore, p^ G i ^ ( n ) for sufficiently large i. Hence u (Pi) ~* °° a s ^ ~^ °°- O n the other hand, w is a continuous function on (£2,i?). Hence {w(pi)}i>i converges to a finite limit as i —> oo. Thus we conclude that TT(O;) ^ ft. If (3.1.2) is not satisfied, again we may repeat the same arguments as in the proofs of Proposition 3.1.8 and Theorem 3.2.4. • Lemma 3.3.7. If (D,r) is not regular, then Q, ^ K. u G T such that suppGV^ \u(p)\ — °°-

Also there exists

Proof. If (D,r) is not regular, there exists r^ > 1. Let u = k. Then Proposition 3.1.8 implies that UJ ^ V. Hence uo satisfies the conditions in Lemma 3.3.6. So TT(UJ) £ Q and Vt^ K. Also, by the proof of Lemma 3.3.6, we can find u G T that is unbounded on V*. • Proof of Theorem 3.3.4- Assume that (D, r) is regular. Then Lemma 3.3.5 implies that {Fi}ies is a system of contractions on a complete metric space (fl, i^). Hence, by Theorem 1.1.4, there exists a non-empty compact subset K of (fi,i?) such that K = UiesFi(K). As 6 is continuous, K is also a

87

3.3 Topology given by effective resistance

compact subset of (K,d). For x G K, Dwew*Fw(x) is a subset of K. It is easy to see that Uwew*Fw(%) is dense in (K,d). Hence K = K. Therefore ft = K, (ft, J?) is compact, and 9 is a homeomorphism between (ft, it!) and (K,d). Obviously (2) implies (3). Also as \u(p) — u(q)\2 < £(u,u)R(p,q) for u G F, we can see that (3) implies (4). The above discussion with Lemma 3.3.7 completes the proof of Theorem 3.3.4. • Next, we consider how large ft is in K if (D,r) is not regular. For instance, we will calculate /x(ft) for a self-similar measure μ on K. Theorem 3.3.8. Let μ be a self-similar measure on K with weight (μ^gs. (2) IfYliesW^l,

then μ^il) = 0.

Remark. A. Teplyaev showed that if Y[ieS(ri)^

— 1, then μ(Q) = 0.

This theorem is somewhat intriguing, in particular, in the second case where μ(Q) = 0. As is shown in the next section, even in such a case, if riμi < 1 for all i G S, then (S^J7) is a local regular Dirichlet form on L2{K^). So u G T is originally a continuous function on (V*,R) and it can be extended to a continuous function on (fi, R). This is not surprising. But, despite the fact that 17 is a null set in K, u can be extended to an L2class function on a far larger space K. This fact might suggest that some of the analysis on a large (higher dimensional) space could be determined by information on a small subset of the space. Lemma 3.3.9. Let u> = n(u) G ft.

UJIUJ2

Proof. Let r e V and set p m = using Lemma 3.3.5, we see that

... G E. // -K(UJIUJ2

Z^OVL^...^™

< °°; then

•. .tc;mr). Then p m G F m , and,

max R(x,y). Hence {pm} is a Cauchy sequence in (ft, i?). Let p be its limit. On the other hand, d(pm,

TT(U>)) —> 0 as m —» oo. Hence #(p) = TT(U;) G

Proof of Theorem 3.3.8. Set nm(i,Lj) = # { j : ^ w G E, i G S and m>1. Define A/"={^G£:

lim n ™( 2 ^) m—>oo

777,

= M

ft.

= z, 1 < j < m} for

. for all 2 G 5}.

D

88

Construction of Laplacians on P. C. F. Self-Similar Structures

Then the strong law of large number implies that μ(.^/') = 1. Note that # ^ logr<. Hence, for w&N, l i m Wu uj m—>-oo

(jj m I

/ m

=

ies

Now, if nieS(ri)^ < l> t n e n X^m>or^i^2..-ojm < oo for Lu e JV. Therefore, Lemma 3.3.9 implies TT(UJ) G ft for any LJ G N. Hence μ^tl) — 1. Next, suppose I l i e s f a ) ^ > ^' ^ e t ^ m = {^1^2 • • -Mm : v G V}. Then we can choose m so that #(Vm) < #(W m ). Set

and ^4 = l i m i n f ^ ^ o o ^ . If min r G p8(a n uj,r) —> 0 as n —>• CXD, then, for sufficiently large k, ujk^k+i • • • ^fc+m-i ^ 'Pm- Hence c<; G A. Now, as Bk+mi for / = 1,2,... are independent and μ^k) = μ(Bk+ml) < 1, we see that μ{(^j kBj) = 0. Hence μ{A) = 0. So μ{N D Ac) = 1. Assume that LO G A/" fl A c . Then limsup m _, 0 0 min rG -p ^(cr m o;,r) > 0. Also, liminf m ^ o o r U ; i a ; 2 . < . a ; m > 0 because lim m _ > o o |r u ; i a ; 2 > .. C t , m | 1 / r n = n*Gs( r i) M i > !• Hence, by Lemma 3.3.6, TT(UJ) £ Q. As μ^f] Ac) = 1, it follows that ^(n) = 0. •

3.4 Dirichlet forms on p. c. f. self-similar sets In this section, we continue with the assumptions of the previous section: (D,r) is a harmonic structure on (K,S,{Fi}ies)Then we have 7 (£:^F) G 1U-(Q), where (S^J ) is the natural limit of the compatible sequence {(K n ,ff m )} m >o. If the harmonic structure (D,r) is regular, it was shown in Theorem 3.3.4 that K = Q. Then, results from Chapter 2, in par7 ticular, Theorems 2.2.6, 2.3.10 and 2.4.1, immediately imply that (S^J ) is 2 a local regular Dirichlet form on L (X, μ) for any Borel regular probability measure μ on K. See B.3 for the definition of Dirichlet forms. On the other hand, by Theorem 3.3.4, fi is a proper subset of K if (D, r) is not regular. In this case, functions in T may not be extendable to continuous functions on K. (In fact, T turns out to contain an unbounded function by Theorem 3.3.4.) In this section, however, we will show that T 2 7 can be embedded in L (K^) for a certain measure μ and that (S^J ) is a 2 local regular Dirichlet form on L (K^). Let M(K) be the collection of all Borel regular probability measures on

89

3.4 Dirichlet forms on p. c.f. self-similar sets K. Also we define M{K) = Jμ G M{K) : μ(V;) = 0 and p(O) > 0 for any non-empty open set O C K.}

Of course, all self-similar measures belong to M(K). The following sufficient condition for μ to be in M(K) is sometimes useful. Lemma 3.4.1. Let μ G M(K). If there exists 7 > 0 such that /yμ(Kw) < ) for any (w,i) eW*xS, then μ G M(K). A measure satisfying the condition in the above theorem is called a 7elliptic measure. Proof. If we choose m large enough, then for any w G W m, there exists w1 G Wm such that Kw n Kw> = 0. So, changing the self-similar structure to Cm = (K,Wm,{Fw}w€Wm) if necessary, without C = (K,S,{Fi}ieS) loss of generality we may assume that for any i G 5, there exists j G S such that Ki D Kj = 0. Then 1 = μ^lt) > μ^i) + μ^j) > 27. Therefore 7 < 1/2. Let 6 = 1 - 7. Then 0 < 8 < 1. Now, for any it; G W* and any

> μ^i)

+ μ^j)

>

where we choose j G S so that KiO Kj = 0 . This implies that m m 6μ^w). Hence 7 < μ ^ w ) < ^ for any w G T ^ . This inequality implies that μ(p) = 0 for any p e K and that μ(O) > 0 for any non-empty open subset O C K. • u ( I n 3-2> p m ^ As in 3.2, set P m : T -> C(tf) by Pmu = T,Pevm (p)^is denoted by it m . We also use this notation in this section.) Recall that Um is an m-harmonic function and it can be regarded as a continuous function on K. Also, define Hi$ = {u : w is a 1-harmonic function and it|v0 = 0}.

L e m m a 3 . 4 . 2 . Then there exists c > 0 such that (UI^μ any u G 7^i,o and for any μ G M(K),

where (u, v)μ = fK

< c£(u,u)

for

uvdμ.

Proof. Note that both (•, -)M a n d £ are inner-products on a finite dimensional vector space Hito- As u = ]CpeVi\Vb U(P)^P

^or

a n

^ ^ ^ ^i,o» we

90

Construction of Laplacians on P. C. F. Self-Similar Structures

have (u, u)p =

Y^

apqu(p)u(q) <-

(u(p)2 + u(q)2)

^

V

where apq = §K'^p1^)qdμ. (Since 0 < i/jp(x) < 1 for any x G K, we have u( 2 0 < apq < 1.) Also there exists c > 0 such that #(Vi\Vb) EPeVi\vb &) c£(^, ix) for any w G Wi,o- This proves the lemma. • Lemma 3.4.3. Let μ G M{K). Sm>o Rm(n) < °°> then {Pmu}

Define Rm(f^) — m a x ^ ^ ^ fwl^{Kw). If 2 converges in L {K^) as m —> oc for any

For ease of notation, we write flm instead of Rm{^) if no confusion can occur. w

Proof Jov w G W*, define ^ ^ by μ (A) = μ{Fw(A))/μ{Kw). Then w /x G Af ( # ) . Note that / K / o F^μ™ = ^ — j ]K^ fdμ. Now, by Proposition 3.3.1 and Lemma 3.4.2, it follows that, for u G J-, -Um) £((um+1 weWm

- Um) O Fw, (Um+1 - Um) O Fw)

w

>c-^7(nm+1-Um)^. tiyn JK

Hence cRm£(um+i - % , V i i. So

- um) > \\um+i - um\\% where ||u||£ =

n-l

- Wfc) > | | w n

fc

- U( m | Iμ u

u

for n > m. By Lemma 3.2.16, we have X]fc=m^(^ +i ~~ ki k+\ 5(ifcn — w m , ii n — Txm). Hence

— u k) =

n-l

c( ^

iifc)5K - «m, ^n - ^m) > I K ~ t ^ | | ^ .

(3.4.1)

91

3.4 Dirichlet forms on p. c.f self-similar sets This shows that {w m } m >o is a Cauchy sequence in L2(K^).

•

Define Lμ(u) for u G T as the limit of {ii m } m >o in L2(K, μ) as m —• oo. Then Lμ'. T ^ L2(K, μ) is a linear map. ^μ will be shown to be an injective map. If swpwew* J2m>o Rm(nw)

L e m m a 3.4.4. Let μ G M(K). Iμ'. T ^ L2(K, μ) is injective.

< oo, then

Proof. By (3.4.1), we have R

k)£(U

C( J2

-Um,U-

Um) > \^(u)

- U^.

(3.4.2)

k>m

Note that c is independent of μ. First we will show the following claim. Claim: There exists c\ > 0 such that, for any μ € M(K) and for any u G T with Lμ (u) — 0, > cx max|^(p)| 2 .

max{l, y^ Rm^)}£(u,u) ^

(3.4.3)

peVo

m>0

Proof of Claim. Let S = max{l, ^ in (3.4.2). Then we have cS£(u,u) > cS£(uo,uo) + /

.Rm(/x)}. If Lμ(u) — 0, set m — 0

m > 0

|^o|2^ 2

/

JKK

^/i > cE(uo,uo) + (tto) 2 ,

(3.4.4)

where u$ = JK U^μ = ^2peVo u(p) JK IJJPC^μ. Let Tio be the collection of all harmonic functions. If ap > 0 and ^2peVo OLV = 1 for a = (a p ) pG vb? w e define 7Va(?;) - -y/c5(i;, v) + ( E p G v 0 a p v < » ) 2 f o r v G Ho- T h e n ^ α is a norm on Ho. Since the parameter space a is compact, there exists c 0 such that Na(v) > c^ maxpGvb l v (p)| f° r a n Y v € Wo and for any a. Hence, it is easy to see that (3.4.4) implies the claim. • Now we prove Lemma 3.4.4 by using the claim. If Lμ(u) = 0, then Lμ™ (u o Fw) = 0 for w G W*. Hence, applying (3.4.3) to μw and u o Fw, c3£(uo Fw,uo 1

Fw) >

max

\u(p)\2,

£F(V) R

where c 3 = c i " max{l,sup w e M / ^ J2m>o m(^w)}Note that c 3 is independent of w G W*. Since £^(ti, u) > rw~1£(u o i ^ , w o F w ) , we have max F(V)

\u(p)\ 2

92

Construction of Laplacians on P. C. F. Self-Similar Structures

So if p = TT(U) G Vm, then Wm

£(u,u) > \u{p)\2.

Using Proposition 3.1.8, we see that r(JJlUJ2,^UJrn —> 0 as m —> oo. Hence tx(p) = 0 for any p eV*. Therefore u = 0. D

Now we may identify ^ " a s a subset of L2(K, μ) through Lμ.

Lemma 3.4.5. Let μ G M(K). Assume sup^GVF+ E™>o ^m(/^w) < oo. // £*(u,v) = £(u,v) + JKUVJ^μ, then Lμ is a compact operator from (T,£*) Proof. By (3.4.2),

k>m

Hence | | ^ — Pm\\(jr1s*)-+L2(K^) -^ 0 as m -^ oo. Since Pm is a finite rank operator (Definition B.1.9), Proposition B.I. 10 and Theorem B.I. 11 imply that Lμ is a compact operator. • Now, we are ready to show that (£,F) is a local regular Dirichlet form on L2(K^). See Definitions B.3.1 and B.3.2 for the definitions concerning Dirichlet forms. Theorem 3.4.6. Assume s u p ^ €^ ^ ^ ^ ( / i 1 " ) < oo. Then {£,J~) is a local regular Dirichlet form on L2(K, μ). The corresponding non-negative self-adjoint operator H^ on L2(K^) has compact resolvent. Remark. If μ is a self-similar measure with weight fai)ies, then the condition s u p ^ t ^ J2m>o R"m{lJ>w) < cxD is equivalent to r ^ < 1 for all i € S. Recall that H^ is characterized as a non-negative self-adjoint operator on L2(K^) that satisfies £ = QHN and T = Dom(i7jv1//2). Existence of such a self-adjoint operator is ensured by Theorem B.1.6. In other words, for u e T, u e Dom(iJjv) if and only if there exists / G T such that £(u, v) = (/, v)μ for any v G T. In this case, / = H^u. The subscript "JV" in HN represents the first letter of "Neumann" because — ifAT corresponds to the Laplacian with Neumann boundary condition. We call —HN the Neumann Laplacian. See 3.7 for details. As we mentioned at the beginning of this section, the above theorem is an easy corollary of theorems in Chapter 2 if (D, r) is regular or, equivalently, K = fJ. In such a case, the condition sup t t ; G ^ Sm>o ^0^™) < °° is automatically satisfied by any μ G M{K) because r^ < 1 for any i G S.

3.4 Dirichlet forms on p. c.f self-similar sets 2

Proof Closedness: Note that L (K^)

C ^(K^).

93

Set

= 0}. JK

Then £ is an inner-product on T and (T, £) is complete. For u^T, define a 7 u = u — JK ^μ. If {i£n}n>o is Cauchy sequence in (.T , £*), then {un}n>o is a Cauchy sequence in (.T7, £). Hence f(ii n — v, un — v) —> 0 as n —> oo 2 2 2 for some v G f . Using the fact that / K ?/ d/i = JK u dμ+(JK ^μ) , we see that {JK undμ}n>Q is a Cauchy sequence. Let c be the limit of JK U^μ as n —> oo and let tx = v + c. Then it is easy to see that {un}n>0 is convergent to u in (.?*,£*) as n —• oo. Hence (^",5*) is complete. Regularity: Set W* = {u : u is an m-harmonic function for some m > 0 } . Then H* C C(K). The maximum principle shows that {P m ii} m >o converges to u uniformly on K for u G C(K). Hence H* is dense in C(K). Also, W* is dense in (.F, £*). Therefore W* is a core for (£^T). Markov property: For u E J7, define u in the same way as in (DF3) of Definition 2.1.1. As (fi,?) is a resistance form, u £ T and f(/u, u) < £(u,u). We need to show that ^Mf2 = T^u. This is obvious for any u G J 7 n C(K) because {P m ^} m >o converges to u uniformly on K as m —> oo. Therefore, (5,FC\C(K)) has the Markov property. Since f n C ( K ) D W*, it follows that (£,.F) is the minimal closed extension of (£,,Fn C(K)). Therefore, by [43, Theorem 3.1.1], (5, J 7 ) has the Markov property. Local property: Assume ^ ( u ) = 0 on a open set O C K. For any p G fiflO, there exists w G VF* such that p G X^ C O. Then ^ ( w o F^) = 0. As w μ satisfies the condition of Lemma 3.4.4, it follows that u o Fw — 0 in T. Hence u = 0 on Q D O. Now if supp(^(ii)) D supp(^(v)) = 0 for it, v G J 7 , then supp('u) D supp(v) = 0. Since Sm(u,v) = 0 for large m, we obtain f(w,v) = 0 . Finally, by Theorem B.I. 13, Lemma 3.4.5 shows that H has compact resolvent. • As an easy corollary of Theorem 3.4.6, we also have the Dirichlet form which corresponds to Dirichlet boundary conditions. Corollary 3.4.7. Define To = {u : u G T,u\yQ = 0}. Suppose that 8\ipweW^ ^ m > 0 Rrn^w) < oo. Then (£,J-Q) is a local regular Dirichlet form on L2(K, μ). Moreover the corresponding non-negative definite selfadjoint operator H& on L2(K, μ) has compact resolvent. The operator is called the Dirichlet Laplacian. Remark. Strictly speaking, as T§ is not dense in C(K) with respect to the

94

Construction of Laplacians on P. C. F. Self-Similar Structures 2

supremum norm, (S^o) is not a regular Dirichlet form on L (if,/i). It is, however, a regular Dirichlet form on L2(AT\Vb,//). As with Hjsr, we see that HD is characterized by £\j?QX:pQ — QHD and 1 2 T§ — Dom(Hz) / ). Also u £ Dom(i7/)) if and only if there exists / £ J-Q such that £(u, v) = (/, v)μ for any v £ T§. In this case, / = Proposition 3.4.8. HD is invertible. Moreover, GD — (HD)~ 2 pact operator on L (K^) characterized by £(Gof,v) = (f,v)μ 2 v £ FQ and for any f £ L (K, μ). X

is a comfor any

GD will be called the Green's operator. See 3.6 for details. We will give two proofs of this proposition below. For those who are not familiar with the subjects in B.I, the second proof may be more accessible. Proof. Note that (To, £) is a Hilbert space. First we will give a proof using some general theory. Since £ is positive definite on To, 0 is not an eigenvalue 1 of Hj). As HD has compact resolvent, GD — HD~ is a compact operator. As £(u, v) = (HDU, v)μ for any u £ DOTII(HD) and for any v £ T§, we have

£{GDf,v) = {f,v)n f° r any v £ T^ and for any / £ L2{K^). Next we give another proof, which is less abstract. By (3.4.2), we have 2 \(u, f)μ\ < c£(u,u) > (iz,iz)/x for any u £ TQ. Hence, for any / £ L (K^), u \\ \\μ\\f\\μ yjc£{u,u) I I/I Iμ for u £ T§. This means that u —• (u,/) M is a bounded functional on a Hilbert space (J~o,£). Therefore, there exists h £ J-Q such that £(u, h) = (u, / ) M for any u £ T§. It is easy to see that h is uniquely determined by / and the correspondence / —> h is linear. Hence we write h = Gz>/. Since (GDf, j % = £(GDf, GDf)

> c(GDf, GDf)^

(3.4.5)

we have CHGD/IU < \\f\\μ^ So GD is a bounded operator from L2(K^) 1 to itself. It is obvious that GD = HD~ - Moreover, we also see that Iμ > \/£{GDf'> GDf) by (3.4.5). Hence GD is a bounded operator from 2 2 L (K, μ) to (Vo, £). So, if U is a bounded subset oiL (K, μ), then GD(U) is > 2 a bounded subset of {To, £). Now, by Lemma 3.4.5, Lμ : (^b? £) — L (K, μ) is a compact operator. Hence GD{U) = tμ{GD{U)) is a relatively compact 2 2 -» L2(K^) is a compact subset of L (K, μ). Therefore GD • L (K^) operator. •

3.5 Green's function We continue to assume that (D, r) is a harmonic structure on a connected p. c. f. self-similar structure (K,S,{Fi}ieS), where S = {1,2,... ,JV}. In

95

3.5 Green's function

this section, we will construct the Green's function for (£, J7) derived from the harmonic structure (D,r). The Green's function will be the integral kernel of Green's operator in the next section. Roughly speaking, the Green's operator is the inverse of the Dirichiet Laplacian, GΒ = ( i J ^ ) " 1 , given in Proposition 3.4.8. More precisely, (GDu)(x) = /

JKK

2

for u G L (K, /i), where g is the Green's function which we will define in this section. The above relation is justified in 3.7, in particular, Lemma 3.7.10 and Theorem 3.7.14. Recall that Hm : e(Vm) -> t{Vm) is given by Hm = Ewewm ^RwDRw, where Rw : £(Vm) —»> £(Vo) is defined by Rwu = u o Fw. Also, Hm can be expressed as ( \

m

™ ) , where Tm : £(Vo) -> £(V0); Jm

: £(V0)

and Xm : ^(KA^o) i(Vm\V0). First we will study a discrete version of Green's function —X^1. For this purpose, we need the following lemma. Lemma 3.5.1. Let X — (Xij) be a symmetric n x n real matrix. Assume that X satisfies the following four conditions. (1) Xα < 0 for any i = 1,2,... , n. (2) X^OifijLj. (3) 5Zr=i Xij < 0 for any j = 1,2,... , n and this sum is negative for some 3(4) If i ^ j , then there exist m>1 and io, • • • , im € {l> 2 , . . . , n} such = that iQ — i, im j and X{kik+1 > 0 for all k = 0 , . . . , m — 1. T/ien X is invertible. Moreover, if G — (G{j) = (—X)" 1 , £/ien Gti > Gij > 0

/or anyij

G {1,2,... , n } .

Proof Define a n n x n diagonal matrix A by Aα = —Xα- Also define an n x n symmetric matrix B by Bij = X^ if i ^ j and i ^ = 0 for any z = 1,2,... ,n. Then - X ^ " 1 = / - BA~X, where / is the n x n unit matrix. Let C = BA~X. Then C^ = Bij/Ajj > 0. By the assumption (3), we see that 0 < 5^=1 ^ j — ^ ^ o r a n y ^ = 1? 2,... , n and that this sum is strictly less than 1 for some j , say, / G {1,2,... , n}. Set \x\ = YZ=1 \xi\ for x = (^)i=i,2,...,n € Mn. Then | • | is a norm of

96

Construction of Laplacians on P. C. F. Self-Similar Structures

R n . It follows that

\cx\ <

it

and that \Cx\ < \x\ if xx ^ 0. For j G {1,2,... ,n}, define e5 G RN by (ej)i = 6ij for all i E {1, 2,... , n}, where 6 is Kronecker's delta. By the assumption (4), we see that there exist rrij and z0, i i , . . . , i m j G {1,2,... , n} such that zo = /, imj = j and Cikik+1 > 0 for any k = 0,1,... , m^ — 1. This implies that (C^e^i > 0. Hence it follows that ( C ^ + ^ 1 < IC^e^l < \ej\ — 1. Let s = maxjG{l52,... ,n} m j + 1- If x E Mn and x ^ 0 , then |C s x| < E ^ i k j l i ^ e ? ! < N- Hence it follows that ||C 5 || < 1, where ||C S || is the operator norm of Cs with respect to the norm | •| on R n . Hence -XA'1 = / - C is invertible and (/ - C)" 1 = / + C + C 2 + • • •. This implies that X is invertible and that m>l(io,...,t m )€5m (t,i)

where Sm(iJ)

= {(*o,... ,*m) :io = «, *m = j and {ik}k=o,...,m C { 1 , 2 , . . . , n } } .

By condition (4), we see that G^ > 0 for any i, j G { 1 , 2 , . . . , n } . Now define Mi = maxfc=i?2,...,nGifc. Suppose that G^ = M^ and that i / j . Then

By condition (3), it follows that ^2k:k^j —X kj/Xjj < 1. Hence G^ = G^ for any k with X ^ > 0. This implies that Gik = Mi if there exists m>1 and io,.. •, i m such that zo = k, im = j and X^il+1 > 0 for any / = 0,... , m — 1. By the condition (4), it follows that G^ = Mi for any i = 1,2,... , n. Hence G^ > G^ for any i,j. • Lemma 3.5.2. X m is negative definite. If Gm = —X^ 1, then (G m ) pp > (Gm)pq 0 for anyp.q G F m \F 0 , ^ f e e ^ G m = ((Gm)Pg r, forp,q G V^\Vfc, (Gm)pq > 0 if and only if p ~ q. m

Recall Definition 3.2.9 for the definition of ~. m

1}

Proof. Let (V m \V 0 )/- = {V4 ,... , vi }, where each V^} C Vm\V0 is an 771

k)

equivalence class. Then Xm is decomposed as X$ : ^(vdi}) -> ^(Vdi}) for i = 1,2,... ,/c. Applying Lemma 3.5.1 to each X^ , we obtain the first

3.5 Green's function

97

half of the lemma. Also, by Lemma 3.5.1, it follows that (Gm)pq > 0 if and • only if p ~ q. m

Set X = Xu Xpq = (X^pq, G = Gi and Gpq = (Gi)OT for any p,q € Definition 3.5.3. Define *(x,y) = J2P,qev1\v0GP
{xy)=

[0

for

*,V

otherwise,

x

for x,yeK.

Write ^ (y) = tf(s,y) and ^ ( y ) = *w(a:,2/).

^ ^ is a non-negative continuous function on K x K. \J/£, is an ra + 1harmonic function if w G T^m-

Lemma 3.5.4. For any u e T, i/a; e Kw, otherwise. Proof. For u

Therefore, if x G /fw, ly G Wm and z = F~1{x), then

= rw~x((u o Fw)i(z) - (uo Fw)o(z)) = r w " 1 (u m + i(a:) -

um(x)). D

Set m-l

and set g^(y) = gm(z, y)- Then g^ is an m-harmonic function and g^(y) = 0 if y G Vo- By Lemma 3.5.4, we see that £{g*m,u)=um{x)-uo{x) for any u 6 f . Therefore, gm(x,y) =

(3.5.1) l n

n

Y^PtqeVm\Vo{Gm)pqi p {x)ipq {y)-

98

Construction of Laplacians on P. C. F. Self-Similar Structures

Now note that ^w(x,y) > 0. Hence {gm(x,y)}m>Q is a non-decreasing sequence. Therefore the following definition makes sense, allowing o o a s a possible value of the limit below. P r o p o s i t i o n 3 . 5 . 5 . For x,y G K,

g(x,y)=

define

lim gm(x,y) = m—>oo

Then g is continuous on {(x,y) € K x K : x ^ y}. Moreover, (1) Let x, y G K\Vo. Then g(x, y) > 0 if and only if x and y belong to the same connected component of K\Vo. (2) // (D,r) is regular, then g is a continuous function on K x K. g is called the Green's function associated with the harmonic structure (AΓ).

We need the following lemma to prove this proposition. Lemma 3.5.6. For any p, q G Vm\Vo, the following three conditions are equivalent. (1) p and q belong to the same connected component of K\Vo, (2) p ~ q, (3) (Gm)Pq > 0. Proof By Lemma 3.5.2, we immediately see that (2) and (3) are equivalent. Also, Lemma 3.2.13 implies that (2) => (1). Now the remaining part is to show that (1) implies (3). Let C be the connected component of K\Vo which contains both p and q. Then since C is arcwise connected by Proposition 1.6.6, there exists a continuous path 7 such that 7(0) = p, 7(1) = q and 7([0,1]) C C. Let {pi,p 2 ,..- ,Pk} = Vn H 7([0,1]), where Pi = p(U) and 0 = ti < £2 < • * • < tk = 1- Note that p\ = p and pk = q. Then there exist w(0),... ,it;(/c — 1) G Wn such that Pi,Pi+i G ^ ( i ) for all i = 1,2,... , k — 1. For sufficiently large n, it follows that Kw^ C C for i = 1,2,... , k — 1. By Lemma 3.2.10-(l), there exists an i7n-path between pi and pi+i for any i = 1,2,... , k — 1. Combining those iJn-paths, we have an i/ n -path between p and q. Hence (Gn)pq > 0. Note that (Gm)pq = grn(p,q) = ^ ( p , ? ) = {Gn)pq H Tu < Ti sad p,q G Vm.

Therefore

we see that (Gm)pq > 0. This completes the proof. Proof of Proposition 3.5.5. Set Ktn,p = UW€Wm,p€^w-Kti; f° r P £ K.

• If

x y^ y, then ifm}a5 fl ifm,3/ is empty for sufficiently large m. Hence the sum in the above definition is really a finite sum if x ^ y. Since tyw is continuous, we can verify that g is continuous on {(x, y) € K x K : x ^ y}.

3.5 Green's function

99

(1) Choose m so that Km,x n Vo = Kmty D Vo = 0. Set Vm,x = Km^ n V^n and V^y = i^m,y ^ Vo- Since K m , x and lfm)1/ are connected, there exist connected components C\ and C2 of K\Vo such that K m ? x G C\ and #m,y € C2 . Note that if ^™(a) > 0, then p € V^>a for a = x,y. First assume that C\ —C^. We may choose p e V^ >a. and g' G F m , y that satisfy ^ ( x ) > 0 and ip™(y) > 0. Then Lemma 3.5.6 implies (Gm)pq > 0. Therefore,

9m(x,y) > (GrJn^ixWfiy)

> 0.

Next suppose that C\ ^ C2- If ip™(x) > 0 and ip™(y) > 0, then p e V m,x and q G Vmty. So p and ^ belong to different connected components of K\Vo. Hence, by Lemma 3.5.6, (Gm)pq = 0. Therefore gm(x,y) — 0 for sufficiently large m. This immediately implies that g(x, y) — 0. (2) As (jD,r) is regular, r = maxiesri < 1. There exists C > 0 such that su C rm Px, 2 / GKS U ;Giy m r ^^( : r '2/) ' - S i n c e Y<wewTnr™^™ i s continuous, r P = T,m>o(T,wewm w^w) is continuous. D If (D,r) is not regular, it could happen that g(x,x) = 00. In fact, we will show that g(x, x) < 00 if and only if x G ft. Theorem 3.5.7. The following three conditions are equivalent. (1) g(x,x) < 00 (2) gx G F, where gx(y) = g(x,y)

(3)

xen.

Moreover, if any of the above conditions is satisfied, then £(gx,u) u{x) — UQ(X) for any u G T. Lemma 3.5.8. There exist ci, c 2 > 0 such that, for any u G E,

m>0 m>0

where u> = '

Proof Let x = ir(u>). Then g(x,x) = Note that there exist ci, C2 > 0 such that, for any u G i(Vi\Vo),

pGVi\Vo

—

100

Construction of Laplacians on P. C. F. Self-Similar

Structures

Note that *Wla;a...Wm(ar,x) = EP,qev1\v0 Gpqipp(n(amu;))p > 0. Also, by the definition of (/?, (3.5.2) implies cM7r(amLj))2

< *Ul»2..Mm(x,x)

< c2
D Proof of Theorem 3.5.7. (3) => (2): Set To = {u G J 7 : u|vb = 0}Then (.Fo,£) is a Hilbert space. If x G ft and p G Vo, then |?x(:z)|2 < £(u,u)R(x,p) for li G ^o- This shows that u —> it(x) is a continuous functional To — > R- Hence there exists h e To such that £(/i, w) = ix(x) for any u G ^b- Now £(hmju) = £{h,um) = um(x) = 8(g^,u) for any u G ToHence hm = g^. So h = gx and gx G To(2) =^> (1): This is obvious by the following equality.

£(gx,gx)=

lim £{gxn,,g^,)= lim gm(x,x)

m—>^oo

=g(x,x).

m—>^cxD

(1) =^> (3): l£g(x,x) < oo, then, by Lemma 3.5.8, J2m>o r^^i-..o;m^(^m)2 < oo, where 7r(a;) = x and # m = 7r(crmc<;). Hereafter we will use \I>m and and r a;ia;2...a;m respectively. r(m) to denote ^u^-ujm First, if lim m ^oo (p(xm) = 0, then m i n r 6 p 6(amu,r) —> 0 as m —• oo, where ^ is a metric on E. Then, by Proposition 3.1.8, we see that Sm>o r ( m ) < °°- So, Lemma 3.3.9 implies that x G ( l . Next assume that limsup m _^ oo (p(xm) > 0. Choose {nii}i>o so that rrii < mi+i and (p(xmi) > limsup m _ >oo ip(xm)/2. We may assume that (j mi u; -+ T £ T>\P as z —^ oo. Set ^ = 7T(T). Then we can choose k so that iffc,g = UW£wk:qeKwKw does not intersect VQ. Also we may assume that xmi G i^ffc,^ for all i > 1. Choose q* G i^fc,9 H V^ and define pi = Claim 1: There exists c\ > 0 such that

^m{puPi) < c1^m(x,x)

and ^m(pux)

<

cl^rn{x,x)

for any i and for any m G [0, m^ — 1] U [m^ +fc,oo). Proof of Claim 1: If m > ra» + fc, then ^m(pi,y) = 0 for any y E K. So the claim is obvious in this case. Assume that m < m^ — 1. If ii(y) = \I/ m (p, FWlW2..iW (y)), -u is a non-negative harmonic function for any p £ K. Set X = i£fc,g. Then the Harnack inequality (Proposition 3.2.7) implies that there exists c > 0 such that u(yi) < cu(y2) for any 2/1,2/2 € X. Hence, we see that ^Tn(p,x1) < c^ m (p,x 2 ) for any x^ x2 G i^i^ 2 ...^m(^M)Therefore

*m(Pi,Pi)

c*m(pi,x)

3.5 Green's function

101

This immediately implies the claim. Claim 2: Let Ni = Ui>i{mi,... ,ra*+fc- 1}. Then XlmeNi r ( m ) < °°Proof of Claim 2: Since liminf^oo ip(xmi) > 0 and Y2m>or(rn)(P(x™>) 2 < oo, it follows that X^>i r{mi) < oo. So y ^ r(m) < V^(l + i? + R2 H mGNi

h Rk~1)r(mi) < oo,

i>1

where i? = maxsG,s r s . Claim 3: lim*—«> g(puPi) = lini^oo g(pi,x) = ^(x, x). Proof of Claim 3: Define ^ci^ m (o:,^)

otherwise.

Then Claim 2 along with the fact that g(x,x) < oo implies X^m>o a™ < °°Also, by Claim 1, it is easy to see that tym(pi,pi) < am and \£m(pi, x) < a m for all m. As ^ m (x,x) - lim i _ oo * m (pi,pi) = lim^oo * m (p»,x), the dominated convergence theorem proves the claim. Claim 4: There exists C3 > 0 such that R(z, y) < 2£(gz - g», g* - g») + c3 m^c \^p{z) - i>p(y)\2. for any z,y eft. Proof of Claim 4: Since u(z) — u(y) = £(gz — gy,u) + (uo(x) — uo(y)) for any u G T, we have |«(^) - u(y)\2 < 2£(gz - g», u)2 + 2\uo(z) - uo(y)\2 <2£(gz - gy,gz - gy)£{u,u) + 2 ( £ \u(p) - «(p,) - gy,gz - gy)£(u,u)

p€Vo

where p* G Vb a n d M = #(Vo). Also recall t h a t £(UQ,UO) = fo( ^D(^|vo» M lvb)j where D G CA(Vo). So there exists c > 0 such t h a t

(max\u(p) -u(p*)\)2 < c£(uo,uo) < c£(u,u) peVo

for any u G T. Hence we have Hz) U{ 2

-

f

< 2£{g> -gy,gz-

gy )

This immediately implies the claim. Claim 5: {pi}i>i is a Cauchy sequence in (

102

Construction of Laplacians on P. C. F. Self-Similar Structures

Proof of Claim 5: Note that {pi}i>\ is a Cauchy sequence in (K, d), where d is a metric on K that is compatible with the original topology of K. Hence maxpGvb \ipp(Pi) ~ i/>p(Pj)\ —> 0 as z, j —• oo. Also, by Claim 3, as i —> oo, i

Pi

- 9^9

X

- 9 ) = g(Pi,Pi) - 2g(Pi, x) + g(x, x) -> 0.

As £{u — v,u — v) < (yj£(u — h,u — h) + y/£(h — v, h — ^)) 2 , it follows that £(gPi — gPj ,gPi — gPj) -^ 0 as i,j —» oo. Therefore, by Claim 4, we have R(pi,pj) —> 0 as i, j —• oo. So {pi}i>i is a Cauchy sequence in (fi, ii). Now as lim^^oo d(x,pi) — 0, the limit point of {pi} in (0, i?) is identified with x. (Recall that Q, is identified as 9(Q) C K. See Proposition 3.3.2.) Hence x e ft. • Remark. The idea of using the Harnack inequality is originally due to Dr. A. Teplyaev.

3.6 Green's operator As in the previous sections, (D, r) is a harmonic structure on a self-similar structure (K, 5, {F;}iGs), where S = {1,2,... , N}. In the last section, we defined the Green's function g. In this section we define the (extended) Green's operator Gμ, which is an integral operator whose kernel is the Green's function. Let μeM(K). r

Theorem 3.6.1. Define R^μ) = maxwewm wV>(Kw) ' for 1 < t < oo. (Ift = oo, Rln = maxwewmrw') Assume ^ m > 0 #m(AO < °°- If s is the constant {including oo) that satisfies 1/t + 1/s = 1, then for f G LS(K^),

(GJ)(x) = f g(x,y)f(yMdy)

(3.6.1)

JK

is well-defined for all x G K and Gμ f G C(K) D T§. Moreover, Gμ : S L {K^) —• C(K) is a compact operator. Also £{u, Gμ f) = [ (u- uo)fdμ

(3.6.2)

JK

for any u G T. The operator Gμ is called the (extended) Green's operator. Remark. If /i is a self-similar measure with weight (/zi,//2,... ,^iv), then the condition J2m>o ^mil2) < oo is equivalent to r^/x^)1/* < 1 for all i G S.

103

3.6 Green's operator

Remark. If (-D,r) is regular, then g is a continuous function on K x K. Therefore the above theorem is almost obvious in this case. In particular, we see that E m > o i ? m < oo. Hence Gμ : Ll(K^) -> C(K). Comparing the above theorem with Proposition 3.4.8, we see that Gμ = 1 w GD = Hr>~ if s\xpweW^ Y^m>0 Rm(v ) < oo. More precisely, (3.6.2) imS plies that Gμf = GDf for / G L (K^) n L2(K^). Hence, if t > 2, then the (extended) Green's operator Gμ is really a natural extension of the Green's operator GD and GD(L2(K^)) C C(K) n T^. Hence Dom(if D ) C C(K) n JF0. Also, if 1 < t < 2, then G D | L S ( K , M ) = GM and GD(LS(K, μ)) C C(K) n J^To prove Theorem 3.6.1, we need several lemmas. Lemma 3.6.2. / / E m > 0 C M < oo, then gx G L\K^) for any x eK. Moreover, x —> gx is a continuous map from K to Ll{K, μ). For u G La(K,μ),

^{"(Iμ)1?0'.

we set | | u | | M j a = ( / K

Proof. For X = -K{UJ\UL)2 . . . ), ^ ( y ) = E m > 0 rwiW2-wm^a;ia;2...Wm (^5 !/)• Hence y ^ ruj1uj2 m>0

11<7^1 Iμ t

Wmll^^ cj

w

Hμ*

m>0

Hence 0*

m>0

eL\K^). Note that x

y

\\g -9 \\^t<

\\9Xm-9Vm\U +

\\9 m-:^lUt+ X

11^-^11

Since | | < ^ - ^ | U t < 2-^k>m+l Rk\lu) and gm is continuous on iC x Bf, we x y see that \\g — ^ || M ,t —> 0 if d(x, y) •

Lemma 3.6.3.

(?/)

"

Proo/. Note that £(ip™,u) = £m(il>™, u) = -(Hmu)(p) Since £(gp,u) = u(p) — uo(p), it follows that q

£( Y^ (Hm)pqg ,u)

for any u e T.

= (Hmu)(p) - (Hmuo)(p) = £(-i>™,u)

peVm r

Since UQ is harmonic, Lemma 3.2.16 implies £{$ ? — il>p, UQ) = 0 for p G VQ.

104

Construction of Laplacians on P. C. F. Self-Similar Structures

Also (Hrnuo)(p) — 0 for p 6 V^\Vb. Hence we have

7

for any u e J . This implies the desired equality.

•

Lemma 3.6.4.

Proo/. Note that (H7n(Gμf))(p)

=

The result follows by Lemma 3.6.3.

•

Proof of Theorem 3.6.1. By Holder's inequality,

Hence Lemma 3.6.2 implies that Gμf G C(K). Let {/ n } n >i be a bounded sequence in LS(K^). Then {G M / n }n>i is uniformly bounded and equicontinuous. So, by Ascoli-Arzela's theorem, {Gμfn}n>l contains a subsequence that is convergent in C{K). Hence Gμ is a compact operator from LS(K^) to C(K). By Lemma 3.6.4, = [ (um- v * ^ .

(3.6.3)

JK

peJm Set u = Gμ?. Then SmiG^Gnf)

= [ (Gμf)rnfdμ

< II/IU,! SUp \(Gμf)(x)\.

(Note that (G M /) 0 = 0.) Hence S(Gμf, G M /) < oo and G M / G To. For any w e f , letting m -> oo in (3.6.3), we obtain (3.6.2). • Next we study the n-th power of the Green's operator Gμ for n G N. The integral kernel #(n) for G^; is given by the following inductive formula. Set g^\x,y) = g(x,y) and

[

K

for n G Z.

105

3.6 Green's operator

Theorem 3.6.5. Assume that s u p ^ ^ E m ^ ^ i ^ 1 " ) < o c - Then, / o r any n, there exist cn > 0 and \I/™ (x,y) swc/i t/iat ^w is continuous on KxK,

ff(s,y)= X) r ^ M ^ r - 1 * ^ ^ ^ ) ,

(3.6.4)

and o < tfW (*, y)

(3.6.5)

CnXKw (x)xKw (y)

for any w G VF*. Moreover, if μ is a self-similar measure on K, then there exists a non-negative continuous function ^^ on K x K such that

[0

»*»«*.. otherwise.

(

3.6.6)

Prom the above theorem, it is easy to see that g^ is continuous on {(x, y) : x,y € K,x ^ y} because the sum in (3.6.4) is, in fact, a finite sum if x T^ y. Note that g^ = g is independent of μ but this is no longer true if n > 2. Corollary 3.6.6. Assume that sup^ eVF ^ E m > o ^ ( ^ ) < °°- V there exists t>1 such that Y^m>0 fl^(^) < oo7 then, for any n > (1 — 1/t)" 1 , #(n) zs continuous on KxK and G™ Z5 extended to a compact operator from LX{K, μ) to C{K). In particular, suppose that μ is a self-similar measure with weight (μ^izs which satisfies μ^i < 1 for any i G S. Then g^ is a continuous function on K x K for any n > max^s log A** / log μ^i. Remark. If ^ is a self-similar measure with weight fai)ies, the quantity maxie^log/ii/log^r^ coincides with ds&(v, μ)/2, where ds is the unique positive number that satisfies Y2ies(Viri)ds^2 = 1, ^ is a self-similar meads 2 sure with weight (^iri) / )ies and <5(i/, μ) is the distortion between v and μ defined in Proposition 4.5.2. In 4.1, ds is called the spectral exponent and it determines the asymptotic behavior of the eigenvalue counting function of the Laplacian. See Theorem 4.1.5 for details. Proof of Corollary 3.6.6. Set

Then hm is continuous on K x K and \hm (x,y)\ < c n (i?^(^)) n , where a = (1 - l / n ) " 1 . As 1 > 1/n -h 1/t, it follows that t' < t and hence E m > 0 < W < °°- S O E m > 0 ^ n ( < ( M ) ) n < C n ( E m > o < ( ^ ) ) n < 00.

106

Construction of Laplacians on P. C. F. Self-Similar Structures

Therefore ^™>o ^™ *s uniformly convergent on K x K. Hence g^ continuous on K x K. The rest of the statement is straightforward.

is •

Proof of Theorem 3.6.5. The proof is by induction on n. For n = 1, we have g(x,y) = Y,weWti rw^w(x,y) and 0 < ^w(x,y) < ciXKw(x)xKw(y), where C\ = maxX)1/Gx ty(x,y). Next assume that the claim of the theorem is true for n. Set ^ ™β = { \ for a,PeW*. Note that

Hence we have

0 < Vffifay) Now, let hw,m(x,y)

< clC^(KanK0)xKa(x)xK0(y).

= E 7 e ^ m rwrwynμ(KW7)n-x¥^)(x,y).

0 < &w,i,\\x,y) < c1cnμ(KW7)xKm(x)xKu,^(y)• \hw>m(x,y)\ < clCnrwn+^(Kw)n

max

(3.6.7) By (3.6.7),

Therefore, ^^{KJT

Since £ m > 0 ( / U / ^ ) ) n < Mn, where M = supw€W, E m > o ^ ( ^ ) . w e see that J2m>o hw,m is uniformly convergent on KxK. Hence, if hw(x, y) = Sm>o ^w,m(x,y), hw is continuous on K x K. Also 0 < hw{x,y) < ClcnMnrwn+^{Kw)nXKw{x)xKw{y)<

(3.6.8)

Next define fw,m(x,y) = Y^ieWrn rwlrwnμ(Kw)n~1^^,w (x,y). Similar arguments to those given above for /i W j m will show that fw = ^2m>1 fw,m is continuous on K x K and that 0 < fw(x,y)

< dc n Mr^ n + V(^) n Xi^Wx^(2/)-

So define ^L n + 1 ) by r ^ 1 μ^v)"V^(x,y) by (3.6.8) and (3.6.9), it follows that ^ + that 0 < ^+l\x,y) < cn+lXKw(x)xKw(y), Now by definition,

1 )

(3.6.9)

= hw(x,y)+fw(x,y). Then, is continuous on K x K and where cn+l = c n ci(M n

107

S.I Laplacians

Finally assume that /i is a self-similar measure. Note that (3.6.6) is true for n = 1. So suppose (3.6.6) is true for n. Then

"

X l

'

e J r

-

(3.6.10)

otherwise. Recall that r^+^iK^V^ix^y) = hw(x,y) + / w (a;,y). By (3.6.10), routine calculations show that (3.6.6) is true for n + 1. • Obviously, g^n\x,y) is non-negative by definition. Moreover, Theorem 3.6.5 implies that g^ (x, y) > 0 if x and y belong to the same connected component of K\Vo. More precisely, we have the following result. Corollary 3.6.7. Let m

£ r for any ra, n G N. If x and y belong to the same connected K\Vo,

then there exists m>0

component

such that g™ (x,y) > 0 for any

of

n>1.

Proof By Proposition 3.5.5-(l), there exists ra > 0 such that gin(x,y) = grn{x,y) > 0 and ^ } ( x , x ) > 0. Suppose that g{m\x,y) > 0. By (3.6.4),

gm(x,z)gW(z,yMdz) < g£+1\x,y).

(3.6.11)

Note that gm and gin are continuous on K x K. Since gm(x,x) > 0 and { { +l g m\x,y) > 0, (3.6.11) implies that g m \x,y) > 0. Now the corollary follows by induction. •

3.7 Laplacians As in the previous sections, (£>, r) is a harmonic structure on a self-similar structure (K, 5, { F j i G 5 ) , where S = {1,2,... , N}. In this section, we define the Laplacian AM associated with (D, r) and μ, w where μ G Af(jRT). If sup l u e V ^ m Xlm>o ^m{l^ ) < oo, we already introduced in 3.4 non-negative definite self-adjoint operators HN and Up associated with Dirichlet forms (S.J7) and (£,.Fo)> respectively, on I?(K^μ). (See Theorem 3.4.6 and Corollary 3.4.7.) We might think of -HN and -HD as Laplacians for the corresponding boundary conditions. However, in this section, we will give a more direct way of defining the Laplacian. Our Laplacian, Aμ, corresponds to the classical second derivative and its domain Dom(AM) corresponds to the C2-class of functions. See Example 3.7.2.

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Construction of Laplacians on P. C. F. Self-Similar Structures

Definition 3.7.1. Let μ G M(K). Define Vμ = {u G C(K) : there exists / G C(K) such that - f(p)\ = 0}, lim max ^]p(Hmu)(p) ™-+oo peV\Vo

where μ-m^p — JK ^β™ Aμ. Then for u G Vμ, we write / = Aμu, where / is the function appearing in the definition of Vμ. Aμ is called the Laplacian associated with (D, r) and μ. Obviously, Aμ : Vμ —• C{K) is a linear map and Vμ — Dom(AM). If u is a harmonic function on K, then it immediately follows that u £ Vμ and Aμ!! = 0. Example 3.7.2 (Interval). Recall Example 3.1.4. Set r = (1/2,1/2). Then (D,r) is a harmonic structure. For this harmonic structure, if p = m i/2 £ Vm, then

{

ix(2-^)-u(0) u(l - 2~m) - u(l)

if p = if p = 1 .

(3.7.1) Obviously, a harmonic function is a linear function. Now let μ be the selfsimilar measure with weight (1/2,1/2). Then μ is, in fact, a restriction of Lebesgue measure on R to [0,1]. For p G V^i\Vo, since ip™ is a piecewise linear function, we see that μrn^p = 2~m. Hence Routine calculus arguments show that Vμ = C2([0,1]) and Aμ?! = d2u/dx2 for u e Vμ. Example 3.7.3 (Sierpinski gasket). Let (D,r) be the harmonic structure on the Sierpinski gasket given in Example 3.1.5. Define a linear operator Lm : £(Vm) -+ e(Vm) by

where, for p G Vm, Vmjp = {q € Vm: q ^ p, there exists w G W^ such that p, g G -F^ (Vb)}.

109

S.I Laplacians Then Hm = (5/3) m L m . Note that

\{P

#{Vm,p) = l

^

e

Now let μ be the self-similar measure on the Sierpinski gasket with weight (1/3,1/3,1/3). Then it follows that JK ^ μ = 1/3 for any p G Vo- Using self-similarity and symmetry, we obtain,

IK

P

[1/3-+1

ifpe^oV

Hence μ^p(Hmu){p)

^5m(Lmu)(p)

=

for any p G Vm\Vo. The associated Laplacian Aμ is called the standard Laplacian on the Sierpinski gasket. Lemma 3.7.4. For v G C(K) and u G Vμ, V

-> / vAμUdμ

v(p)(Hmu){p)

as m —• oo.

Proof. If fm(x) = ^2peVrn\vQv(p)^m)p(Hmu)(p)ip^l(x), then fm —»- vA^.w as m —> oo uniformly on any compact subset of K\Vo. Also {/m}m>o is uniformly bounded. Hence, fK / m d ^ —> / K vAμudμ as m —• oo. • Lemma 3.7.5. For / G Pμ and pG^o? lim -(H

Proo/. Recall that Hm = (T™

*^ m ), where T m :

i{Vo) - ^(^m\Vo) and X m : £(Vm\V0) Tm - tJmX^Jm, hence we have

where / 0 = /|y 0 and /i = /Iv^vb-

T n i s

-> ^ m \ V b ) -

Note that D =

implies

(^m/)|Vo = ^rn^m

(Hmf)\vm\V0-

110

Construction of Laplacians on P. C. F. Self-Similar Structures

Since ( V ^ " 1 ) ^ = -i/>p(q) for psV0

and q G Vm\V0, we have

Letting ra —> oo, Lemma 3.7.4 implies the lemma.

•

Definition 3.7.6. For / G Vμ and p G Vo, we define

p is called the Neumann derivative of / at p G VoFor the case of Example 3.7.2, (3.7.1) implies that (d/) 0 = ~/'(0) and (df)i = f ( l ) . Lemma 3.7.7. For v G C(K) and u G Vμ, lim fm(i;,ifc)= y^v(p)(dw)p- / vAμUdμ.

m—>^oo

'

J

JITS K

pevo

Proof. Note that

^2

Y^ v(p)(Hmu)(p). pevm\Vo

Apply Lemma 3.7.4 to complete the proof.

•

Using the above lemma, we can immediately obtain the following theorem. Theorem 3.7.8 (Gauss-Green's formula). Vμ C T. For any v G T and u G Vμ, S(v,u) = ] T v(p)(du)p - J vAμudμ.

(3.7.2)

If both u and v belong to Pμ, then p

- u(p)(dv)p) = / (vAμU - uAμv)dμ. J K

(3.7.2) is a counterpart of the Gauss-Green's formula in ordinary calculus on R n and Riemannian manifolds. For example, let U be a bounded domain in R 2 and assume that dU is a smooth curve. For u,v € C2(f7), set

, dv du

dv du x

S.I Laplacians

111

Then the Gauss-Green's formula says £(v,u) = / v—ds — I vAudxdy. Jdu un Ju w h p r p L\U A?/ — @^ -t-Iwnere — dx2

Next we will study relations between the Laplacian Aμ and the abstract non-negative definite self-adjoint operators HN and HoTheorem 3.7.9. Assume that μ G M(K) and s u p ^ G ^ S m > o oo. // Z>D,M = {u G Vμ : u\v0 = 0}, then VD^ = Dom(HD) n Vμ, Ho\vD p — ~^μ\'DD,μ and Ho is the Friedrichs extension of—Aμ onV-p^. Also define VN^ = {u e Vμ : (du)\p = Ofor all p G Vo}. Then VN^ — Dom(H]s^)^Vμ, HN\VN^ = ~^μ\vN,μ and HN is the Friedrichs extension of —Aμ on Visf^. This theorem shows that — H^ is the Neumann Laplacian associated with (S, J-) and μ and that —Ho is the Dirichlet Laplacian associated with {£,J~) and μ. We write Vo and £>TV instead of Vo^ and X>AT^ respectively if no confusion can occur. Lemma 3.7.10. For any f G C(K), G M / G VD and AM(G?M/) = - / . Proo/. By Lemma 3.6.4, for p G Fm\Vb,

where e m = max p G v m (sup 2 / e j f t : m \f{y) — f(p)\)- As / is uniformly continuous on K, em —> 0 as m —> oo. • Proof of Theorem 3.7.9, Part I. Recall that, for u G To, u G Dom(# D ) and Hou = f if and only if £(v,u) = (v, / ) M for any i; G ^o- By (3.7.2), if u G DDJ £(V,U) = (v, —Aμu)μ for any i; G ^>• Hence D ^ C Dom(Ho) l and F D | P D = - A M | p D . Since Gμ\c^K) = Hp \C(K), li u e VD and -AμU = /, then Gμ? = u. Hence P D C Gμ(C(K)). On the other hand, by Lemma 3.7.10, we have VD D G^CiJK)). Hence VD = Gμ(C(K)). set Hou = f for u G Dom(if/p). Then, there exists {fn}n>o C such that / n ^> / as n —> oo in L2(K^). If wn = Gμfn, then u n G ). Recall that G^) = Ho~x- By (3.4.5) (and the discussions following it), < c(ft, ft)M for any ft G Z/2(i^, //), where f* = 5 + (•, -)M. As () |c(K), wehavef*(u n -Tx,w n -w) < c(fn-f,fn-f)μ -> 0 as n —>• oo. Hence the completion oiVo with respect to £* contains Dom(if^). Now obviously, the completion of Dom(Ho) with respect to £* equals Jo

112

Construction of Laplacians on P. C. F. Self-Similar Structures

because the Friedrichs extension of Hp is HD and T§ Therefore the completion of Vp with respect to £* equals T§. This means that HD is the Friedrichs extension of — A M | p D . • To prove the rest of Theorem 3.7.9, we need several preliminaries. For / G L\K^)Z define Pf = f -JKfdμ. Set IP(K^μ) = P(If(K^)\ f = P(J=), VN = P(VN)znd C(K) = P(C(K))L Note that (J^£) is 2 a Hilbert space, hence {£,F) is a closed form on L (K, μ). Let H^ be the non-negative self-adjoint operator associated with (£, OF) on L2(K, μ). Then, for u G T and / G L 2 (if,/i), w e pom(HN) and #7v^ = / if and only if £(v,u) = (v,/) for any v e J7. So it is easy to see that Dom(HN) = Dom(iJjv) H ^ and HN = ^iv| D o m ( j f f i v ) n ^- Note that HN is injective because £ is positive definite on T. The following lemma is obvious from Lemma 3.6.4. Lemma 3.7.11. For f G L°°{K^), peV0.

(d(G M /)) p - -JK'^βpfdμ

for any

Lemma 3.7.12. For any f G L2(K, μ), there exists a unique u G f such that £(v, u) = (vo, / ) M /or an?/ v e T, where v0 = J2pev0 v{v)^p' Moreover, define Q : L2(K, μ) —> L2(AT,/i) 6?/ Q/ = u. T/ien Q/ is harmonic and (d(Qf))p = $K^vWv f°r anype Vo. Also Q : L2(K^) -> L 2 (X,^) is a compact operator. Proof Let ap = / x ^ p / d ^ for any p G V^. Then (v0, / ) M = Z)p€Vb Since / G L2(K^)y

J2peVo ap = 0. Choose any # G fi. Then

Therefore, v —> (VQT^μ is a continuous functional on (iF,£). Hence there exists u G T such that £(v, u) = (i>o, /) for any v G J^. Let it = Q/. Then, for p G Vm\Vo, -(Hm(Qf))(p) = eU>?,Qf) = S(Py>?),Qf) = ( ( P « ) ) O , / ) M = 0, because (P(^^))o is a constant. Hence Qf is harmonic. Since A^Q/) = 0, (3.7.2) implies £(«,Q/) = E p€vb v(p)(d(Qf))p. Therefore (d(Q/)) p = ap. Since Qf is harmonic, (3.7.3)

113

S.I Laplacians

Let Ho = {Pu : u is a harmonic funtion on K}. As dimW0 < oo, there exists c > 0 such that c{u,u)μ < £o(u,u) for any u G Ho- Hence, by (3.7.3), we have c||Q/|| M < ||/||M.^Therefore Q : Z 2 ( i < » -> L2(K^) is a bounded operator. Also, as dim'Wo < °°5 Q is a finite rank operator and hence it is a compact operator by Proposition B.I. 10. • Proof of Theorem 3.7.9, Part 2. For / G Z 2 (if,/i), define GNf = P(GDf + Qf). Then for any iz G J 7 , £(u, GAT/) = (u - u0, /) M + (u 0 , /) M = (u, /) M Hence Gjv/ G Dom(iJjv) and -ffjv(Gjv/) = /. As G^ and Q are compact operators, we see that GN ' L2(K^) —> L2(K^) is a compact operator. Since i7/v is injective, it follows that H^ is invertible and GN = H^1. Also we obtain £(GNf,GNf)

= {GfffJ^

IIGAT/IUII/IU

< cWfWl

(3.7.4)

Next for, u G T>N-, (3.7.2) implies that £(v, u) = (v, —Aμu)μ for any v G J^. Also, by (3.7.2), 0 - £(l,u) = - fK A^μ. This implies A ^ G C(i^). Therefore, we have -u G Dom(i?^) and Hjyu = —Aμu. Thus we obtain that Vjsf C Dom(ifiv) and that ^T/vlp^ = —AM|p . On the other hand, for / G C(K), GNf = P(G£f + Qf). Hence, it follows from Lemma 3.7.11 and Lemma 3.7.12 that Gjv/ € £>w Therefore GN(C(K))=VN. Now for ti G Dom(iifjv), let / = Hjyu. Then there exists {fn}n>i C C(K) such that / n —* / as n -^ oo in L2(K^). If wn = GNfm then, by (3.7.4), £(un—u,un—u) < c\\fn — /I| 2 . This implies that itn —* iz as n —> oo in (J7, £*). Since un G Pjv? we have shown that the completion of V^ with respect to £* contains Dom(if/v). Since i7^ is itself a non-negative selfadjoint operator, the Friedrichs extension of H^ on Dom(i/^) is HN and T — Dom(Hj ). Hence, the completion of Dom(ifjv) with respect to £* equals to T. Therefore the completion of V^ with respect to £* is j ^ . Finally, note that T = T 0 {constants} and Piv = T^N © {constants}. Also we obtained H^ = HN\Dom(HN\n Since V^ C Dom(ifjv) and w e seet h a t HN\X>N = ~ A M I P N ' ^ C Dom(ifiv) and HN\VN = -A M |p J v . Moreover, since the completion of Djv with respect to £* is J^, it follows that the completion of T>N with respect to £* equals J7. Therefore iJ^v is the Friedrichs extension of — A μ ^ . • By the arguments in the above proof, we also see the following fact.

114

Construction of Laplacians on P. C. F. Self-Similar Structures

Corollary 3.7.13. For b = D and N, if f G Dom(Hb) and Hbf G C(K), then f G Dom(Hb) D Vμ = Vb^. To end this section, we study the Dirichlet problem for Poisson's equation. Theorem 3.7.14. For any f G C(K) and p G £{Vo), there exists a unique ii G Vμ such that (3.7.5) u\Vo

= p.

Moreover, u is given by u = J2Pev0 Pip)^

~ Gμ!•

Proof. If u = Y2pev0 P(P)^P ~~ G>/> then, by Lemma 3.7.10, u satisfies (3.7.5). Assume that both u\ and u^)" 1 , it follows that t> = 0. Therefore U\ = U2' n Corollary 3.7.15. The collection of harmonic functions on K coincides with {u : u G Pμ, Aμ^ = 0}. Combining Lemma 3.6.4, Corollary 3.7.13 and Theorem 3.7.14> we obtain the following theorem. Theorem 3.7.16. Let

) - /

9(x,

JK

pevo (4) u G C(K) and, for any m>1

and any p G Vm\Vo,

(Hmu)(p) = f

JKK

(5) u G T and, for any v G Vμ^, £{v,u) = — JK

3.8 Nested fractals

115

3.8 Nested fractals In 3.1, we introduced the notion of harmonic structures. In the following sections, we then constructed Dirichlet forms, Green's functions and Laplacians under the assumption of the existence of a harmonic structure. In this section, we will show that there exists a harmonic structure on a class of highly symmetric p. c. f. self-similar structures based on the idea of Lindstr0m [116], where the notion of nested fractals was introduced. Recall 3.1, where we observed that the existence problem for harmonic structures is equivalent to the eigenvalue problem (or fixed point problem) of the renormalization operator lZr. In [116], Lindstr0m showed the existence of a fixed point of a probabilistic counterpart of the renormalization map lZr for nested fractals. In this section, we will extend the notion of nested fractal and introduce strongly symmetric p. c. f. self-similar structures. In Theorem 3.8.10, we will show existence of harmonic structures for strongly symmetric p. c. f. self-similar structures. Throughout this section, £ = (K, 5, {Fi}ies) be a connected p. c. f. selfsimilar structure, where S = {1, 2,... , N}. First we introduce the notion of symmetry of a p. c. f. self-similar structure. Definition 3.8.1. A homeomorphism g : K —» K is called a symmetry of C if and only if, for any m > 0 , there exists an injective map g^ : Wm —• Wm such that g(Fw(V0)) = Fg(m){w)(V0) for any w G Wm. In this section we also assume that K cRd and that i^ is the restriction of a similitude with Lip (i^) = Q on K for any i G S. Recall that a map / : Rd -> Rd is called a similitude with Lip(/) = c if there exist U G 0(d) and a eRd such that f(x) = cUx + a for any x G R. See Definition 1.1.2 and the following remark. Let #(Vb) = M and let Vo = {Pi}i=i,... ,M- Then, without loss of generality, we may assume that X^=i Pi = 0Definition 3.8.2. For x,y G Rd, x ^ y, let Hxy = {zeRd:\x-z\

=

\y-z\}.

(HXy is the hyperplane bisecting the line segment [#,y].) Also let gxy : Rd -> Rd be reflection in Hxy. Remark. Let g be an affine transformation from Rd to itself. Suppose that g\x is a symmetry of C. Then g(Vo) = Vo- Since Y^L^Vi — 0, it follows that #(0) = 0. Hence g is a linear transformation.

116

Construction of Laplacians on P. C. F. Self-Similar

Structures

Definition 3.8.3. (1) Let ra* = # { | x - y\ : x,y € V0,x ^ y}. Define l0 = min{|a; - y\ : x,y G Vb,x ^ y}. Also we define {Zt}t=o,...,m«.-i inductively by U+\ = min{|x — y\ : x, y G Vb, \x — y\ > k}. (2) Let Xi G Vm for i = 1,2,... , n. Then (xi)i=i,2,... , n is called an m-walk (between #i and x n ) if and only if there exist u> 1 ,... , u>72"1 G W m such that #i, Xi+\ G i ^ t (Vo) for alH = 1,2,... , n — 1. (3) A 0-walk (xi)i=i j2,... ,n is called a strict 0-walk (between x\ and x n ) if and only if \xi — Xi+\ \ — lo for any z = 1,2,... , n — 1. (4) Qs = {g : ^ G 0(d), # | K is a symmetry of £ } . ^ s is a group with respect to the natural composition. Recall the remark after Definition 3.8.2. Then g G Qs is equivalent to the condition that g : Wd —• W* is an affine transformation which preserves the Euclidean metric and g\x is a symmetry of C Definition 3.8.4. (1) We say that C = (K, 5, {Fi}ieS) is strongly symmetric if it satisfies the following four conditions: (551) For any x ^ y G Vb, there exists a strict 0-walk between x and y. (552) If £,2/, z G Vb and \x — y\ = \x — z\, then there exists g G Qs such

that g(x) = x and g(y) = z. (553) For any i = 0 , . . . , ra* — 2, there exist x, y and z £ Vo such that |x — 2/| = /;, |x — 2| = Zi-i-i and ^ ^ I A : is a symmetry of £. (554) Vb is ^-transitive, that is, for any x ^ y G Vb, there exists g € Qs such that #(:E) = 7/. (2) We say that C = (K, 5, {Fi}iEs) is an affine nested fractal if gxy\K is a symmetry of C for any x ^ y G VoIt is easy to see that if C is strongly symmetric, then it is weakly symmetric, which has been defined in 1.6. The following proposition is obvious by (SS4). Proposition 3.8.5. Let Vb = {Pi}i=i,2,...,M- V^ then |pi| = . . . = \pM\.

^s strongly symmetric,

It follows that an affine nested fractal is strongly symmetric. In fact, the conditions (SS2), (SS3) and (SS4) are obvious. We can verify (SSI) as well by virtue of the following lemma. L e m m a 3.8.6. Let C = (K,S,{Fi}ies) be an affine nested fractal. there exists a strict 0-walk between x and y for any x ^ y EVQ. Proof For p eVo, set Bp = {q G Vb : there exists a strict 0-walk between p and q}.

Then

3.8 Nested fractals

111

Define A — {{x,y) : x,y G VQ,BX ^ By}. (Note that there exists no strict 0-walk between x and y if (x, y) G A ) Assume that A ^ 0. Set a = min{|x — y\ : (x, y) G A} and choose (x, y) G A satisfying \x — y\ = a. (Note that a > Zo-) Let z e Bx with |z — x| = fo- Then (z,y) G A and therefore |y — z\ > \y — x\ = a. Since \z — x\ = lo < \x — y\ = a, it follows that x and z belong to the same side of the hyperplane Hyz. Hence, if e = (y — z)/\y — z\, then (x,e) < 0. Let 7 = —(x,e) and let XQ = gyz(x). Then \y — xo\ = lo and x$ G By. Note that xo = x + 2je. Using Proposition 3.8.5, we obtain (y -x,e)

= - ( x , e) H- (y, e) = 7 + (2/ - (y + *)/2, e) = 7 4- |y - 2;|/2.

Therefore, |y - x| = a > (y - x, e) = 7 + \y - z\/2 > 7 4- a / 2 .

(3.8.1)

This implies that \y — x\ > 27 = |x — xo| > a — \y — x|. Therefore equalities hold in (3.8.1). Thus we obtain that \y — x\ = (y — x, e) and that \y - z\ = \y - x|. So e = (y - z)/\y - z\ = (y - x)/\y - x\ = (y - x)/\y - z\. Hence x = z. This contradiction immediately implies that A = 0. • So we obtain the following proposition. Proposition 3.8.7. If C = (K, S, {i^}ies) is an affine nested fractal, then it is strongly symmetric. The converse of the above proposition is not true. See Example 3.8.13 below for an example. Originally Lindstr0m introduced the notion of nested fractals in [116]. Afterwards, the notion of affine nested fractals was introduced as an extension of nested fractals in [37]. The definition of (affine) nested fractals in [116] and [37] contained the following conditions: Connectivity: For any i,j G 5, there exist z i , . . . ,im such that i\ — z, im = j and Fik(V0) n Fik+1(V0) ^ 0 for any k = 1 , . . . ,ra - 1. Nesting: For any m and any w, wr G Wm with w ^ wf, Kwr\Kw,

=

Fw(Vo)nFw,(Vo).

Since we assume that K is connected, the connectivity condition is satisfied by Corollary 1.6.5. Also the nesting condition is immediately verified by Proposition 1.3.5-(2). Since the boundary VQ of an affine nested fractal is highly symmetric, the geometry of VQ seems quite restricted. In fact, it is conjectured in [6,

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§5] that Vo is a regular planar polygon, a d-dimensional tetrahedron or a d-dimensional simplex. Hereafter in this section, we always assume that C = (X, 5, {Fi}ies) is strongly symmetric. We also impose the following assumption on C: #(Fi(Vo) n Fj(Vo))

1 whenever i ^ j .

(3.8.2)

This may be regarded as a technical assumption in order to avoid nonessential difficulties. The forthcoming results in this section may hold without (3.8.2). In [37], it is claimed that (3.8.2) holds for any affine nested fractals. However, this does not seem quite clear at this moment. See also [6, §5] for some comments. Definition 3.8.8. (1) on Vb, Cs(Vo), by

We define the collection of symmetric Laplacians

£s(Vo) = {DG CA(Vo) : Dxy = Dx,y, if \x-y\

= \x' - yf\}.

(2) Let r — ( r i , r 2 , . . . , r ^ ) £ (0,00)^. We say that r is ^-invariant if Ti — Tj whenever there exists g G Gs such that g(Fi(Vo)) = FJ(VQ). (3) For a symmetry g, D G CA(Vo) is said to be ^-invariant if Dxy = Dg(x)g(y) f° r any x, y G Vb. Also, if D G CA(Vo) is ^-invariant for any g G Gs, then D is said to be ^-invariant. Obviously, if r = ( 1 , . . . ,1) then r is ^-invariant. P r o p o s i t i o n 3.8.9. The collection of all Gs-invariant Laplacians equals Proof. Since Gs C O(d), D is ^-invariant for any D G Cs(Vo). Conversely, suppose that D G CA(Vo) is ^-invariant. Let x, y,x',y' G Vb and assume that \x — y\ = \x' — y'\. By (SS4), there exists g\ G Gs such that gi(x) = x'. Let y" = gi(y'). Then \x' - y'\ = \x' - y"\. By (SS2), there exists g2 G Gs such that ^2(^0 — x' and g2{y") = 2/'- Set g — g^°9\- Then p(a:) = x' and = 2/;. Hence Dxy = Dx*y>. So D G £ s (Vb). D Theorem 3.8.10. Le£ £ = (iiT, 5, { F j ^ s ) 6e strongly symmetric. If T = {riir2T • • ,TN) ^ (0,00)^ is Gs-invariant, then there exist D G £ s (Vb) and A > 0 such that (D,Ar) is a harmonic structure on C Moreover, the resistance form {£,T) constructed from the harmonic structure (D,Xr) is Gs-invariant, i.e., u o g G T and E{u,u) — £ (u o g, u o g) for any g G Gs and u G T'. Remark. In 3.1, we have seen that (J9, Ar) is a harmonic structure if and only if Ur{D) = XD (or equivalently H\r{D) = D), where Hr : CA(V0) -+

119

3.8 Nested fractals

CA(Vo) is the renormalization map on CA(Vo) defined by lZr(D) = [i?i]vbIt is easy to show that such a A is uniquely determined, (i.e., if there exist f D, D' e CA(Vo) and A, A' > 0 such that nr(D) = XD and 1Zr(D ) = \'D', then A = A'.) Note that Ur{Cs{Vo)) C C8(V0), which will be shown in Lemma 3.8.22. Using Sabot's arguments in [160], we can also show that the fixed point of 7l\v\cs{v0) 1S unique if (K, 5, {Fi}ies) is an affine nested fractal. See also Metz [129] for another proof of the uniqueness of the fixed point of Uxr\cs(Vo)' Remark. The harmonic structure (D, Ar) obtained in the above theorem is not always regular. For example, in Example 3.2.3, the self-similar structure (K, S, {Fi}ies) is an affine nested fractal and the harmonic structure (D, Ar) is (?5-invariant. In this example, (D, Ar) is not regular if t > y/b. Example 3.8.11 (Pentakun). Let S = {1,2,... ,5}. For k e 5, let pk = e2fc7Tv/=rr/5 e c a n d d e f i n e a contraction Fi : C -^ C by

where we naturally identify C with R 2 . The pentakunf is the self-similar set with respect to {Fi}ies- See Figure 3.2. The self-similar structure that corresponds to the pentakun is post critically finite. In fact

k=1

V — {k}kes and pk = 7r(h) for k e 5, where [i]5 e S is defined by [i]5 = i mod 5. It follows that Vo = {pk}kes is the collection of vertices of a regular pentagon. The pentakun is an affine nested fractal. In fact, Qs coincides with {g e O(2) : g(Vo) = F o }, which is generated by the reflections {gxy : x,y G Vo,x ^ y}. D e CA(Vo) belongs to CS(VO) if and only if

PiPj

Jα

if |i — jl = ± 1

mod 5

~ \b

i f | i - j | = ±2

mod 5,

Therefore,

where a, b > 0 with a + b > 0. In this case, r — (vk)k=i121... ,5 is ^-invariant if and only if n = ••• = r 5 . Let r = ( 1 , . . . ,1). By Theorem 3.8.10, there exist A > 0 and D £ Cs(Vo) such that 1Zr(D) = AD, or equivalently t In the same way, we can also define hexakun, heptakun, octakun and so on. 'kun' is 'Mr' in Japanese.

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Construction of Laplacians on P. C. F. Self-Similar

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Fig. 3.2. Pentakun (D, Ar) is a harmonic structure. In this case, it is also easy to check uniqueness of such A and D directly. (Use a symbolic calculus program, such as Mathematica or Maple V.) Example 3.8.12 (Snowflake). Let pk = e 2fc7r ^ /irT / 6 G C for k = 1 , . . . , 6 and let p7 = 0. We naturally identify C with R 2 . Define F{ : C -> C by Fk(z) = (z — Pk)/3+Pk for k — 1,2,... ,7. The snowflake is the self-similar set with respect to the contractions {Fk}k=i,2,...,7- See Figure 3.3. The self-similar structure associated with the snowflake is post critically finite. In fact, pk = IT(h) for k = 1,2,... ,7,

C=

(J

, k[k + 2] 6 , k[k + 3] 6 , k[k + 4 ] J ,

fc=l,2,...,6

fc=i,2,...,6, where [i]6 G { 1 , 2 , . . . ,6} is V = {fc}fc=i,2 6 and Fo = defined by [Z]Q =i mod 6. The snowflake is an affine nested fractal. In fact, Qs coincides with {g € 0(2) : g(V0) = Vo}, which is generated by the reflections {gxy : x,y G Vo,x ^ y}. In this case, we see that # { | x — y\ : x,y G Vo,x ^ y} = 3 and hence Cs(Vo) is a 3dimensional manifold. Also r = (rk)k=i,2,...j is ^-invariant if and only if n = • •. = r 6 . Let r = ( 1 , . . . , 1, i) for t>0. Then, by Theorem 3.8.10, for any t > 0 , there exist A > 0 and D G CS(VO) such that 7lr(D) = A£>, or equivalently, (D, Ar) is a harmonic structure.

3.8 Nested fractals

121

Fig. 3.3. Snowflake The next example is strongly symmetric but it is not an affine nested fractal. Example 3.8.13. Let (K, 5, {Fi}ies) be a connected p. c. f. self-similar set, where K C R3 = {(x,y,z) : x,y,z G R} and Fi : R3 -• R3. We assume that Vo is the set of vertices of a cube, that is, Vo = {((-1)*, ( - I F , (-1)*) : i,j, k € {0,1}}. This case is not included in the class of affine nested fractals, because 9pq(Vo) ^Voifp = ( - 1 , - 1 , - 1 ) and q = (1,1,1). Let Q = {g e O(3) : (y ) = V }. Then it follows that #(£) = 48. We define three subgroups g 0 o 1 2 ^°,G and Q of Q. First, let Q° be the group generated by the three reflections in the xy, yz and xz-planes. Then #(£°) = 8. Next, let Q 1 be the group generated by the four rotations by 2TT/3 around the lines x = y = z,x 2

= y = —z, x — —y = —z and x = — y = z. T h e n #{G X) 1

2

= 12.

Also let g be the group generated by G° and Q . Then #(£ ) = 24. Note that g2 is a proper subgroup of Q For example, gxy £ Q2 if x = (1,1,1) and y = (1,-1,-1). Also the rotation by TT/2 around the x-axis does not belong to Q2. Now if Qs includes Q2, then (K, 5, {Fi}ies) is strongly symmetric. Let us present a concrete example. Let {pi}i=i,2,... ,8 be the vertices of a cube. Also let PQ = 0. For any i = 1,2,... ,9, define Fi(a) = (a—pi)/3-\-pi for any a G R3. If K is the self-similar set with respect to {Fi} i=i)2,...,9, then (K,S,{Fi}i(zs), where S = {1,2,... ,9}, is a connected p.c.f.self-

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Construction of Laplacians on P. C. F. Self-Similar

Structures

similar structure satisfying Vo = {pi}i=i,2,... ,8- It is easy to see that Qs 2 contains Q . In fact, Qs = Q. Moreover, r = (r^)j=i?2,... ,9 is ^-invariant if and only if r 1 = • • • = rg. In the rest of this section, we will prove Theorem 3.8.10. The idea of the proof is essentially due to Lindstr0m [116]. L e m m a 3.8.14. Let C*(VQ) be the collection of symmetric linear operators D from £(Vo) to itself which satisfy the following three conditions: (VI) DXy >§ ifx^y and ^2qeVo Dpq = 0 for any p G Vo. (V2) If \x-y\ = l0, then Dxy > 0. (V3) // \x - y\ > \x' - y'\ > 0, then Dxy < Dx>y>. Proof. Let D G £*(Vo). Let £D be the quadratic form on £(Vo) defined by £D(U,V) = —(u,Dv). Then obviously £& satisfies (DFl) because £D(U,U) = J2p qeVo Dpq(u(p) ~U(Q))2/^(Recall Definitions 2.1.1 and 2.1.2.) Assume that Du = 0 for u G £(V0). Then u(p) = u(q) if Dpq > 0. Now by (SSI), for any x, y G Vo, there exists a strict 0-walk between x and > 0 for any i = 1,2,... , n - l . 2/, say, (fft)i=i,2,...,n. Then, by (V2), Dx.Xi+1 Therefore u{x) = u(y). This shows (DF2). Obviously D satisfies (L3) and hence £D satisfies (DF3). By Proposition 2.1.3, it follows that D G £A(V0). f Moreover, by (V3), Dxy = Dx>y> if \x-y\ = \x -y'\. Hence D G CS(VO). D For ( ^ Γ ) G CA(VO) X (0,OO)^, we defined H± G CA{Vi) in Definition 3.1.1. Recall that the renormalization operator lZr is defined by nr{D) = [H^. D e f i n i t i o n 3.8.15. For any x,y G V\ a n d n > 0 , define Pn(x,y)

= { x = (xo,xu...

, x n , x n + i ) :x0 = x,xn+i

=y,

x is a 1-walk between x and y and xi G Vi\Vb for any i = 1, 2,... , n.} Moreover, let D G £A(Vo). Pn(x,y) by

Define /i(x) for x = ( x o , x i , . . . ,xn,xn+i)

, / x _ hXQXlhXlX2

•-•

hXri_lXnhXnXri+1

r*'X\X\'*'X2X2 ' ' ' '*'XriXn

where =

Pq

f(i?i)p 9

if p^ q,

l - ( F i ) p p if p = q.

In particular, if n = 0, then Po(x,y) = {(x,y)}

and h((x,y)) = hxy.

G

3.8 Nested fractals

123

The following lemma holds for any connected p. c. f. self-similar structure and any (L>,r) G CA(V0) x (0,00)*. Lemma 3.8.16. For any x 7^ y G Vo, ^—> n>0

^—> x.ePn(x,y)

Proof. We decompose Hi into the following form: Hl

=

(j

x

where T : t(V0) - *(Vb), J : *(Vb) - €(Vi\V 0 ) and X : £(Vi\Vo). Then, by Theorem 2.1.6, it follows that [H1]v0=T-tJX~1J. Note that hpp = —(Hi)pp > 0 and hpp > hpq for any p,q G Vi\Vb. Define A : ^(Vfc) -+ ^(Vb) by A p p - hpp = -Xpp for any p e V^Vo and Apq = 0 if p / g. Also define L : ^V^Vb) -> ^(F^Vb) by L pq = / i p q / / i ^ for any Pi Q € ViVK)- Then X = - 2 ( 7 - L/2)A, where / is the identity map. Therefore, ( - X ) " 1 - A " 1 ^ Lβ)"1^. Recall the proof of Lemma 3.5.1. Let C be the counterpart of the C defined there. Define an equivalence relation ~ on Vi\Vb by p ~ q if and only if there exists x G Un>oPn(x, y) such that ft(x) > 0. Let (Vb\Vi)/~ = {[/i,... , C/m}, where C/"i C Vb\Vi is an equivalence class. Then X and C are decomposed as J*Q : £(Ui) —• ^(C/"i) and C* : ^(f/i) —> £{Ui), respectively, for i = 1, 2 , . . . , m. Then, by the argument in the proof of Lemma 3.5.1, it follows that | | ( C t ) n i | | < 1 for some m G N. This implies that | | C n | | < 1 for some n. Note that | | C | | < 1. Since L = (C + J), we obtain | | ( L / 2 ) n | | < 1. Hence,

(-X)-1 = A-\I - L/2)"V2 = A-1 J2 2-(fc+1)Lfe, where L° = /. Hence, for p, g G Vi\Vb, )pq

This immediately implies the desired expression of TZr(D)xy.

D

Remark. Let (K, S, {Fi}ie$) be strongly symmetric and let D G £*(Vb). Then we can show that X itself satisfies the assumptions of Lemma 3.5.1. (This fact is not used afterwards.)

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Construction of Laplacians on P. C. F. Self-Similar

Structures

L e m m a 3.8.17. Let D G £*(Vo) and let x,y,z G Fs(Vo) for some s G S. If \x -y\ <\x - z\, then hxy > hxz. Proof. First suppose that x — y. Then hxy

— hxx — —\Hi)xx

—

/ \H\)px ^ \Hi)xz pev1\{x}

—

hxz.

Next suppose that x ^ y. Then, by (3.8.2), there exist XQ, yo,zo £ Vo such that Fs(x0) = x,Fs(y0) = y,Fs(z0) = z,hxy = (ra)'1DXoyo and hxz = (7's)~1-Dxo«o" Since DXoyo > DXoZo, it follows that hxy > hxz. D Lemma 3.8.18. Let w G Wm and let x G K\Vm. Then x G ^ \ ^ ( V b ) if and only if J D Vm — ^ ( V o ) , where J is the connected component of K\Vm containing x. Proof. Suppose that x G Kw\Fw(Vo) and let J be the connected component of K\Vm containing x. Then, by Proposition 1.6.9, it follows that J — Kw\Fw(Vo). Hence, 7 n Vm = FW(VO). Conversely, if x £ Kw\Fw(Vo), there exists w' G Wm such that w ^ w' and x G Kw,\Fw,(Vo). Then In Vm = FW>(VO). Hence FW(VO) = FJ{V0). This is impossible because #(Fw(Vo) fl Fwt{Vo)) < 1 by the assumption (3.8.2). • Proposition 3.8.19. Let g be a symmetry of C. If Fwf(V0) = g(Fw(V0)) for w,w' G Wm, then Kw> — g(Kw). Proof. Let x G Kw\Fw(Vo) and let J be the connected component of K\V m containing x. Then g( J) is the connected component of K^m containing g(x). Since g{J) C\ Vm = Fw'(Vo), Lemma 3.8.18 implies that g{x) G Kw>. Hence g(Kw) C Kw>. Note that g~l is also a symmetry and g~1(Fwf (Vo)) = Fw(Vo). Using entirely the same arguments, we see that g(Kw) 3 Kw>. • Originally Fi was a map from K to itself. Prom now on, we think of Fi as a similitude on Rd. Proposition 3.8.20. For any w G W* and any g G Gs, set Ug,w = FW>~1

ogoFw,

where w' G W* is the unique word that satisfies Fw'(Vo) = g(Fw(Vo)). Ug,w egs. Note that Ug$ = g.

Then

3.8 Nested fractals

125

Proof. By Proposition 3.8.19, we see that g(Kw) = Kw>. Therefore, it follows that, for any v € W*, there exists v' G W* such that g(Fwv(Vo)) = Fw'v'(Vo). This implies that U9iW(Fv(Vo)) = Fv'(Vo). Hence U9jW is a symmetry. Note that U9iW is a linear transformation. Also, from the definition, U9iw preserves distance. Hence U9iW G O(d). D Let A(-, •) G VT{ym)' We say that A(-, •) is ^-invariant if and only if A{u ° g,v o g) = A(u, v) for any u , u G i(Vm) and any g € GsCorollary 3.8.21. Let D G C8(V0). Ifr is gs-invariant, then £<m)(-, •) is Qs-invariant, where S^ G VFiVm) is defined in Definition 3.1.1. Proof. We use induction on m. First £(°' = £& is <5S-invariant. Now assume that £( m ) is ^-invariant. Let g G Qs. By (3.1.1), iV

£

(m+1)

(wo^,vo^) = y

—f(m) {uogoFuvogoFi)

(3.8.3)

for ix,i; G £(Vm+i). Now let g(Ki) = Kj. Then r^ = r^ because r is ^-invariant. Also the induction hypothesis along with Proposition 3.8.20 implies

£ ( m ) (uogoFi,vogoFi)=

£{m) (u o Fj o Ug4,v o F, o U9ii)

Again, by (3.1.1), it follows that ^ m + 1 ) ( « o 5 , U o 5 ) = S^+^^u).

•

Lemma 3.8.22. Ifr is Qs-invariant, then fcr(£s(Vo)) C £ e (Vb). Proof. Let D G £ S (F O ). By Corollary 3.8.21, Hi is ^-invariant. Hence (Hi)pq = (Hi)g{p)g(q) for any p,q G Vi and any g e Gs. Now, as we have seen in the proof of Proposition 3.8.9, if x,xf,y,y' G Vo and |x — y\ = |x' — 2/'|, then there exists # G Qs such that ^(x) — x' and ^(y) = y'. Define #(x) = (^(xi))i=o,i,..., n +i for x = (xi)i=o,i,..., n +i € Pn(x,y). Then ^ is a bijective mapping between Pn(x,y) and P n (x / ,y / ). Also it follows that ft(x) = h(g(x)) because i?i is ^-invariant. Hence, by Lemma 3.8.16, y,. D Lemma 3.8.23. Let p^ q G Vo and assume that gpq G Qs. If x,y G i for some w e W*, \x — q\ < \x—p\ and \y — q\ > \y—p\, then gpq(Kw) = Kw. Proof. Let J = K-u;\Ft;(;(Vro). Then, as K\Vo is connected, J is a connected component of K\Vm. If gpq{Kw) = Kw>, then ^ ( J ) = KW\FW>(VQ) and 9pq(J) is also a connected component of K\Vm. By Proposition 1.6.6, there exists a continuous path between x and y contained in J U {x,y}. Since

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Construction of Laplacians on P. C. F. Self-Similar

Structures

x and y belong to different sides of the hyperplane Hpq, this continuous path between x and y intersects Hxy. Hence JnHpq ^ 0. By the fact that gpq(JnHpq) = JnHpq, we see that Jngpq(J) ^ 0. Hence J = gpq(J) and therefore w = w1. • Lemma 3.8.24. Let a{ G R for i = 1, 2,... , n. Then,

J:JC{l,2,...,n}

jeJ

i=1

Lemma 3.8.25. If r is Qs-invariant, then TZr(C*(Vo)) C £*(Vb). Proo/. Let D G £*(Fo)- Since £> G £ e (Vb), Lemma 3.8.22 implies that Tip(D) G £s(Vfc). Therefore we need to show (V3). Now by (SS3), there exist x, y, z G VQ such that \x — y\ — Z&, |a: — J2T| = Z^+i and ^ G ^ s . Let V = {p G Rd : \p - y\ < \p - z\} and define T : Rd -+ Rd by

1

(9yz(p) otherwise. Note that x G V and T(x) = x. Also for x = (#i)i=o,...,n+i €Pn(x,z) U P n (x, z), define T(x) = (T(xi))i=0,... ,n+i- Then, by Lemma 3.8.23, T(x) G Pn(a;,y)Now let x = (^)^=o,... ,n+i ^ ImT and let us consider T~1(x). Define /(x) — {i G {1,... , n + 1} : there exists s € S such that Xi-ugyz(xi)

e ^(Vb) and ^ ^ i7 y2 }.

For J C /(x), we define r(x, J) = (2/i)i=0|... ,n+i by ^ = ( 5 (i) = # { j £ J : j
127

Notes and references

where α ^ x ) = hXj_igyz{xj)/hXj_lXj. Since \xj-! - x5\ < \xj-X - gyz(xj)\, Lemma 3.8.17 implies that c*j(x) < 1. So, using Lemma 3.8.24, we obtain

E

xGlmT

E

J:JC/(x) #(J) is even.

M

xGlmT

J:JC/(x) #(J) is odd.

= E Mx) E (xelmT

J:JC/(x)

We also see that /ix?/ > /i x z . (If / i ^ > 0, then x,z G Fs(Vo) for some 5 G T^). Lemma 3.8.23 implies y G Fs(Vb). By Lemma 3.8.17, we obtain hXy > hxz.) Hence, by Lemma 3.8.16, we verify that 7lr(D)xy > lZr(D)xz. This immediately shows (V3). • Proof of Theorem 3.8.10. Choose x,y G Vo so that \x — y\ — /0-

c*(Vo) = {D G cjy*V

D

*y =}}-

Define

K(Dl=

Let

nr(D)/nr(D)xy.

Then we see that IZ : £*(Vb) —> £*(Vo) and that IZ is continuous. Now C*(Vo) is homeomorphic to a compact ball in E 7 7 1 " 1 , where m = #{\p — q\ : p,q €Vo,p y^ q}. By Brouwer's fixed point theorem, it follows that IZ has a fixed point D G £*(y 0 ). Let A = 1Zr(D)xy. Then 7£r(£>) = A£>. Hence (L), Ar) is a harmonic structure. By Corollary 3.8.21, f ^ ) (= VTiVm) induced by the harmonic structure (Z), AΓ) is C/s-invariant for any k>0. Letting k —• oo, we verify that (f, J7) is ^-invariant. D

Notes and references The use of Dirichlet forms to study analysis on fractals was first developed in Fukushima & Shima [44], where they studied the asymptotic behavior of the eigenvalues of the standard Laplacian on the Sierpinski gasket. Then, in [107] and [83], such an approach using Dirichlet forms was extended to more general classes of (finitely ramified) self-similar sets. See also Fukushima [41], Kusuoka [108], Kumagai [100] and Metz [125, 128]. Although the main source of this chapter is [83], we add many results for the case of non-regular harmonic structures. 3.1 The notion of harmonic structures was introduced in [83]. The counterpart of the renormalization map lZr on £>.F(Vo) had already been intro-

128 Construction of Laplacians on P. C. F. Self-Similar Structures duced in Hattori, Hattori & Watanabe [70], Kusuoka [107] and Lindstr0m [116]. In particular, Kusuoka constructed a self-similar Dirichlet form from a fixed point of such a renormalization map in [107]. Also, Lindstr0m constructed a diffusion process (called Brownian motion) on a nested fractal from decimation invariant random walks in [116]. The fixed point problem of lZr has been studied by many authors, for example, Lindstr0m [116], Metz [126, 127, 130, 129] and Sabot [160]. All of those works focus on (affine) nested fractals. The existence of the fixed point was shown by Lindstr0m in [116]. Moreover, Sabot proved the uniqueness of the fixed point in [160]. See Theorem 3.8.10 and the remark after it. Also, Metz obtained a kind of stability of the fixed point in [129]. 3.2 Theorems 3.2.8, 3.2.11 and 3.2.14 have not been explicitly stated elsewhere before. 3.3 Self-similarity of Dirichlet forms, Proposition 3.3.1, was first observed in Fukushima [41] for nested fractals. Theorem 3.3.4 is an extension of [85, Theorem 3.1]. Theorem 3.3.8 is a new result. 3.4 The key ideas of the proof of Lemmas 3.4.3, 3.4.4 and 3.4.5 are due to Kumagai [100]. Those lemmas play an essential role in proving Theorem 3.4.6 when the harmonic structure is not regular. 3.5 The explicit definition of the Green's function (as we see in Proposition 3.5.5) was given in [83]. Theorem 3.5.7 is a new result. In [94], one can find a figure of the graph of the Green's function for the standard harmonic structure on the Sierpinski gasket. 3.7 The explicit definition of the Laplacian in Definition 3.7.1 was given in [83]. Theorem 3.7.9 has been known to specialists for sometime. In [44], Pukushima & Shima showed this fact for the standard Laplacian on the Sierpinski gasket with Dirichlet boundary condition. Exercises Exercise 3.1. Let C be the harmonic structure associated with the Sierpinski gasket. (See Example 3.1.5.) Set D= and r = (s,st,st), where /i, s and t are positive real numbers. Show that if we fix h > 0, there exist unique s and t such that (D, r) becomes a harmonic structure on the Sierpinski gasket. Also prove that (D, r) is a regular harmonic structure.

Exercises

129

Hint: let R = 1/h. Then calculate the effective resistances for D and Hi by using the A-Y transform (Lemma 2.1.15). Then apply Corollary 2.1.13. You will find that the condition for (D, r) being a harmonic structure is

Exercise 3.2 (modified Sierpinski gasket). Let {pi}i=i,2,3 be the vertices of a regular triangle in the complex plane C. Set p4 = (p2 +ps)/2, Ps — (Ps + Pi)l^ a n d PG = {Pi H-p2)/2. Choose real numbers a and P so that 2a + ft = 1 and a > /? > 0. We define F;(z) = a(z - pi) + Pz for i = 1,2,3 and Fi(z) = β^z—pi) -\-pi for i = 4,5,6. Let X be the self-similar set with respect to {Fi}ies, where S = {1,2,... , 6}. (1) Prove that the self-similar structure C associated with K is post critically finite with Vo = {pi,P2,P3}> (2) Define D € CA{V0) by

Let r = (r, r, r, re, rs, rs) where r, 5 > 0. Show that if we fix s, then there exists an unique r such that (D,r) is a harmonic structure on C Is this harmonic structure regular? Hint: use the A-Y transform and calculate effective resistances as in Exercise 3.1. Exercise 3.3. Calculate A\ for the harmonic structures on Hata's tree-like set (Example 3.1.6) and the modified Sierpinski gasket (Exercise 3.2).

Exercise 3.4 (Neumann problem for Poisson's equation). Prove that, for / G C(K) and rj G ^(Vo), there exists u G Vμ such that

\{du)p

= Tj(p) for all peV^

if and only if JK ^μ = T,peVo rj(p)Exercise 3.5. Let V C Vo and assume that V i=- 0. Define Tγ = {u G T : (1) Prove that (Tγ, £) is a Hilbert space. Also show that (£, JV) is a regular Dirichlet form on L2(K, μ). (2) Show that, for any / G L2(K^), there exists ix G ^V such that £(v,u) = (y^ Dμ for any v G JV- Denote tx by Gy/. Then prove that

130 Construction of Laplacians on P. C. F. Self-Similar Structures 2 1 Gγ is a bounded operator from I? [K^μ) to L (lf, μ) and Gγ = Hγ , where Hγ is the non-negative definite self-adjoint operator associated with the Dirichlet form (£,JV) on L2(K^). (3) Set Vγ = {u e Vμ : u\v = 0, (du)p = 0 for any p e V0\V}. Then prove that Vγ C Dom(ifvr) and Hv\vv = — A M |x> v . Also show that Hγ is the Friedrichs extension of —/S.μ\x>v^

4 Eigenvalues and Eigenfunctions of Laplacians

In this chapter, we will study eigenvalues and eigenfunctions for the Laplacian AM associated with (D, r) and μ. In particular, we will be interested in the asymptotic behavior of the eigenvalue counting function and present a Weyl-type result (Theorem 4.1.5) in 4.1. It turns out that the nature of eigenvalues and eigenfunctions of Aμ is quite different from that of Laplacians on a bounded domain of R n . For example, we will find localized eigenfunctions in certain cases. More precisely, in 4.3, we will define the notion of pre-localized eigenfunctions, which are the eigenfunction of Aμ satisfying both Neumann and Dirichlet boundary conditions. It is known that such an eigenfunction does not exists 1 for the ordinary Laplacian on a bounded domain of W . Proposition 4.3.3 shows that if there exists a pre-localized eigenfunction, then, for any open set O C K, there exists a pre-localized eigenfunction whose support is contained in O. One important consequence of the existence of pre-localized eigenfunctions is the discontinuity of the integrated density of states. See Theorem 4.3.4 and the remark after it. We will give a sufficient condition for the existence of pre-localized eigenfunctions in 4.4. In particular, we will see that there exists a pre-localized eigenfunction for the Laplacian on an affine nested fractal associated with the harmonic structure appearing in Theorem 3.8.10. See Corollary 4.4.11. Throughout this chapter, C — (K, 5, {Fi}i£s) is a connected p. c. f. selfsimilar structure with S = {1, 2,... , N} and (D, r) is a harmonic structure on £, where r = (ri)ies. Also μ e M(K) and s u p ^ G ^ I2m>oR™(vw) < oo. Note that if // is a self-similar measure with weight (/i^)^^, then the 7 above condition on μ is equivalent to μ^i < 1 for any i G S. We use {£, J ) to denote the resistance form associated with (Z), r). Also d is a metric on 131

132

Eigenvalues and Eigenfunctions of Laplacians

K that is compatible with the original topology of K. (Note that (K, d) is compact.)

4.1 Eigenvalues and eigenfunctions In this section, we first define eigenvalues and eigenfunctions of the Laplacians Aμ. Then we will define the eigenvalue counting function and present a theorem which gives the asymptotic behavior of the eigenvalue counting function.

Definition 4.1.1. Let EN{\) = {ipeVN

: A^

= -A
>

1, then A is called a Neumann eigenvalue (N-eigenvalue for short) of Aμ and any non-trivial ip G E^ (A) is called a Neumann eigenfunction (Neigenfunction for short) of Aμ belonging to the Neumann eigenvalue A. Also define ED(X) = {

1, then A is called a Dirichlet eigenvalue (D-eigenvalue for short) of Aμ and any non-trivial (p G Ep (A) is called a Dirichlet eigenfunction (D-eigenfunction for short) of Aμ belonging to the Dirichlet eigenvalue A. Of course, eigenvalues and eigenfunctions of Aμ are expected to coincide with those of the non-negative self-adjoint operator H^ (or Hp, depending on the boundary condition) associated with the local regular Dirichlet form

(£, f) (or (£,f0)) on

L2(K^).

P r o p o s i t i o n 4.1.2. Suppose that X^m>o-^m(/i) < ° ° f Then the following three conditions are equivalent. (Nl) if G Dom(iifjv) o/ad Hjycp = \tp, (N2) (f G T and S(cp, u) = X((p, ^) M for any u G T, (N3) ipeEN(\). Also, the following three conditions are equivalent. (Dl) (f G Dom(#£)) and HDV = X(p, (D2) (f G TQ and £(, u) M for any u G T§, (D3)
o r s o m e

t > 0 .

r Remark. Let /z be a self-similar measure. Then 5 ] m > 0 ^ M < °° f° some t > 0 if and only if r ^ < 1 for all i G S. See the remark after Corollary 3.6.6.

Proof. By Corollary 3.6.6, # ( n ) is continuous on K x K for n > (1 -

Recall that G^c^d) (GDnu)(x)

l/t)~l.

= GD\c(K,d)- Hence, = (G^u)(x) = f gW(x,y)u(yMdy) JK

(4.1.1)

4-1 Eigenvalues and eigenfunctions

133

U 2 for u G C{K,d). Since GD is a bounded operator from L (K, μ) to itself and g^ is continuous, it follows that (4.1.1) is true for all u G L2(K, μ). This implies that GDnu G C(K, d) for any u G L2(K, μ). Now the equivalence between (Dl) and (D2) is obvious from the characterization of a self-adjoint operator associated with a closed form in Lemma B.1.4. Also, by Theorem 3.7.9, (D3) implies (Dl). Now suppose (f G Dom(Hz)) and Hrup = X(p. Then, as XnGDny> = <£> we see that if G C(K,d). Hence XGDV = XGμ(p = (p. So applying Lemma 3.7.10, we see that (p G Vp. Therefore (p G ED(X). Next we consider the Neumann case. It follows immediately that (Nl) and (N2) are equivalent by Lemma B.1.4. Assume that ip G Dom(if/v) and H]y(p = $f. Let (po = Y^pev0 ^p(p)^p' Then we see that (P—
By Theorem 3.4.6 and Corollary 3.4.7, the self-adjoint operators and HD have compact resolvents. Therefore, N-eigenvalues (and also Deigenvalues) are non-negative, of finite multiplicity and the only accumulation point is oo. In other words, for b = D and N, there exist {A^}^>i and ip\ G Eh(\$ such that

2

and {<£i}i>i is a complete orthonormal system for L (K, μ). So we can define eigenvalue counting functions as follows. Notation. Hereafter, the symbol b always represents a boundary condition: b is either N or D, where N represents the Neumann boundary condition and D represents the Dirichlet boundary condition. Definition 4.1.3 (Eigenvalue counting function). Define p&(x,/x) for b = D and N by pb(x^) = ^ A < x ^ i m ^ ( ^ ) * Pb{x,v) is called the beigenvalue counting function. Obviously, p\,(x^) = max{A: : Xbk < x} and pb(x^) —> oo as x —» oo. For ordinary Laplacians on bounded domains in R n , we have Weyl's famous theorem on the asymptotic behavior of eigenvalue counting functions. Theorem 4.1.4 (WeyPs theorem). Let Q, be a bounded domain in Rn. Let Xi be the i-th eigenvalue of the Dirichlet eigenvalue problem of —A on

134

Eigenvalues and Eigenfunctions

of Laplacians

Q, that is, ( Au = -Xu where A = Y^=1 d2 /dx2.

Set p(x) — # { i : A^ < x). Then as x —> oo,

n — (97r\~ lOl Jb srn/2 _|i_ OyX n(^rn/^\J , n(rr\J — P\Jb \^" J R&n\^L\ n

w/iere | • \n is n-dimensional

Lebesgue measure

and Bn — \{x : \x\ < l } | n .

Remark. Weyl [182, 183] proved the above result under some extra conditions on the domain D. It is now known that the above result is true for any bounded domain. See Lapidus [111]. We are interested in an analogue of Weyl's theorem for our Laplacian Aμ, or more specifically, the asymptotic behavior of the eigenvalue counting functions p&(x,/i) as x —> oo. If /i is a self-similar measure, we have the following result. Theorem 4.1.5. Let μ be a self-similar measure with weight (^i=1,2,... ,NAssume that μ^i < 1 for all i G {1,2,... , iV}. Let ds be the unique real number d that satisfies N

2=1 /

where 7^ = y r^7- Then 0 < liminf ph(x^)/xds/2

< limsuppb(x^)/xds/2

< 00

for b = D and N. ds is called the spectral exponent of (£,T^μ). Moreover (1) Non-lattice case: If Y^i=1 Zlog7i is a dense subgroup ofR, then the limit limx_^oo pb(x^)/xds/2 exists and is independent of the boundary conditions. (2) Lattice case: IfY^=1 Zlog7; is a discrete subgroup ofR, let T > 0 be its generator. Then, as x —> 00,

where G is a right-continuous, T-periodic function satisfying 0 < inf G(x) < supG(x) < 00 and o(l) is a term which vanishes as x —> 00. Moreover, the periodic function G is independent of the boundary conditions. Remark. In the proof, we will get more concrete expressions for the limit for the lattice case and the periodic function G for the non-lattice case. See (4.1.4) and (4.1.5).

135

4-1 Eigenvalues and eigenfunctions

Remark. If we use the full version of the renewal theorem (Theorem B.4.3), ds 2 in the we get more detailed information about the error term o(l)x ^ lattice case. For example, if the 7^ are all equal, then pb(x^)

= G(logx/2)xds^2 + O(1)

(4.1.3)

a s x - ^ o o , where O(l) is a term which is bounded as x —» 00. Remark. According to [41], G((logx)/2)xds^2 is the integrated density of states of K associated with the Laplacian A^. Example 4.1.6 (Sierpinski gasket). For the standard Laplacian on the Sierpinski gasket in Example 3.7.3, r* = 3/5 and μ^ = 1/3 for all i = 1,2,3. Let ds be the spectral exponent. Then, by (4.1.2), we have ds = log 9/log 5. In this case, ds is also called the spectral dimension of the Sierpinski gasket. Note that ds differs from the value of the Hausdorff dimension of the Sierpinski gasket, which is log 3/log 2. Obviously we are in the lattice case. Taking the above remark into account, we have p^(x,^) = G(logx/2)x~ds^2 + O(l) as x —> 00, where G is a log 5/2-periodic positive function. Fukushima and Shima [44] showed that 0 < liminf pb(x,/x)/xds//2 < limswppb(x, μ)/xds/2

< 00

by using the eigenvalue decimation method. See Notes and References of this chapter about the eigenvalue decimation method. Hence we see that G is a non-trivial (i.e., non-constant) function. The rest of this section is devoted to proving Theorem 4.1.5. First we give a comparison theorem for eigenvalues. Theorem 4.1.7. Let H be a separable Hilbert space. Let Si be a closed form on H for i = 1,2. Assume that E\ is an extension of 82- Set T{ = Dom(£i) for i = 1,2. Also let Hi be the non-negative self-adjoint operator associated with Si for i = 1,2. Assume that Hi has compact resolvent for i = 1,2. Let {A^}n>i be the set of eigenvalues of Hi appearing in Theorem B.I. 13. Then A* < A^ for all n > 1 . Moreover, if dim^1/^*2 < oo, then A^ < A ^ + M for all n> 1, where M = d i m ^ i / ^ Proof. For any finite dimensional subspace L of Ti, set

where || • || is the norm of H. Since Si is an extension of £2, we see that

136 X

Eigenvalues and Eigenfunctions of Laplacians 2

A (L) = \ {L) if L C F2> So the variational formula (Theorem B.I. 14) implies A^ < A^ for all n > 1. Next, assume that d i m ^ i / ^ < oo. Then, for any subspace L C T\, there is a natural injective map from L/(L n^2) —• ^l/C^i n^2) = ^l/^bThis shows dimL fl J ^ dimL — dim ^1/J^- Hence, if £fc,z = {L : L is a subspace of ^ 1 , dimL = A;, dimL D ^ 2 = 0? then {L : L is a subspace of T\, dimL = n + M} = u££0 For L G £ n+ M,n+i, the variational formula shows that

Again, by the variational formula, we have A^ + M > A^.

•

Immediately, from the above theorem, we have the following comparison of eigenvalue counting functions. Corollary 4.1.8. Under the same assumptions as in Theorem 4-1-7, der fine pi(x) = #{n : A^ < x}. If dim J 1/F2 = M < 00, then p2(x) < Pi{x) < P2(x) + M for any x > 0 . For ease of notation, we write pb{x) instead of pb(x^) Lemma 4.1.9. Set M± = #(^1^0) and Mo = #(V0). xeR,

hereafter. Then, for any

N

pD(x) - Mi < ^pD^ix)

< pD(x) < pN(x) < pD{x) 4- Af0.

i=1 w e

Proof. First applying Corollary 4.1.8 to (£,J~) and (£, ^0)5 immediately obtain pr> (x) < PN{%) PD(X) + M). Next, define T\ = {u : u G T,u\Vl = 0}. Then it follows that ( f , ^ i ) 2 is a closed form on L (iiT,/i). Note that dim^o/^i = #(Vb\Vi) = M1. Hence, if p\ is the eigenvalue counting function associated with (f,J*i), then Theorem 4.1.7 implies p£>(x) — M\ < pi(x) < PD{^)- Let Hi be the non-negative self-adjoint operator associated with (5,^1). Then H\u = Xu for u G Dom(iJi) if and only if u G T\ and £{v, u) = A(v, w)M for all ?; G T\. By the self-similarity of both £ and /i, the last equation is equivalent to EiLi rcx£(v oFi,uo Fi) = A£)ili ^ ( v o i^, w o F»)M. Hence, it follows that H\u = Xu if and only if HD(U O Fi) = μiriX(u o F^) for any i Thus Pi( x ) — 2z=i PD^iTix). This completes the proof. •

137

4-2 Relation between dimensions Proof of Theorem 4-1-5. Let R(x) = PD(%) — S i = i PD^ifix).

Then

N

PD(X) = 2_j PD(Hiri%) + R(x). i=1 dst

2t

Defining f(t) = e- pD(e ),u(t) equation becomes

= e-dstR(e2t)

and p{ = ^ d s , the above

•u(t). i=1

Note that XliliPi — 1 by (4.1.2). This is the renewal equation (B.4.1). Recalling Proposition 3.4.8, we see that the first eigenvalue Ai of Hp is positive. Since PD(X) — 0 for x < Ai, it follows that there exists t* £ R such that f(t) = R(t) = 0 for all t < t*. Also, Lemma 4.1.9 implies that \u(t)\ < M\e~dst for any t. Hence all the assumptions of the renewal theorem (Theorem B.4.2) are satisfied. So, for the non-lattice case, we have AT i

/

_

(4.1.4) as t —» oo. For the lattice case, we have /(t) — G(t) —> 0 as t —• oo, where T is the positive generator of the discrete additive group X^=i

i=1

This implies the theorem for the Dirichlet case. The Neumann case immediately follows as well by Lemma 4.1.9. •

4.2 Relation between dimensions In Example 4.1.6, it was observed that the spectral dimension of the Sierpinski gasket is different from the Hausdorff dimension with respect to the restriction of the Euclidean metric. (Note that so far the terminology "spectral dimension" is merely a synonym of spectral exponent. Recalling Theorem 4.1.5, we find that the spectral exponent ds depends on (D,r) and μ. We will define the spectral dimension of the resistance metric (K, R) later.) In any case, the spectral dimension (exponent) is defined through the eigenvalue distribution of the Laplacian. Hence we may think of the spectral dimension as a dimension from the analytical point of view. On

138

Eigenvalues and Eigenfunctions of Laplacians

the other hand, the Hausdorff dimension is a dimension from the geometrical point of view. In this section, we study the relation between those two dimensions. Theorem 4.2.1. Assume that (D,r) is regular. Let R be the effective resistance metric on K associated with (D,r). Let Hd be the d-dimensional Hausdorff measure of (K, R). If du is the unique positive number that satisfies

i=1 dH

then 0 < H (K) < oc. In particular, dimn(K,R) = d # . Moreover, let v dH dH be the normalized djj-dimensional Hausdorff measure: v = 7i /Ti (K) dH Then there and let μ* be the self-similar measure with weight ((ri) )ies> exist positive numbers c\ and C2 such that ci^,(A) < u(A) < c2v*(A)

(4.2.2)

for any A e B(K). Also, if

then there exist positive numbers c 3 and c 4 such that c3xds/2

< PD(X,V) <

c4xds/2

for sufficiently large x. It is natural to think of v as the intrinsic probability measure on (X, R). If we define the spectral dimension of (K, R), ds {K, R), by

then ds(K,R) = ds and (4.2.3) gives the relation between the Hausdorff dimension and the spectral dimension of (if, i^). In [93], the quantity du given by (4.2.1) is called the similarity dimension of the harmonic structure (D, r). Also, if we denote the solution of (4.1.2) by G^μ), then ds(K,R)

= ds^*)

= max{ds(//) : ^ is a self-similar measure}.

(4.2.4)

On the contrary, if (D, r) is not regular, then the supremum of the spectral exponent ds(y) is infinite. In the rest of this section, we will prove Theorem 4.2.1.

4-2 Relation between dimensions

139

Definition 4.2.2. Let A be a partition. / G T is called a A-harmonic function if and only if / o Fw is harmonic for any w G A. For p G ijjp is the A-harmonic function with ^ I V ^ A ) — XP • Recall Lemma 1.3.10 for the definition of V(A) = V{A, C). L e m m a 4.2.3. Let A be a partition. Set Aw = {v G A : Kw D Kv ^ 0} /or any w G A. T/ien

/or an?/ W G A . Proof. It is enough to show that #( / 7T~ 1 (p)) < #(Cc) for any p £ K because #(AW) < Y,peFw(Vo) #(7r~1(p)). For ease of notation, we use C instead of Cc hereafter. Let k = #(C) and assume that 7r~1(p) 2 {a; 1 ,... ,ujk+1}. Let m = min{n : uoln ^ cj^for some i , j G { l , . . . ,fc + 1}}. If t> = uj\ .. . ^ - i and q = F ^ " 1 ^ ) , then q £ Cc and 7r~1(g) C C. Hence < fc. Since TT"1^) D l ^ - 1 ^ 1 ) , . . . ^ ^ ^ o ; ^ 1 ) } , it follows that a/ = a;-7 for some i < j . Hence # ( T T ~ 1 ( P ) ) < k. D Let A(a) = A(r, a), where A(r, a) is defined in Definition 1.5.6. Note that A(a) is a partition for small a > 0 . L e m m a 4.2A. There exist positive constants c and M such that, for sufficiently small a > 0 and for any x G K, #{w : w G A(a), d(x, Kw) < ca} < M, where d(x,Kw)

== min 2/G x w

(4.2.5)

R(x,y).

Proof. For tt; G A(a), define w = ^2pejrw(v0) ^P • Suppose that v G A(a) and Kw n Kv = 0. Then u\Kw = 1 and u\Kv = 0. Hence Theorem 2.3.4 implies that R(x,y) > £'(ix,'a)~1 if x G Kw and i/ G X v . Recalling (3.3.1) we have heA(a)

Tfl

£ ( F F )

£ { F F )

^ 2 heAw(a)

^ (4.2.6)

where Aw(a) = {h G A(a) : i ^ n Kh ^ 0}. Set r = min{ri : i = 1,2,... , AT}. Then it follows that r^ > ar_ for any h G A(a). Also define c\ — m a x { £ ( ^ ^ p , ^

^ p ) : V is a non-empty subset of Vb.}

140

Eigenvalues and Eigenf unctions of Laplacians

As u o Fh = ^2pey i/jp for some V C Vo, we obtain £(u o F^, u o F^) < c\. Thus, applying those estimates along with Lemma 4.2.3, it follows from (4.2.6) that

Using the above estimate, if w, v G A(a), KwnKv then we have l

R(XJV) > £{u,u)~

= 0, x G i ^ and ?/ € ifv,

> c2a,

1

where c<2, = (#(C)#(Vb)ci)~ r. Thus we see that for any x E if, if d(x,Kv) < c2a/2 for v G A(a), then there exists u> G A(a) such that v e Aw(a) and x G i ^ . Hence, if c = c 2 /2, Lemma 4.2.3 implies

#{v G A(a) : d(x,Kv) < ca} < #{w G A(a) : x G KW}#(C)#(VO). As is shown in the proof of Lemma 4.2.3, # ( 7 T ~ 1 ( X ) ) < #(C) for any x G K. Hence #{u? G A(a) : x G X ^ } < #(C). This completes the proof of this lemma with M - #(C) 2 #(Vb). • of Theorem 4-2.1. We will apply Theorem 1.5.7 for the first part of the theorem on the Hausdorff measure. By Lemma 3.3.5, there exists C > 0 such that dia,m(Kw) < Crw for any w G W*. This is the condition (1.5.2). Also, by Lemma 4.2.4, we can verify the other condition (1.5.3). Hence Theorem 1.5.7 implies the first part of Theorem 4.2.1. Next we show the second part of the theorem on the asymptotic behavior of PD(X, ^). Let Ai(/i*) and \i(v) be the i-th eigenvalue of — A ^ and —A^ with Dirichlet boundary conditions, respectively. By (4.2.2), ci||^||Mj(<52 < IM|i/,2 C211^1 Iμ*,2' Therefore, the variational formula, Theorem B.I. 14, implies that (ci)2Ai(i/) < A,(/i*) < (c2)2A,(i/) for i > 1. Hence we obtain PD((CI)2X,

μ^t) < pD(x,

v) < PD{{C2)2X, /i*)

(4.2.7)

for any x > 0 . By Theorem 4.1.5, it follows that 0 < liminf pD(x^^)/xds/2 x->oo

< limsupPD(X,μ*)/xds/2

< oo,

a;-»-oo

where ds is the unique positive number that satisfies (4.1.2). By (4.2.7), the

141

4-3 Localized eigenfunctions above inequality holds if we replace p£>(x, μ*) by PD{X,V). dH (ri) , we see that ds = 2dH/(dH + 1).

Since {μ^ji = •

4.3 Localized eigenfunctions In [44], it was observed that there exists an eigenfunction of the standard Laplacian on the Sierpinski gasket whose support is confined in a small subdomain of the Sierpinski gasket. Such an eigenfunction is called a localized eigenfunction, which never appears in the case of the ordinary Laplacian on a connected domain of R n . So the existence of a localized eigenfunction is one of the unique features of the Laplacian on fractals. In this and the following sections, we study localized eigenfunctions of the Laplacian Aμ. Hereafter we assume that /i is a self-similar measure on K with weight First we give a definition of localized eigenfunctions. It is, however, difficult to say that the support of an eigenfunction is "small". So we define the notion of a pre-localized eigenfunction instead. We will see that a pre-localized eigenfunction produces genuinely "localized" eigenfunctions in Proposition 4.3.3 below. Definition 4.3.1. u G T is called a pre-localized eigenfunction belonging to the eigenvalue A if u G ED{\) Pi EN (A) and u^0.

of Aμ

In other words, a pre-localized eigenfunction is both a Neumann and a Dirichlet eigenfunction of Aμ: Aμ^ = —\u, u\v0 = 0 and (du)p = 0 for all p G VQ. Proposition 4.3.2. Define Pw : T^ -• T by

(T> $ \ \<^-\x)) (Pwu)(x) = < ^0

ifxeKw, m

otherwise.

Let u be a pre-localized eigenfunction belonging to the eigenvalue A. Set uw = Pw(u). Then uw is a pre-localized eigenfunction belonging to the eigenvalue Proof. Let w G Wm. The self-similarity of (£,.F) (Proposition 3.3.1) along with Proposition 4.1.2 implies

£(uw,v)=

]P

—£(uwoFT,voFr)

= —£(u,voFw ) = —(u,vo

142

Eigenvalues and Eigenfunctions of Laplacians

for any v G T. On the other hand, VriUw O FT,VW O F^μ

=

Therefore S(uw,v) — X(^wrw)~1(uw^v) for any v G T. Hence, by Proposition 4.1.2, uw G Ew(^wrw)~1$. Also, obviously uw\y0 = 0. This completes the proof. • By the above proposition, once we find one pre-localized eigenfunction, then there exists an infinite sequence of pre-localized eigenfunctions, which are actually "localized" eigenfunctions. Proposition 4.3.3. There exists a pre-localized eigenfunction of —Aμ if and only if for any non-empty open subset O C K, there exists a prelocalized eigenfunction u such that supp(w) C O. Proof. For any non-empty open subset O C K, there exists w G W* such that Kw GO. If there exists a pre-localized eigenfunction tt, then uw is a pre-localized eigenfunction with supp(ii^) C Kw C O. The converse direction is obvious. • A pre-localized eigenfunction causes several notable phenomena. First we consider the lattice case in Theorem 4.1.5. Theorem 4.3.4. For the lattice case, the following four conditions are equivalent: (1) There exists a pre-localized eigenfunction of —Aμ. (2) G is discontinuous. (3) For any M G N , there exists a (Neumann or Dirichlet) eigenvalue of —Aμ whose multiplicity is greater than M. (4) There exists a (Neumann or Dirichlet) eigenvalue of —Aμ whose multiplicity is greater than #(Vb). ds

2

is called the integrated density of Remark. Recall that G((logx)/2)x ^ states. (See the third remark after Theorem 4.1.5.) So if there exists a pre-localized eigenfunction, then the integrated density of states is discontinuous. By the above theorem, the T-periodic function G in Theorem 4.1.5 turns out to be non-constant if there exists a pre-localized eigenfunction. So, we have the following corollary. Corollary 4.3.5. For the lattice case, if there does exist a pre-localized eigenfunction, then pb(x^)/xds/2 does not converge as x —> oo for b =

143

4-3 Localized eigenfunctions

So far, this corollary is the only known result which can be used to show that the T-periodic function G is not constant. We will give a proof of Theorem 4.3.4 after stating another consequence of the existence of a pre-localized eigenfunction. The second theorem concerns approximation by polynomials. Notation. Define spaces of multi-harmonic functions n m for m > 1 inductively by fill

= {ueVIA:AIJlu

= 0}

\nm+i

={ueVfi:

G nm}

A^

for m > 1 .

Also set IIoo = U m >iII m . Obviously, III is the collection of harmonic functions. In the case of the ordinary Laplacian on [0,1], n ^ corresponds to the collection of all polynomials. So we think of u G n ^ as a polymonial on K with respect to the Laplacian Aμ. By Weierstrass's theorem (see Yosida [186, §0.2] for instance), the collection of all polynomials on [0,1] is known to be dense in L2([0,1]). On the contrary, the following theorem tells us that a pre-localized eigenfunction cannot be approximated by polynomials in L2(lf, μ). In other words, n ^ is not dense in L2(K^) if there exists a pre-localized eigenfunction. Theorem 4.3.6. Let Eloc be the L2-closure of the linear space spanned by all pre-localized eigenf unctions. Also let n ^ be the L2-closure of U^. Then Eloc is the orthogonal complement oflloo in L2(if, μ), so that L2(K^) = OO VJ7

JOJ

The rest of this section is devoted to proving these theorems. First we prove Theorem 4.3.4. Recall the notation for the lattice case in 4.1. T is the positive generator of Z ^ i ^ l ° S 7 ^ where 7^ = y/r~jJTi. Let mi = — T " 1 Iog7^ for i G S. Then the m^ are positive integers whose largest common divisor is 1. If we define M(n) = #{w = wxw2 ...WitW+i

—— r

11

1

wH'w

=

ke2nT}

for n G N, then /

M(n) = #{w — W1W2 .. .wi

EW*

:^ m

W i

— n}.

i=1

We also define M(0) = 1 and M(n) = 0 for any negative integer n.

144

Eigenvalues and Eigenfunctions of Laplacians

Lemma 4.3.7. If p = eTds, Note that p~ m * = e-miTds

then l i m ^ ^ M(n)/pn

= ( X ^ m^r™*)" 1 .

= 7 ^ s . Hence Y^=1p~mi

Proof Let pi = p ~ m i . Set M(rc) = M(n)/pn lemma follows immediately by Lemma B.4.5.

= 1.

for any n G Z. Then this •

Proof of Theorem 4-3-4- We write p(x) instead of p& (#,//). (1) => (2) : Let u be a pre-localized eigenfunction belonging to the eigenvalue k. Then, by Proposition 4.3.2, kn = ke2nT is a (Dirichlet and Neumann) eigenvalue with multiplicity at least M(n). Hence p(kn)-p(kn-)>M(n),

(4.3.1)

where p(k-) = limxVip(x). As lim^oo \p(x)/xd^2 and log kn = log k + 2nT, it follows that lim (p(kn) - p{kn-))kn-ds'2

n-^oo

- G((logx)/2)| = 0

< G(a) - limG(t),

(4.3.2)

t\a

where a = ^logk. Combining (4.3.1) and (4.3.2), Lemma 4.3.7 implies that G(a) — lim^a G(t) > 0. Hence G is discontinuous. (2) => (3) : Let Gn(t) = p(e2(t+nT)ye(t+nT)da for 0 < t < T. Also let e n = s u p 0 < t < 1 \Gn(i) — G(t)\. Then, by Theorem 4.1.5, e n —• 0 as n —• oo. Now, if G is discontinuous at a, then we can choose am > 0 and 6 m > 0 so that limm_^oo a m = lim m ^oo 6 m = 0 and L = liminfm-^oo \G(a -f a m ) — G(a - bm)\ > 0. It follows that liminf \Gn(a + a m ) - G n ( a - 6 m )| > L - 2e n . m-^oo

This implies that e 2 ^ a + n T ^ is an eigenvalue whose multiplicity is no less than 2e(a+nT>>d8/L for sufficiently large n. (3) => (4) : This is obvious. (4) => (1) : Assume that dimi£;v(&) > #(Vo). Define a linear map r : EN(k) -+ £(Yo) by T(U)(X) = u(x) for x G Vb- As dimEN(A:) > dim^(F 0 ) = #(Vb), the kernel of r is not trivial. Hence there exists a non-trivial u G Ex(k) that satisfies u\y0 — 0, and so it is a pre-localized eigenfunction. A similar argument works for the case of D-eigenvalues. • Remark. Note that the implication (4) => (1) is true even in the non-lattice case. Next we prove Theorem 4.3.6. Lemma 4.3.8. u is a pre-localized eigenfunction if and only if u G (Hi)- 1 and u is a D-eigenfunction.

145

4'S Localized eigenfunctions

Proof. Let u G ED(X) fl E^(A) for some A > 0. Also, let v b e a harmonic function. Then, by the Gauss-Green's formula (Theorem 3.7.8), E(u, v) = 2_] u(p)(dv)p — / uAμvdμ = y ^ v(p)(du)p - / VA^μ JK pevo

=0 = X(u,v)fjL.

Hence (w, v)M = 0. This implies that u G (111)-1. Conversely, assume that u G ED(X) and (u,v) = 0 for any harmonic function u. Then using the Gauss-Green's formula in a similar way to the above, we see that (du)p = 0 for any p G % . Hence it is a pre-localized eigenfunction. • Proof of Theorem 4.3.6. Step 1: GoiUoo) C n ^ , where GD = Ho'1 is the Green's operator defined in Proposition 3.4.8. Proof of Step 1: Let u G n ^ . Then there exists {un}n>i C n ^ such that un —• u as n —> oo in L2(K^). By Lemma 3.7.10, it follows that GD^TI £ IIoo. Since G^ is a bounded operator from L2(K^) to itself, GD^TI —• G D ^ as n —> oo. Hence G^u G n ^ .

Step 2:

ii

Proof of Step 2: Let u G n ^ . Note that GΒ is symmetric. By Step 1, for any v G n ^ , (GDU,v)μ = ( w , G ^ ) M = 0. Hence G#w G n ^ . Now, by Step 1 and Step 2, it follows that ^ ( n ^ n Bom(HD))

= U^

and iJ£>(noonDom(.ff£>)) = n ^ . Since if^ is a self-adjoint operator having compact resolvent, H& \^± nDom(H

\ is a self-adjoint operator from n ^ to

itself having compact resolvent as well. Hence there exists a complete orthonormal system {/ n } n >i of n ^ consisting of eigenfunctions of HDBy Proposition 4.1.2, fn is a D-eigenfunction.

As fn

G n ^ C (Hi)- 1 ,

Lemma 4.3.8 implies that fn is a pre-localized eigenfunction. Hence n ^ C loc

E . Next, let / be a pre-localized eigenfunction belonging to an eigenvalue A > 0 . Then, by the Gauss-Green's formula, A m ( . / » M = - A — \f, ^v)»

= ••• = ( - i n / , (A M ) m «) / i = 0 loc

for any v G n m . Hence (/, v)yi = 0 for any v G n ^ . This implies E

C

146

Eigenvalues and Eigenfunctions of Laplacians 4.4 Existence of localized eigenfunctions

In this section, we give some sufficient conditions, in terms of the geometry of K, for the existence of localized eigenfunctions. Those conditions will require a certain kind of strong symmetry for the self-similar set, the harmonic structure and the measure. This is rather restrictive and seems a little far from a necessary condition. By using those conditions, however, we can show the existence of a pre-localized eigenfunction for the symmetric harmonic structure on (affine) nested fractals obtained in Theorem 3.8.10. See Corollary 4.4.11 for details. Notation. If g : K -> K is a bijection, and / : K -> R, define Tgf : K -> Definition 4.4.1. Define Q by Q — {g : g is (!) (2) (3)

a bijective homeomorphism from K to K that satisfies g:Vo-+ Vo, μof1=μ, Tg(f) C T and £(Tgu, Tgv) = £{u, v) for any u, v G J7}.

Obviously, Q is a group. Q is called the symmetry group of (£, (D, r), /x). Remark. Note that g G Q is not necessarily a symmetry of C as defined in 3.8. Let {(fn}n>i t>e the complete orthonormal system for L2(K^) consisting of the Dirichlet eigenfunctions of Aμ. Also, let us assume that (Pn £ Ef)(Xn). (In 4.1, we used (p^ and A^ in place of (pn and An, respectively.) Lemma 4.4.2. Let R = Y^=1 otiTgi, where a^ G R and gi G Q for % = 2 1,2,... , n. If Ruj^O for some u G L (K, μ) and R*v G T^ for all v G T, a e n where 1C — ^ZlLi ^ o ~ ^ ^ there exists a pre-localized eigenfunction. 2

Proof. If R(pn — 0 for all n > 1, then Ru = 0 for any u G L (K, μ). Hence there exists n > 1 such that i?
Again, by Proposition 4.1.2, we see u G EN^U)that wis a pre-localized eigenfunction.

Since w G ^b, it follows •

For g G
4-4 Existence of localized eigenfunctions

147

Let i be the identity map of K. Proposition 4.4.3. (1) // there exists h G G\{t>} such that Vo C S(h), then there exist pre-localized eigenf unctions. (2) If #(G) is infinite then there exist pre-localized eigenf unctions. Proof. (1) Let R = I — Th- As h ^ L, there exists x G K such that h(x) T^ x. Since h is continuous, there exists a neighborhood A of x such that h(A) n A = 0. Set it = XA, where XA is the characteristic function of the set A. Then we have Ru ^ 0. As y0 C S(h), /i" 1 ^) = x f° r a ^ x € Vb- Hence v - Th-iv = R*v e T§ for all v G T. Now the proposition follows from Lemma 4.4.2. (2) If Q is infinite, then, since Vo is finite, a counting argument shows that there exist distinct elements #i, #2 of G with the same action on Vo. Hence Vo C S(g^1g2), and the result is immediate from (1). • There do exist p. c. f. self-similar sets for which G is infinite (the Vicsek set is one), but this is a little exceptional. We now turn to the more complicated situation when G is finite. Theorem 4.4.4. Suppose G is a finite subgroup of G which is vertex transitive on Vo, and that there exists h G G, h £ G, such that

SG(h) = (J Sih-'g) ? K.

(4.4.1)

geG

Then there exist pre-localized eigenf unctions. Proof. Set RG = E9eGT9 = E9eGTg-^ a n d R = Ro(Th-i - / ) . Let x G K\Sc(h). Then \g(x) : g G G} is finite and does not contain h(x). Hence there exists a neighborhood A of x such that h(A) C\g(A) — 0 for all geG. Set u = XH{A)- If 2/ G A, then u(g(y)) = 0 for g G G, and so

(Ru)(y) = Y^ u(h(g(y))) - £ u(g(y)) > u(h(y)) = 1, geG

proving that Ru ^ 0. Let v G T. As G is vertex transitive, if y G Vo then

y) = Tl v(d(y)) = which is independent of y. If x G Vo then h x{x) G Vb, and therefore i r v = (T,, - I)RGv G JT0. Now, using Lemma 4.4.2, we can complete the proof. •

148

Eigenvalues and Eigenfunctions of Laplacians

As the condition (4.4.1) is a little troublesome to verify, one might hope that this symmetry argument would work under the weaker condition that G contains a vertex transitive subgroup G and there exists h G G\G. (4.4.2) However, it is easy to see that (4.4.1) is equivalent to the statement that RG(Th-i - I) ^ 0. See Example 4.4.9. On the other hand, note that if (4.4.1) fails, then S ^ 1 g) has an interior point for some g G G. Hence (4.4.2) along with S(h~1g) has no interior point for any g G G

(4.4.3)

implies (4.4.1). In fact we will see below (Lemma 4.4.5-(3)) that for nested fractals (4.4.3) always holds, so that (4.4.2) is all that needs to be verified. Next, we will discuss the case where K is a subset of Rd for some d G N and the Fi are restrictions of similitudes of Rd. In such a case, we can assume, without loss of generality, that M

5>i = 0,

(4.4.4)

i=1

where Vo = {pi,P2j • • ->PM} • with M = #(Vb), and that {x - y : x, y G K}

span

Rd.

(4.4.5)

Under these assumptions, Lemma 4.4.5. (1) / / / is an affine map from Rd to itself with /(Vo) = Vo, then /(O) = 0. (2) LetfiRd -> Rd be linear fori = 1,2. If h\K = HK then fx(x) - f2(x) for allx eRd. (3) / / / : Rd —> Rd is a linear map with f(K) = K, then f is linearly conjugate to an orthogonal transformation ofRd. Moreover, if f is not the identity map, then S(f) = {x G K : f(x) = x} contains no interior point in the intrinsic topology K. Proof. (1) Let f(x) = Ax + b, where A is a d x d-matrix and b G Rd. As Ef=i f(Pi) - Ef=i Pi = 0, we have A ( £ £ i Pi) + Mb = 0. Hence b - 0. (2) This is immediate from (4.4.5). (3) As K C Im / and (4.4.5) holds, it follows that / is invertible. Note that K is bounded. We can easily see that fn{x) and f~n(x) remain bounded as n —• oo for any x G Rd. Hence, if we extend / to a map from Cd to itself, / is semisimple and the absolute values of its eigenvalues are all equal to 1. Therefore / is linearly conjugate to an orthogonal transformation.

4-4 Existence of localized eigenfunctions

149

Next suppose there exists a non-empty open subset O of K such that f(x) — x for any x G O. By (4.4.5), there exist Xi,yi G K for i = 1,2,... ,
Theorem 4.4.6. Define go = {g eg : g = f\K for some linear map f : Rd -> Rd}. If there exists a proper subgroup G of Qo which is vertex transitive on Vo, then there exists a pre-localized eigenf unction. Proof As G is a proper subgroup of <7o> there exists h G Go\G. By Lemma 4.4.5-(3), we see that S{h~1g) has no interior point for any g G G. Hence we can verify (4.4.2) and (4.4.3). As is mentioned above, this implies (4.4.1). Hence, by Theorem 4.4.4, there exist localized eigenfunctions. • The corollary below is sometimes easier to apply to examples than Theorem 4.4.6. Let Py0 be the group of permutations of Vb- We define a natural restriction map K : Q —> Py0 by K,(g) = g\v0Corollary 4.4.7. (1) If K is not infective then there exists a pre-localized eigenf unction. (2) Set Go = K{GO). If there exists a proper subgroup of GQ which is vertex transitive on V$, then there exists a pre-localized eigenf unction. Proof (1) If K is not injective then there exists g ^ i G Q with g{x) — x for all x G Vo. The result is now immediate from Proposition 4.4.3-(l). (2) We can find a proper subgroup of QQ which is vertex transitive on Vb by using z^"1. Then use Theorem 4.4.6. • Here are some cases where we can apply Corollary 4.4.7-(2). Example 4.4.8. (1) If Go = PVo and #(Vb) > 3, then the group of even permutations is a proper subgroup of Py0 which is vertex transitive on V^. This includes the case of the standard Laplacian on the Sierpinski gasket. (2) Let Vo be a regular n-sided polygon for n > 2 and suppose Go contains Dn, where Dn is the symmetry group of the regular n-sided polygon. We may write Vo = {(cos (27rj/n), sin (2nj/n)) : j = 1,2,... , n}. Let g be the rotation by 2?r/n around (0,0). Then {gi : j = 1,2,... , n } is a proper subgroup of Dn and is vertex transitive on VQ.

150

Eigenvalues and Eigenfunctions of Laplacians

Next we give an example where we can apply Corollary 4.4.7-(l). Example 4.4.9. Set Fx(z) = $z + $,F2{z) = ±(z-l),F3(z)

=

1) and F±(z) = ^^(z — 1) for z e C. K is the self-similar set with respect to {Fj i= i,2,3,4. It is easy to see that (K, S, {Fi}ieS) where 5 = {1, 2,3,4} is a p. c. f. self-similar structure. In fact, Cc — {0}, Cc = TT~ 1 (0) = {2i, 12, 32,4i} and Vc = {i, 2}. As 7r(i) = 1 and TT(2) = - 1 , Vo = {-1,1}. Let £ G (0,1), and set

D=

{~i

-l) and r = ( i i M ) -

r

Then (£), ) is a regular harmonic structure. Also let /i be a self-similar measure on K with weight (/Xi)ies, where μ\ = μ2 and μs — μ^. The reflections in the real axis and the imaginary axis, denoted by g\ and #2 respectively, belong to Q, and QQ is the group generated by {#i,#2}Obviously n(g\) is the identity map on VQ. Hence, by Corollary 4.4.7(1), there exists a pre-localized eigenfunction of —Aμ. (Note also that Q contains infinitely many elements, and so the existence of a pre-localized eigenfunction also follows from Proposition 4.4.3-(2).) Now let G = {^<72}, and let h : K —> K be defined by h(x) = x for x e KiU K2 and h(x) — g2(x) for x G ^ U ^ . It is not hard to check that heQ. Then h £ G but RG(Th-i -1) = 0. Recall the discussion after Theorem 4.4.4. In the rest of this section, we will consider the case of affine nested fracd tals. Recall Definition 3.8.2. For x,y G M , x ^ y, Hxy is the hyperplane d d bisecting the line segment [x,y\. Also gxy : IR —• R is the reflection in Theorem 4.4.10. Assume that #(Vb) > 3. / / g ^ G ^0 /or all x, y e Vo with x 7^ y, then there exists a pre-localized eigenfunction on K. Proof. Let Q\ be the subgroup of Go generated by {gxy : x, y G Vo, x ^ y}. In view of Lemma 4.4.5-(2) we may identify gxy and gXy\K, and so regard Q\ as a subgroup of 0(d), the d-dimensional orthogonal group. Note that, as gxy is a reflection, detgxy — —1. Let Q2 be the set of g G Q\ which are the product of an even number of the gxy. Then every element g of G2 has det<7 = 1, and so Q2 is a proper subgroup of Q\. Furthermore, if #(Vb) 3 then Q2 is vertex transitive. For, if x, y G Vo, let z G V0\{x, y}: then gyzgxz{x) — y- The result now follows from Theorem 4.4.6. • If #(Vo) = 2 then examples show that both possibilities (i.e., existence

4-4 Existence of localized eigenfunctions

151

or non-existence of localized eigenfunctions) can arise. Recall the case of [0,1], Example 3.7.2. If we choose a self-similar measure μ with weight (1/2,1/2) (// is, in fact, Lebesgue measure on [0,1]), then obviously goi belongs to Q^. The Laplacian is equal to d2 jdx2 and of course has no localized eigenfunctions. On the other hand, Example 4.4.9 gives an example where #(Vo) = 2 and a pre-localized eigenfunction exists. Now let C = (K, 5, {Fi}ieS) be strongly symmetric. By Theorem 3.8.10, there exists a harmonic structure (D,r), where D £ £s(Vb), on C if r is ^-invariant. (Recall that Qs is the group of linear symmetries of C and that Cs(Vo) is the collection of ^-invariant Laplacians.) Let (S.J7) be the quadratic form associated with the harmonic structure (D,r). Then we see that (£,J-) is ^-invariant by Theorem 3.8.10. Now let /x be a selfsimilar measure on K with weight ^ ) ^ G s . Assume that μ is ^-invariant, that is, μi = μj if there exists g £ Qs such that g(Ki) = Kj. Then we see that Qs C Qo. So if C is an affine nested fractal, the assumption of Theorem 4.4.10 is satisfied. Corollary 4.4.11. Let C — (K, S, {Fi}ies) be an affine nested fractal with #(Vb) > 3. Let (D,r) be a Qs-invariant harmonic structure {i.e., (-D,r) is a harmonic structure, D G £s(Vb) and r is Qs-invariant.) Also, let μ be a Qs-invariant self-similar measure on K with weight fai)ies. Then the associated Laplacian Aμ has a pre-localized eigenfunction. Moreover, in the lattice case of Theorem 4-1-5, it follows that pb{x^)/xds^2 does not converge as x —> oo for b = N,D. For the general strongly symmetric case, if there exists a proper subgroup of Qs which is vertex transitive on Vo, then the associated Laplacian AM possesses a pre-localized eigenfunction. For example, Example 4.4.12. Recall Example 3.8.13, where Vo is the vertices of a cube. In this case, we may write

Vo = {((-i)M-i)M-i) f c ): U, * e {o, l}}. Suppose that Q2 is contained in Qs. (The group Q2 is defined in Example 3.8.13.) Let (D, r) be a ^-invariant harmonic structure and let /i be a ^-invariant self-similar measure. Then since G° is vertex transitive on Vb, we see that Aμ has a pre-localized eigenfunction. Unfortunately, it is not known whether Qs has a proper subgroup which is vertex transitive on Vb for the general strongly symmetric case.

152

Eigenvalues and Eigenfunctions of Laplacians

4.5 Estimate of eigenf unctions In this section, we will estimate the supremum norm of an eigenfunction 2 of A M compared to the L -norm. More precisely, let ip G Pμ and assume that Aμ^ = —X

0. Then, if a = ma,Xies fog l^i)/(logμ^i), we will see that IM|oc
(4.5.1)

where c is a positive constant that is independent of A and (p, |
v^) = min μ^j and h(X:v^) where A(i/, A) = {w = W\w2 .. .wm G W* : isWl...w ...

=

Wl

max /i w ,

-i

m

By Definitions 1.3.9 and 1.5.6, A(i/, A) is a partition for 0 < A < 1. The quantities h(X : V^μ) and h(X : v, \i) measure the distortion between the two self-similar measures μ and v.

Proposition 4.5.2. Set o(u^)

m a x a n and d ies logUi

o(i/,a)

m i n . ies logi/»

Then there exists c>0 such that c\*(">ri < h(X : i/, /i) < \

6

M

and cX^^

< h(X : i/, μ) < \&">ri

for any X G (0,1). Obviously, by the definition, (S(z/, μ) > 1 > ^(z/, μ) and any of the equalities holds if and only \i μ = v. To show the above proposition, we need the following lemma. L e m m a 4 . 5 . 3 . Let a\ and bi be positive numbers for z = l , 2 , . . . , n .

mm

bi

, bx + --- + bn

—<

i=l,2,... ,n Cii

<

a\ -f • • • + an

Then

bi

max — .

2=1,2,... ,n a^

One can easily prove this lemma by using induction on n. We use h(X),h(X),I and 6_ in place of h(X : i/, μ),/^(A : z/,//),5(z/, ^) and 8_(v, μ) respectively.

153

4-5 Estimate of eigenfunctions

Proof of Proposition 4.5.2. If a = m i n ^ s z/f, then vw < X < vw/a for any it; G A(z/, A). Hence (aA)(log/x-)/(log^) < ^

< x^st*")/^"").

(4.5.2)

Set c = a6. Then Lemma 4.5.3 implies that cA6 < μw < A-. Choose k £ S so that £ = (log/x/e)/(logi/fc). Then there exists w* G A(i/, A) such that k G £«,„. Hence w* = k...k. By (4.5.2), cA^ < μWiti < X1. Hence 5 s cA < h(X) < X for any A G (0,1). Exactly the same argument shows that cX^ < h(X) < A^. D Now let us return to eigenfunctions of the Laplacian Aμ. Assume that ^ is a self-similar measure on K with weight (μ^)^ €5. Then, recalling Theorem 4.1.7, we have Z^esO^ r *) d s / / 2 = l> where ds is the spectral exponent of {E,T, //). Define V{ = ^iri)ds/2 and let v be the self-similar measure on K with weight (vi)ies. The following is the main theorem of this section. Theorem 4.5.4. There exist positive constants c\ and c^ such that

IMIoo < c i | A | d ^ ^ ) / 4 | | u | | 2

•

(4.5.3)

for any u G Vμ with AM-u = —A^ and \X\ > c 2 . Moreover, if there exists a pre-localized eigenfunction of Aμ, then (4.5.3) is best possible. More precisely, there exist {un}n>i C £VA{0} and c$ > 0 such that — X n u n , limn—j-oo A n = oo and

By the definition of 8(is, μ), it is easy to see that a = ds&(v^)/2 and that (4.5.1) and (4.5.3) are the same. For D-eigenfunctions (and also N-eigenfunctions), (4.5.1) is easily derived from the Nash inequality (5.3.3) along with Corollary B.3.9. We need several lemmas to show Theorem 4.5.4. Notation. Let A be a partition. For u : K —• R, uA = ^2peVrA\ In particular, if A = {0}, we write u$ in place of ^Α-

u{p)ij)p.

Recall Definition 4.2.2 for the definitions of V(A) and ij)^. u\ is the unique A-harmonic function that satisfies I/A|V(A) — u\v(A)- In particular, UQ is the unique harmonic function that satisfies UQ\YQ = u\yQ. L e m m a 4 . 5 . 5 . There

exists

c > 0 such

b>:

that

154

Eigenvalues and Eigenfunctions of Laplacians

for any partition A and any A-harmonic function

f.

Proof. Since the space of harmonic functions is finite dimensional, there exists c > 0 such that c||it||2 > |M|oo for any harmonic function u. Now let / be a A-harmonic function. Then

> min μw weA

Since

\\f\\n =

inequality.

V^ /

ii * j

(min μw

\\J

weA

weA

foFu •IIL.

m a x ^ G^WfoFw lloo, the lemma follows from the above D

By Proposition 3.4.8 and Theorem 3.6.1, it follows that the Green's operator Gμ is a bounded operator from LV{K, μ) to LP(K, μ) for p — 2, oo. Let 11 Gμ Up be the operator norm of Gμ as a bounded operator from LP(K^) to itself for p = 2, oo. Lemma 4.5.6. Set a = max{||G M ||2, HG^Hoo}. If u £ Uμ and AμU = —Xu and \X\ > (2a)~1, then

for p = 2,oo, where A = A(V, (2a|A|)~ d s / 2 ). Proof. Note that \\^wrwa

< 1/2 for any w G A. By Theorem 3.7.14,

u o Fw = XμwrwGμ(u

o Fw) + (wo Fw)0.

This immediately implies that l l h x o i ^ l l p - I K U O F ^ O I I P I < \X^wrw\\G μ^u for p = 2, oo. Hence, if \X^wrwa

o Fw\\p

< 1/2,

\\(uoFw)o\\p/2< \\uoFw\\p <2||(woF u ; )o||p for p = 2, (X). This implies the desired inequality. 1

• ds 2

Proof of Theorem 4.54. Let c 2 = ( 2 a ) " and let A = A(z/, (c2/a) ^ ). Also assume that u G Vμ and AM = —au. Then, by Lemmas 4.5.5 and 4.5.6, we have

where h(-) — h(- : ^ μ ) . Hence, by Proposition 4.5.2, it follows that

Notes and references

155

ds<5//4

where c\ — 4c(c2)~ and 6 = ~8(v^). So we have shown (4.5.3). Next assume that there exists a pre-localized eigenfunction u and /S.^ = —au. Then, as in Proposition 4.3.2, if uw — Pwu, Aμuw = ^u. Note that H ^ l ^ = v T ^ I M ^ and H^Hoo = IMIoo- Choose k so that 5(z/,/i) = log/ii/logi/i. Let w(m) = k...k G Wm for all m > 0 and let T n e n 2m/ds ^m = ww(m)Aμ^™ = -am(pm, where a m = a(i/fc)~ . Also, if m/2 C4 = IMI2/IMI00, then ll^mlloo = c 4 ( ^ ) " | | ^ m | | 2 - Hence we obtain

• Notes and references 4.1 Theorem 4.1.5 was obtained in [93]. One of the key ideas in proving this theorem is the use of Dirichlet forms, in particular, the self-similarity of Dirichlet forms, Proposition 3.3.1. This idea was first used by Fukushima in [41], where he studied the spectral dimension of nested fractals. See also Kumagai [100]. There have been a number of related works before Theorem 4.1.5. In the late 1970s and the early 1980s, physicists studied the integrated density of states on fractal graphs, in particular, the graphs associated with the Sierpinski gasket. Although they did not give any precise definition of Laplacian, they obtained the so called "spectral dimension" of fractals, which is log 9/log 5 for the Sierpinski gasket. The spectral dimension corresponds to the spectral exponent for the standard Laplacian on the Sierpinski gasket. They also suggested the existence of localized eigenfunctions. See Dhar [30], Alexander & Orbach [1], Rammal & Toulouse [154] and Rammal [153]. Also see the series of papers by Gefen et al. [47, 49, 48]. One can find a review of those works in [118] and [75]. On the mathematical side, Kusuoka [106] constructed Brownian motion on the Sierpinski gasket and obtained the value of ds for that case. Also Barlow & Perkins [20] studied the asymptotic behavior of the heat kernel associated with Brownian motion on the Sierpinski gasket by a probabilistic approach. Later, Fukushima & Shima [44] obtained detailed information about eigenvalues and eigenfunctions for the standard Laplacian on the Sierpinski gasket by using the eigenvalue decimation method. In particular, they found localized eigenfunctions and proved that the integrated density of state is discontinuous. (This result immediately implies that the periodic function G appearing in Theorem 4.1.5 is discontinuous in this

156

Eigenvalues and Eigenfunctions of Laplacians

case.) Roughly speaking, the eigenvalue decimation method is based on the following property. Eigenvalue decimation property Let <£(#) = x(b — x). Define to(Vm) = {ue e(Vm) : u\Vo = 0}. (We may identify eo(Vm) with £(Vm\V0) by restricting the domain of u G £o(Vm) to V^\Vo.) Also define L^ : to(Vm) -> io(Vm) by letting L^u = (Lrnu)\Vrn\Vo for any u e to{Vm), where Lm is defined in Example 3.7.3. Then, if A ^ {2,5,6}, L^ +1 iz = — \u for u e £o{ym+1) if and only if L^{u\Vrn) = -$(\)(u\Vrn). See Shima [166] for the proof. L^ is the discrete version of the Dirichlet Laplacian. By virtue of the eigenvalue decimation method, we can trace all the eigenvalues of L^ and the eigenfunctions starting from m = 1,2. And hence we obtain the exact locations of the eigenvalues of the standard Laplacian. In [24], Dalrymple et al. studied (numerically in part) eigenvalues and eigenfunctions of the standard Laplacian on the Sierpinski gasket by using the eigenvalue decimation method and obtained figures of eigenfunctions. See also [50] and [90] for detailed structure of eigenvalues and eigenfunctions of the standard Laplacian on the Sierpinski gasket. Unfortunately, the eigenvalue decimation method only works for quite limited class of Laplacians on p. c. f. self-similar structures. See [169] for details. In [44], Pukushima & Shima also studied the spectrum of the standard Laplacian on the infinite Sierpinski gasket and obtained the discontinuity of the integrated density of states. See also [41] and [45]. In [178], Teplyaev determined the complete structure of the spectrum of the standard Laplacian on the infinite Sierpinski gasket with Neumann boundary condition. In particular, he showed that there exists a complete orthonormal system consisting of localized eigenfunctions and that the spectrum is pure point. However, this problem is still open in the case of Dirichlet boundary condition. Recently Sabot extended Teplyaev's results to infinite nested fractals without boundary points. See [165] for details. 4.2 The results in this section was obtained in [85]. 4.3 and 4.4 The results in those sections, except Theorem 4.3.6, were obtained in [18]. Theorem 4.3.6 is a new result. 4.5 The results in this section are new.

5 Heat Kernels

In this chapter, we will study the Neumann heat kernel ppj(t,x,y) associated with the Neumann Laplacian Hjy and the Dirichlet heat kernel P£)(t,x,y) associated with the Dirichlet Laplacian Hp. Formally, the heat kernels p& and pjy are defined as the integral kernels associated with the tHN and e~tH° respectively. More precisely, in the Neumann semigroups e~ case, for example, tHN

(e- u)(x)=

/

JK

In other words, the heat kernels are the fundamental solutions of the heat equation, du with corresponding boundary conditions. We will first define p^ and po as formal sums with respect to the complete orthonormal systems {^pf}i>i and {^Pji^i respectively. (See Definition 5.1.1.) Using estimates in 4.5, the formal expansions are shown to yield a continuous function on (t,x,y) G (0, oo) x K2. Then p^ and po are shown to be the integral kernels of the semigroups e~HNt and e~Hot respectively. In 5.2, we will prove the parabolic maximum principle for solutions of the heat equation. This implies that 0 < pn(t,x,y) < pN(t,x,y) when (D,r) is a regular harmonic structure. (As we will explain in 5.2, the restriction that (D, r) is a regular harmonic structure is purely technical. In fact, by using probabilistic methods or the general theory of Dirichlet forms, it is known that this holds even if (D, r) is not regular.) In 5.3, we will study the asymptotic behavior ofpN(t, x, x) and £>£>(£, x, x) as t —> 0. If {D,r) is regular and μ = /i*, where μ* is the self-similar 157

158

Heat Kernels

measure defined in Theorem 4.2.1, then our estimate will be sharp. See Corollary 5.3.2. As in the last chapter, we suppose that (K, 5, {Fi}ies) is a connected p. c. f. self-similar structure with S = {1,2,... , iV}, (D,r) is a harmonic o n K structure on C and /z is a self-similar measure with weight (μ^ies 7 with Tiμi < 1 for any i G S. We use (S^J ) to denote the resistance form associated with (D,r). Also, d is a metric on K which is compatible with the original topology of K. (Note that (K,d) is compact.) We will use the convention from the previous chapters that the symbol b represents the boundary conditions, i.e., b G {iV, D}, where N represents the Neumann boundary condition and D represents the Dirichlet boundary condition.

5.1 Construction of heat kernels In this section, we will construct heat kernels pr>(t,x,y) and p]y(t,x,y) corresponding to Dirichlet and Neumann boundary conditions respectively. By the results in 4.1, for b = D,N, there exist {A^}n>i and { ^ } n > i C C(K, d) n £>6,M such that 1, 0

\b1<^^^<\brn<\bm+1<^^^

and {^n}n>i is a complete orthonormal system for Definition 5.1.1 (Heat kernel). For b = D,N, we define the 6-heat kernel pb(t, x, y) for (£, x, y) G (0, CXD) X K X K by

pb(t,x,y) = J2 e-X"Vn(x)
(5.1.1)

n>1

The right-hand side of the above definition is only a formal sum so far. We will show, however, that the sum converges uniformly on [T, oo) x K x K for any T > 0. Hence pft(£, x, ?/) is continuous on (0, oo) x K x if. The following gives some important properties of the heat kernels. In the rest of this section, the symbol b always represents D or N. Proposition 5.1.2. (1) Pb(t,x,y) is non-negative and continuous on (0,oo) xK xK. (2) For any (t,x) G (0, oo) x K, define p\x byp\x(y) =pb(t,x,y) for any x y eK. Then p\ G Dom(Hb) nVfl= Vb^ for any (t, x) G (0, oo) x K. (3) For any (x,y) eKxK, ph{-,x,y) G C ^ o o ) ) . (4) For any (t, x, y) G (0, oo) x K x K, dpb(t,x,y)

159

5.1 Construction of heat kernels (5) For any s, t > 0 and any x,y G K, t

/

Pb(t + s,x,y)=

l

f

ph(t,x,x )ph(s,x ,y)μ(dx ).

JK

We need several lemmas to prove this proposition. Lemma 5,1.3. There exist positive constants c\ and c2 such that cxn2'ds for

<Xbn<

c2n2'ds

n>2.

Remark. Recall that H& is invertible and that Af* > 0. Therefore, if b — D, then the above inequality holds for n > 1 as well. However, this is not the case if b = JV, since A^ = 0. Proof. By Theorem 4.1.5, there exist positive constants c3 and c^ such that c3xds/2 for sufficiently large x > 0 . 2 ds

b

As n < p ^ ( A ^ μ ) < c±\ n 2/ds

b

, we have t h a t d s

cxn l < \ n, w h e r e c1 = c4~ . F o r a n y e > 0 , c3(X n - e ) ^ 2 < pb(Xbn e,/x) < n. Letting e | 0, we see that \bn < c2n2lds, where c 2 = c3~2/ds. • Lemma 5.1.4. For any a,/? > 0 and any T > 0 ; E n > i n O ! e " n * ^s formly convergent on [T, oo).

un

^~

Proof. Choose ?n G N so that ^m — a > 1. Let 7 = /3ra — a. Since n0k atk kl w e s e et h a t a n/3 ~ / i ^ e " * < An-^ for any t > T n-ocenH = J2k>o rn and n > 1 , where A = m\T~ . As J2n>1 n~ 7 < 00, the lemma follows immediately. • Now we prove the proposition except the non-negativity of

pb(t,x,y).

Proof of Proposition 5.1.2. (1) By Theorem 4.5.4 and Lemma 5.1.3, 4

2

Halloo < c ( A ^ / < cV/ , where 6 = 8(v^).

(5.1.2)

Again, using Lemma 5.1.3, we obtain \e-^Vn(x)vbn(y)\

(cOVe-"^,

where $ = 2/ds and c" is a positive constant which is independent of n. t Now, by Lemma 5.1.4, J2n>i(cf)2n6e~c"n converges uniformly on [T, 00) for any T > 0. Therefore, the right-hand side of (5.1.1) converges uniformly on [T, 00). So pb(t, x, y) is continuous on (0,00) x K x K.

160

Heat Kernels

(2) Note that oo. Also, if / mas 5.1.3 and X. Therefore,

/ = £ n > i anipbn G Dom(Hb) if and only if Zn>i lAn«n|2 < G Dom(# 6 ), then Hbf = E n > i A n a n ^ - Now > bY Lem5.1.4, £n>i( A ne~ A ' VnO*O)2 < °° for any (t,x) G (0,oo) x p];x G Dom(# 6 ) and

(Hhpl>x)(y) = Y^ \bne-x*VnM(y)-

(5.1.3)

n>1

The same argument as in the proof of (1) implies that the right-hand side of (5.1.3) converges uniformly on [T, oo) x K x K for any T > 0. Therefore Hbpbx e C{K,d). Corollary 3.7.13 implies that p\x G Dom(Hb) H X>M = (3) As the right-hand side of (5.1.3) converges uniformly on [T, oo) x K x if for any T > 0, we see that 2

^

x,y) -ph(tux,y)

(5.1.4)

for any £i < t 0}. Then, we can extend the heat kernel pb(t, x, y) to a holomorphic function on HR as follows. Proposition 5.1.5. Define pb(z,x,y) by substituting z G HR for t in (5.1.1). Thenpb(z, x, y) is a holomorphic function on HR for any x,y G K. Later, this proposition will play a key role in proving Proposition 5.1.10, where pb(t, x, y) is shown to be strictly positive for any t>0. Proof Note that n>1

n>1

where t = Re z for z G HR. The same discussion as in the proof of Proposition 5.1.2-(1) implies that £ n > 1 e~x™ 2'iPnWVniy) converges uniformly on {z G C : Re z > T} x K x K for any T > 0. Since e~x^z is holomorphic, it follows that p(z, x, y) is holomorphic on HR. • Next we construct the heat semigroup associated with pb(t,x,y). B.2 for a definition and basic properties of semigroups.

See

5.1 Construction of heat kernels

161

Definition 5.1.6. For u G L^i^/i), define (Ttbu)(x) =

[ JK

for t > 0 and x G K. Note that L 1 (#,//) 2 LP(K^)

for p G [l,oo].

Theorem 5.1.7. (1) T^ is a bounded operator from LP(K, μ) -> C(K, d) for any t>0 and any p G [1, oo]. (2) Ttb o T> = Ttb+S for any t,s>0. (3) T*(Ll{K^)) C 2>6iM /or any t > 0. (4) Let u € V-{K, μ). Set u(t, x) = (T?u)(x) for (t, x) € (0, oo) x K. Then u(-,x) € C°°((0, oo)) for any x S K. Moreover, = (A,ut)(x)

(5.1.5)

for any (t, x) G (0, oo) x K, where ut = (5) {T^}t>o is a strongly continuous semigroup on L2(K, μ). The generator of {Tj>}t>o is-Hb. Proof (1) This is obvious since pb(t, x, y) is continuous on (0, oo) x K x K. (2) This follows immediately by Proposition 5.1.2-(5). (3) and (4) Let u G L^K,μ). Choose h so that 0 < h < t Set s = t - tx. Since m = T^u G C(K,d) C L2(K^), we see that m = T,n>i ani with 5 Z n > 1 | a n | < oo. Therefore T^u = Tbu\ = a

n e ~ A n V t Making use of Lemmas 5.1.3 and 5.1.4, we obtain that ne" A - s A^) 2 < oo. This implies that T%u G Dom(Hb) and

n>1

By Lemmas 5.1.3 and 5.1.4 along with (5.1.2), Hh(T%u) G C(K,d). Corollary 3.7.13 implies that T%u G Vb^. Also, it follows that - f\Hb(Ttbu))(x)

Now,

= (Tbu)(x) - (Tbtlu){x).

Hence u(t,x) is a C1-class function and (5.1.5) holds for t> t\. (5) If u G L2(K^), then u = Y,n>1anipbn, where ^ > 1 l«n|2 < oo. So T{*u = Y2n>iane~~Xbnt- Hence||T t 6 w|| 2 < | M | 2 and \\Tbu-u\\2 ^0ast->0. Therefore {Tb}t>o is a strongly continuous semigroup on L2(K, μ). Now let A be the generator of {Tb}t>o> Then, by the definition of the generator in

162

Heat Kernels b

b

b

b

b

Theorem B.2.2, it follows that ip n G Dom(A) and Aip n = -\ nip n = -Hb(p n. 2 Since {iPn}n>i is a complete orthonormal system for L (K, μ), we deduce that A = -Hb. D b

As the generator of {T }t>o is —if6, the corresponding closed form on L (K^) is the Dirichlet form (£,.F) if 6 = N or (5, J^) if & = £>. Theorem B.3.4 implies that {Tb}t>o has the Markov property. One important consequence of this fact is the non-negativity of the heat kernel. See Lemma B.3.5 and Proposition B.3.6 for further properties of strong continuous semigroups with the Markov property. 2

Proof of Proposition 5.1.2-(1). The remaining property is that pb(t,xyy) b is non-negative. By Theorem B.3.4, {T }t>o has the Markov property. Therefore, if u G C(K,d) and u{x) > 0 for any x G K, then (Ttu)(x) > 0 for any x G K. This implies that pb(£, x, y) > 0 for any (£, x, y) G (0, oo) x KxK. D Here we have shown non-negativity of the heat kernels by using the Markov property of {Xf }t>o, more precisely, by Theorem B.3.4. However in this book we will not give a proof of Theorem B.3.4. There is another way of showing non-negativity of the heat kernels by using the (parabolic) 1 maximum principle, as in the classical theory of the heat equation on W . See the next section for details. Unfortunately, however, so far this alternative proof only works in the case of a regular harmonic structure. Prom Theorem 5.1.7, the heat kernel pt>(t,x,y) is the fundamental solution of the heat equation (5.1.5). Definition 5.1.8. Let u : (0, oo) x K -> R. For t G (0,oo), set ut(x) = u(t,x) for any x G K. u is called a solution of the heat equation (5.1.5) on (0, oo) if and only if u is continuous on (0, oo) x K, u(-, x) G C 1 ((0, oo)) for any x G K, ut G Vμ for any t G (0, oo) and u satisfies (5.1.5) for any (t,x) G (0,oo) xK. D

2

Corollary 5»1*9. (l) Set u(t,x) = (Tt (p)(x) for ip G L {K^). Then u(t,x) is the unique solution of the heat equation (5.1.5) on (0, oo) with lim \\ut — ip\\2 = 0

and

u(t,p) = 0

for any t>0 and any p G Vo. N 2 Then u(t,x) is the unique (2) Set u(t,x) = (Tt (f)(x) for cp G L (K^). solution of the heat equation (5.1.5) on (0, oo) with lim \\ut — (p\\2 = 0

and

(dut)p = 0

5.1 Construction of heat kernels

163

for any t > 0 and any p G VQ. Proof. We only need to show uniqueness of the solution, which follows from Theorem B.2.6. • If (f is a continuous function on K, we naturally expect that V^lloo —> 0 a s t I 0. (If b — D, we need to assume that ip\y0 — 0 as well.) However, such continuity of a solution up to t = 0 is not straightforward in general. In the next section, we can show such continuity of a solution only when (Z), r) is a regular harmonic structure. In fact, this problem is closely related to the proof of the non-negativity of the heat kernels through the parabolic maximum principle mentioned above. See the next section for details. By Proposition 5.1.2, we know that pb(t,x,y) > 0 for any (t,x,y) G (0, oo) x K x K. In fact, we can show a stronger result as in the next proposition. Proposition 5.1.10. (1) Piv(t, x, y) > 0 for any (t, x, y) G (0, oo) x K x K. (2) pE)(t,x,y) > 0 for any t>0 if x and y belong to the same connected component of K\VQ. In this section, we only give a proof of (1) of the proposition above, while (2) will be proven in 5.3. Proof. Step 1: For any x,y G K, there exists t > 0 such that PN(t, x, y) > 0. Proof of Step 1. Since A^ = 0 and (p^ = 1, it follows that pjv(s, a, a) > 1 for any (s, a) G (0, oo) x K. Let us fix s > 0. For a G K, if U(a) = {b G K : PN(S, a, b) > 0}, then U(a) is a non-empty open subset of K. As K is compact and connected, we can choose a?o, • • • > xm so that XQ = x, x m = y and Xi+i G U(xi) for all z = 0 , 1 , . . . ,ra. Applying Proposition 5.1.2-(5), we obtain pN(ms,x,y) = /

PN(S, X, VI)PN(S,

2/1, y2) • • • PAT(S, 2/m-i, 2/)Md2/i)Md2/2) * • * M d 2/m-i)-

Since x i + i G f/(^) for any z, ^ ( s ^ x i ^ j v ^ ^ i , ^ ) • • -pjv(s,a:m-i,2/) > 0. Hence p^ (ms,x,y) > 0. • Step 2: If £>jv(£, x, y) > 0 for t > 0 , then PN(S, X, y) > 0 for any s > t.

164

Heat Kernels

Proof of Step 2. If s > £, Proposition 5.1.2-(5) implies that pN(s,x,y)=

f

/

l

pN(s-t,x,x')pN(t,x ,y)μ(dx ).

JK

Since PN{$ — t, x, x')pN(t, x', y) > 0 for x' — x, it follows that pjv(s, x, y) >

o.

n

Now, by Step 1 and Step 2, for any x, y G K, there exists T > 0 such that PN(t,x,y) = 0 for £ G (0,T) and pw(t,x,y) > 0 for £ G (T, oo). Suppose T > 0 . By Proposition 5.1.5, p^(z,x^y) is a holomorphic function on HR. Hence, if PN(t, x, y) = 0 for t G (0, T), then pjv(^, x,y) =0 for any z G HR. Obviously this contradicts the fact that jpiv(t, x, y) > 0 for t > T. Therefore T = 0 and pN(t, x, y) > 0 for any t > 0 . • 5.2 Parabolic maximum principle In 5.1, we defined the notion of solutions of the heat equation

(5.2.1)

g

on (0, oo) in Definition 5.1.8. In this section, we study the continuity of the solution to (5.2.1) at t = 0. More precisely, Definition 5.2.1. Let u : [0, oo) x K —> R. u is called an Z/°°-solution of the heat equation (5.2.1) on [0, oo) if and only if u is continuous on [0, oo) x K and is a solution of the heat equation (5.2.1) on (0, oo). A remarkable property of an L°°-solution of the heat equation on [0, oo) is the parabolic maximum principle. In the classical theory of heat equations n on R , the parabolic maximum principle plays an important role. For example, it is often used to show uniqueness of a solution of a certain heat equation. See, for example, [77] and [152] for details. Theorem 5.2.2 (Parabolic maximum principle). Let u : [0, oo) x K —>• R be an L°° -solution of the heat equation (5.2.1) on [0, oo). Then for any T>0, max u(t, x) =

(t,x)euT

max u(t.x)

{t,x)eduT

and

min u(t,x) =

{t,x)euT

min

(t,x)edUT

u(t,x),

where UT = [0,T] x K and dUT = {0} x K U [0,T] x Vo. Moreover, if ut G T>N^ for any t>0, where ut{x) = u(t, x) for any t G [0, oo) and any x G K, then, max (t,x)e[0,oo)xK

u(t. x) = maxu(0,x) and xEK

min (t,x)e(0,oo)xK

u(t, x) = minu(0,x). XEK

5.2 Parabolic maximum principle

165

To prove the above theorem, we need the following lemmas. Lemma 5.2.3. Let u G Vμ. If (Aμ!!,)^ u{x) < maXpGVo u(p) for anV % € K-

> 0 for any x G K\V0, then

Proof. By Theorem 3.7.16, we obtain (Hmu)(p) = / '^|)™Aμudμ

(5.2.2)

JK

for any p G Vm\Vo. By the assumption of the lemma, the right-hand side of (5.2.2) is positive. Since Hm G CAiVm), there exists q G Vm such that u(q) > u(p). Hence, if u(p*) = maxg€vm u(q) for p* G Vm, then p* G Vo. Therefore u(p) < maxqev0 u(q) for any p G Vm\Vo. Now, let x ^ V*. Choose w G W* so that x G i ^ and Fw(Vo) fl Vb = 0. Using the same argument as above, we have u(p) < maxq^Fw{y0) li(^) for any p G if^nV*. Therefore it follows that u(x) < maxgG^(vr0) u(o)- Again the above argument implies that maxgG^iy(vb) u(o) < maxpGvb w(p)- Hence we obtain u(x) < maxpGvb u(p) for any x G K\Vo• Lemma 5.2.4. Let u G Dμ. If u(x) = maxy^K u(y) for x G i^\Vb, then (Aμu)(x) < 0. Moreover, ifue Vjy^, then the statement above is true for x EVO as well. Proof. Suppose that (Aμ!!)^) > 0. First assume that x G V*. Then, using (5.2.2) in the same way as in the proof of Lemma 5.2.3, we deduce that there exists y £ V* such that u(y) > u(x). This is a contradiction. Hence x £ V*. Then choose w G W* so that x G Kw and (Aμu)(y) > 0 for any y G -K^. Set v = u o i 7 ^. Then obviously t? satisfies the assumptions of Lemma 5.2.3. Therefore, u(x) < max p€^(vb) u(p)- This contradicts the assumption of the lemma. Next assume that u G Vjsf^, p G VQ and (Aμu)(p) > 0. Recalling the proof of Lemma 3.6.3, we see that 5(/0^l,u) = —(Hmu)(p). The GaussGreen's formula (Theorem 3.7.8) implies that (5.2.2) holds for any p G VbTherefore, (Hrnu)(p) > 0 for sufficiently large m. Then there exists q G V^ such that u(q) > u(p). Hence u(p) < max y e Ku(y). • Proof of Theorem 5.2.2. Let u : [0,oo) x K —> R be an L°°-solution of the heat equation (5.2.1) on [0, oo). For n > 1 , define un by un(t,x) = u(t,x)—t/n for any (£,x) G [0,oo) x K. First, we show the following claim. max u (t,x)= max u (t,x). (t,x)euT n (t,x)eduT n

166

Heat Kernels

Proof of the claim. Suppose that (£*,p) G UT\OUT and that u>n(t*,p) = max (t,x)euT

un(t,x).

This implies that d u ^ ^ > 0. Note that A^n = A ^ and that ^ = If - 1/n. Since AMiz = §f, it follows that (AMwn)(t*,p) > 1/n > 0. Hence, Lemma 5.2.4 implies that un(t*,p) < max xG x i£n(£*,x). This contradicts the fact that un(t*,p) = max(t,x)Gf/T ^n(£>#)• D Now, letting n —> oo, we have max u(t,x) — max u(t,x). (t,x)euT (t,x)eduT The minimum also follows if we apply the maximum case to — u. If Ut G T^N^ f° r a n y t > 0, we may repeat the above argument by replacing UT and <9!7T by [0, oo) x K and {0} x K respectively. • Later in this section, we will apply the parabolic maximum principle to show non-negativity of the heat kernels pp (£, x, y) and PAT(£, X, ?/). For that purpose, we need to know whether or not the solutions of the heat equations constructed by the heat kernels in 5.1 can be extended to L°°-solutions on [0, oo). More precisely, the problem is whether or not Tfy is a Z/°°-solution of the heat equation on [0, oo). This is still an open problem in general: we have only a partial answer if the harmonic structure is regular. Theorem 5.2.5. Suppose that (D,r) is a regular harmonic structure. (1) Let 0. In particular, \\ut — ^o||oo —» 0 as t -> 0. (2) Let
5.2 Parabolic maximum principle

167

We need several results to prove this proposition. For the moment, we show the following special case. Lemma 5.2.7. Suppose that (D,r) is a regular harmonic structure. (1) Let if G FQ. Then \\TtD(f - (fW^ -> 0 as t -> 0. (2) Let (p eJ7. Then \\TtN(p - ^||oo - • 0as t -> 0. To prove Lemma 5.2.7, we need the following lemma. Lemma 5.2.8. Suppose that (D,r) is a regular harmonic structure. there exists c > 0 such that

Then

for any u G T. Proof. Since (D, r) is a regular harmonic structure, Theorem 3.3.4 implies that £1 = K and that (K, R) is a compact metric space. Hence, by (2.2.6), \f(x)-f(y)\2<£(fJ)R(x,y)

(5.2.3)

for any x, y G K and any / G T. Let v G To. Then, by (5.2.3), it follows that \v(x)\2 <£(v,v)R(x,p) for a n y x G K , w h e r e p G Vo- T h e r e f o r e , t h e r e e x i s t s c > 0 s u c h t h a t D < cy/£{v,v) for any v G TQ. Let u G T. Define u0 =

(5.2.4)

u(p)ijjp. Then, by (5.2.4),

< Cy/E(u — Uo,U- U0) < Cy/£(u,u). Also, there exists c' > 0 such that ||wo||oo c'H^olb for any u e J7. Also note that \\u — wo 112 ||tx — ^olloo- Combining all the inequalities, we obtain

< Cy/£(U,U) + C'\\U - U0II2 D Proo/ of Lemma 5.2.7. Let b = D or AT. By Proposition B.2.4,

as t I 0. Hence, by Lemma 5.2.8, it follows that \\T^(p —
0 as D

168

Heat Kernels

By Lemma 5.2.7, Tfy is a L°°-solution of the heat equation on [0, oo) under the corresponding assumption on (p in Lemma 5.2.7. Applying the parabolic maximum principle to those solutions, we can obtain nonnegativity of the heat kernels without using Theorem B.3.4. Theorem 5.2.9.

For any (t, x, y) G (0, oo) x K x K, 0
In the following, we will prove this theorem assuming that (D, r) is regular. For the general case, the fact that pr>(t,x,y) < pisr(t,x,y) can be shown by considering the Markov processes associated with pz> (t, x, y) and pw(t,x,y). See A.3 for details. Proof. First we show that pz>(t,x,y) > 0. Suppose that p£>{s,a,b) < 0 for some (s, a, b). As po is continuous, we can choose ^™ so that PD(s,a,b)ip™(b) < 0 and pD(s,a,y)>ip™(y) < 0 for any y G if. Define D u(t,x) = (Tt if;™)(x) for any (t,x) G (0, 00) x K and u(0,x) = i(>™(x) for any x G K. Then, as ip™ G T§, Lemma 5.2.7 implies that u is an L°°-solution of the heat equation on [0,00). Note that U\QUT ^ 0 for any T > 0. Hence, by the parabolic maximum principle, we see that u(t,x) > 0 for any (t,x) G [0,00) x K. This contradicts the fact that < 0. Therefore, pD(t,x,y) > 0 for any u(s,a) = fKpD(s,aJy)1p™(y)μ(dy) (t,x,y) G (0,oo) x K x if. Next let (p e J7. Define w(t,x) = (TtN(p)(x) for (*,z) G (0,oo) x if and ix(0,x) = <^(x) for any x G if. Then it is an L°°-solution of the heat equation on [0,00). Moreover, by Corollary 5.1.9, (dut)p = 0 for any (t,p) G (0,00) x VQ. Hence ut G T>N^ for any t>0. Therefore, we can apply Theorem 5.2.2 to u(t,x) and obtain that u{t,x) > 0 for any (t, x) G [0,00) x if if (f(x) > 0 for any x G if. Using entirely the same argument as in proving non-negativity of P£>(£, x,y), we can verify that PN(t, x, y) > 0 for any (t, x, y) G (0,00) x if x if. Finally, we show that pr> (t, x, y) < Piv(^, x, y) for any (t, x, y) G (0, CXD) X N D if x if. Let (f G T^. Define v(t,x) = (T t ^)(x) - (Tt (p)(x) for any (t, x) G [0,00) x if and i>(0, x) — 0 for any x G if. Since

=/ JK

v is an I/°°-solution of the heat equation on [0,00). Suppose that (p(x) > 0 for any x G if. Then the above argument implies that T/V 0- Since T/Vl(o,T)xVb = 0, it follows that v|ac/T 0 for any T > 0. Hence, by the parabolic maximum principle, v(£,a:) > 0 for any (t,x) G [0,00) x

5.2 Parabolic maximum principle

169

K. Taking this fact into account, the same argument as in proving nonX > 0 for any negativity of PD(t,x,y) implies that pjsi{t^x^y) — PD{1, ,V) • (t,x,y) G (0,oo) x K x K. Now we are ready to give a proof of Proposition 5.2.6. Lemma 5.2.10.

For any (t,x) G (0, oo) x K,

Proof. Theorem 5.2.9 implies that | | p ^ | | i < | | p ^ | | i = fKPN{t,x,y)μ(dy). a n orthonormal system of L2(K^) and ip^ = 1, we see Since {iPn}n>i is that fK ^(Iμ = 0 for n > 2. Therefore, by the definition of pjv(£, x, y).

J

K

n

>x

D Remark. As in Theorem 5.2.9, this lemma holds even if (-D,r) is not a regular harmonic structure. Proof of Proposition 5.2.6. Let y> G C(K, d) and assume that ip\vQ = 0 . As J"o is dense in {u G C(K,d) : u|vb = 0} with respect to the supremum norm, there exists {un}n>i C J~o such that \\un — y||oo —> 0 as n —> oo. Note that D D | | 2 f ^ - PIIOO < ||T t ((^ - ^ l l o o -f \\Tt Un - UnWoo + I K - ^lloo.

(5.2.5) On the other hand, by Lemma 5.2.10, \(TtD(un D

- ip))(x)\ < l l ^ l l l I K -
Hence ||r t (w n — y?)||oo obtain

ll^n —
\\TtD

0. Therefore, using a routine argument, it follows t h a t | | T / V — <^||oo—^Oast—>0. • We can repeat exactly t h e same argument in t h e N e u m a n n case. r

Remark. The key element of the above proof is that HT^Hoo < ||y?||oo f° any (p G C(K,d). Even if (£>, r) is not a regular harmonic structure, this follows from Proposition B.3.6, which is a consequence of Theorem B.3.4. Using Theorem 5.2.9, we obtain the following corollary of Theorem 5.2.5.

170

Heat Kernels

Corollary 5.2.11. Let (D,r) be regular. For any u G C(K,d), T®u converges to u uniformly on compacts in K\VQ as t [ 0. To prove this corollary, we need the following lemma. Lemma 5.2.12. Let U be a closed subset of (K,d) and let u G C(K,d). //supp(ii) n U = 0, then T^u converges to 0 uniformly on U as t [0. Proof. First assume that u is non-negative on K. Then, by Theorem 5.2.9, (TtNu)(x) > (TtDu)(x) > 0 for any x G K. Theorem 5.2.5 implies that T^u —• 0 uniformly on U as 11 0. Therefore, T®u —• 0 uniformly on U as 11 0. For general u, let w + (x) = max{w(x), 0} and U-(x) = — min{w(x), 0} for any x G K. Then ix = ii+ — U- and supp(tx+) fl U = supp(u_) H C/ = 0. So the general case follows on applying the result on non-negative u to u+ and u-. • Proo/ o/ Corollary 5.2.11. Let C7 be any compact subset of K contained in K\Vo. Define it m = J2xev0 u{x)^Then supp(u m )n£/ = 0 for sufficiently large m. By Lemma 5.2.12, T^Um —> 0 uniformly on [/ as t | 0. Now r t D ii = r t D (w - Um) H- T t D w m . By Theorem 5.2.5, TtD(w - w m ) -> u - u^ uniformly on K as £ [ 0. Since wm(o:) = 0 on £/, we see that T®u —> w uniformly on [/ as 11 0. • Using Corollary 5.2.11, we obtain an explicit expression for the difference pN(t, x, y) — PD(t, z, y), which was shown to be non-negative in Theorem 5.2.9. Recall that p^ (y) = pr>(t,x,y). Note that po(t,x,y) > 0 and pD(t,x,y) — 0 if x e Vo or y G V^. Hence it follows that (dpt^c)q < 0 for any q G Vo. Theorem 5.2.13. Let (£>,r) be regular. (1) For any x, y G K\Vo and t>0, pN(t,x,y)

= - 2_^ /

-pD(t,x,y)

where Jo p^(t — 5, x, q){dps^)qds

means, in fact, the following limit:

t-e

/ _

pN(t-s,x,t

pisr(t —

s,x,q)(dpSpy)qds.

(2) For any q G Vo, any x G K\Vo and any / pD(t,x,y)'^pq(y)μ{dy)

JK

f*(dpsjf)qds

t>0,

= ipq(x) - ( - /

means l i m ^ o J€ (dpSjf)qds.

JO

(dpSjf)qds),

171

5.3 Asymptotic behavior of the heat kernels

Proof. First we prove (1). Using the Gauss-Green's formula, we obtain

= [ / PN(t-

s1x1z)pD(s,z,y)μ(dz)]

JKK

=

/ Je

t

Ui

K

l,x,z)pD(s1z,y))μ(dz))ds

=£

rt-e s

/

y

pN(t-s,x,q)(dp £ )qds.

Je

i

Applying Theorem 5.2.5 and Corollary 5.2.11,

as e I 0. Thus we have shown (1). For (2), by the Gauss-Green's formula, /

S

X

*){^P D ){z)^dz)

K

s x

= (dp 6 )q.

By integrating this, \dp'6x)q

= (TeDtl>q)(x) -

(TtDi>q)(x).

Applying Corollary 5.2.11, we can immediately obtain (2).

•

Summing the equalities in Theorem 5.2.13-(2) for all q G Vo5 we obtain the following corollary. Corollary 5.2.14. Let (D,r) be regular. Then for any x G K\Vo and any t>0, \\PDX\\I

= f PD(t,x,yMdy) = 1 - (- £

f\dp%x)qds).

5.3 Asymptotic behavior of the heat kernels In this section, we will study asymptotic behavior of the heat kernel Pb(t,x,y) as t i 0. Recall that v is the self-similar measure with weight ((^i^i)ds^2)ieSi where ds is the spectral exponent defined in (4.1.2). Let J = ~8{y, μ) and 6 = £(z/, μ), where
172

Heat Kernels

Theorem 5.3.1. (1)

For b = D,N,

there exists c\ > 0 such that

sup pb{t,x,y)
(5.3.1)

x,yeK

for any t G (0,1]. (2) There exists C2 > 0 such that C2t-dsS/2

<

pN(t^

^ ^

(5>3>2)

for any t G (0,1] and any x € K. The following corollary is immediate. Corollary 5.3.2. If μ^i = v% for any i G S, then there exist positive constants c\ and C2 such that

for any t G (0,1] and x G K. Since Vi = ^ ? " i ) d s / 2 , we see that μ^ = Vi for any i G S if and only if the dH harmonic structure (-D,r) is regular and μ^ = (r^) for any z G *S, where dH is defined by (4.2.1). By Theorem 4.2.1, ds = 2dH/(dH + 1). Also, in such a case, the spectral dimension of (K,R), ds(K,R) equals dsfa). See (4.2.3). First, we prove the upper bound (Theorem 5.3.1-(1)) through the following Nash inequality (5.3.3). T h e o r e m 5 . 3 . 3 . Let 6 = dsS. Then

there

exists

c > 0 such

\\u\\l+4/e < c(£(u,u) + \\u\\l)\\u\\t/e

that

(5.3.3)

for any f G T. L e m m a 5.3.4. There exists c > 0 such that

for any u e T'. Proof Letting m = 0 in (3.4.2), we see that there exists c\ > 0 such that H^ - U0W2 < Ciy/£(u - Uo,U - Uo) for any u G J7, where UQ = Y2Pev0 U^P)'(I)P' Also, since the dimension of the space of harmonic functions is finite, there exists C2 > 0 such that C 21l^o111 f° r a n y v> £ J7. Also note that ||TX||I < \\u\\2 f° r a n y

173

5.3 Asymptotic behavior of the heat kernels

u G L2(K,μ) and that £(u,u) = £{u — UQ,U — U0) + £(u0, u0). Combining those inequalities, we obtain

where C3 = max{c2, C1C2 + ci}. This implies the desired inequality.

•

Proof of Theorem 5.3.3. Let A be a partition of E. Then, Lemma 5.3.4 along with the self-similarity of μ and £ implies that

I μw(£(u

u o Fw\2dμ Fw) + \\u o Fw\\\)

oFw,uo x

I // T* IT* \r"W' IV J' w

_.

,

x

V^

r i l l O T^ 7/ O f' I —I— f > ^ \ w' if / i ^ /

-1/ // Γ^IV

/* ,

ill I \H"iv I

|

: c(maxμ^i;r^)5(^,^) + Let A = A(i/, A), which is defined in Definition 4.5.1. Then, by Proposition 4.5.2, \u\\l

< c'\2/ds£(u,u)

+ c'\-~6\\u\\l

(5.3.4)

for any u G T and any A G (0,1). Now, suppose that £(u,u) > \\u\\2 and choose A so that \2/ds+fi — |M|?/£(u,u). Then, by (5.3.4), 11 ||2+4/0 ^

IHI2 for any ueF.

net

Ml ll 4 /0

This implies (5.3.3).

If £(u, u) < I Ml?, then, by Lemma 5.3.4, it follows that

IMl! < c(£(u,u) + \\u\\l) < 2c\\u\\l Obviously this suffices for proving (5.3.3) in this case.

•

Proof of Theorem 5.3.1-(1). By Theorems 5.3.3 and B.3.7, we know that e 2 there exists c > 0 such that HT^Hi-^ < ct~ / for t G (0,1]. Since lllJlli-,00 = supxyeKpb(t,x,y), we can immediately verify (5.3.1). • m

Now we show Proposition 5.1.10-(2). First, recall that g^ \xyy) continuous function for m > dsS/2 by Corollary 3.6.6.

is a

174

Heat Kernels

Lemma 5.3.5. Let m > ds^/2. Then, for any x,y G K,

9{m){x, y) = ^— )J^ r-ipDit, x, y)dt, where T is the gamma function defined by T(s) = /0°° xs~1e~xdx. Proof. Note that Af* > 0. Applying Lemmas 5.1.3 and 5.1.4 to the infinite

En>l e " (A "" A?) VnWrf(2/)> sup

(t,x,y)e[l,oo)xKxK

We See t h a t

eAl tpD(t, x, y) < oo.

Moreover, by Theorem 5.3.1-(1), sup^yeK^^PDfax.y) for t G (0,1], Therefore, if ?

<

fort >

!' for 0 < t < 1,

for sufficiently large c > 0 , then Jo°° F(t)dt < oo and f 7 1 " 1 ? ! ^ x, y) < F(t) f o r a n y ( t , x , y ) G ( 0 , o o ) x ^ x ^ . If h(x,y) = T(m)- 1 /0°° trn-1pD(t,x,y)dt, then Lebesgue's convergence theorem implies that h(x,y) is continuous on K x K. Now, by Fubini's theorem and the definition of g^m\ K

JK

for any n > 1 . As both g^ and h are continuous on K x K, it follows that g(m) (x, y) = h(x, y) for any x,y € K. • Proof of Proposition 5.1.10-(2). Assume that x and y belong to the same connected component of K\Vo. Let m > dsS/2. Then, by Corollary 3.6.7, g(m\x,y) > 0. Hence, Lemma 5.3.5 implies that p£>(t,x,y) > 0 for some t > 0 , and in particular that pr>(t,x,x) > 0. This means that ip%(x)2 > 0 for some n > 1 . Therefore we have that PD{t,x,x) > 0 for any t > 0 . We have therefore proved the following. (i) p£>(£, x, y) > 0 for some t > 0 . (ii) PD(t,x,x) > 0 for any (t,x) G (0, oo) x K\VoUsing these, we can repeat the same argument as in the proof of the Neumann case after Step 2 to deduce the desired conclusion. • Next we prove the lower estimate, Theorem 5.3.1-(2). Definition 5.3.6. Let A be a partition of E and let A be a subset of K. Write (1) AA = {w:weA,KwnA^ 0}.

5.3 Asymptotic behavior of the heat kernels

175

(2) (3) AA =

ANA{A)\AA.

(4) UA(A) = NA(NA(A)). (5) dUA(A) = UA(A) n K\UA(A). For x e K, we use A^, NA(x) and so on instead of A{xy, NA({x}), etc. Now let us choose m so that {if m)P } pe vb are mutually disjoint. (Note that Km,p = NWrn(p)-) Set Ko = K\ UpEv0 Km,P- Then Ko is a non-empty compact subset of K. Also define <9ifo by 3KQ = Ko fl (Upevb^m,p)- Note that if p, q G Vo and p ^ g, then p and # belong to different connected components of K\8KQ. Lemma 5.3.7. For t>0, define q(t)=

inf /

pD(t,x,y)μ(dy).

Then q(t) > 0 /or any t > 0 and q(-) is monotonically non-increasing. Proof. Let u(t,x) = fKp^>(t,x,y)μ(dy). Then u(t,x) is continuous on (0, (X)) x K. By Proposition 5.1.10, we also see that u{t,x) > 0 for any x e KQ. Since Ko is compact, it follows that q(t) = mf xG ^ 0 u(t,x) > 0. Since T^_spSpX = p£ for t> s, Proposition B.3.6 implies that u(s,x) = \\pSDX\\i > IbrTlli = u(tix)(This can be deduced from Lemma 5.2.10 as well.) Hence, u(t,x) is a monotonically non-increasing function of t. Therefore q(-) is monotonically non-increasing. • To prove Theorem 5.3.1-(2), we also need results from A.2, where we study general boundary conditions. Let B be a finite subset of V*. By Theorem A.2.1, if =

{ueT:u B

= 2

then (£, TΒ) is a local regular Dirichlet form on L (K\B, μ). As μ(B) = 0, 2 2 L (K\B^) can be identified with L (K^). If B = 0, then TB = T and this is just Theorem 3.4.6. Also, if B = Vb, then this is Corollary 3.4.7. Exactly the same way as for p&(£, x, y), we can define the heat kernel associated with (£,TB) and μ, which is denoted by £>#(£, x,y). See Definition A.2.12 for details. For example, the Neumann heat kernel pisr(t,x,y) coincides with P0(£,x,y) and the Dirichlet heat kernel coincides with pvo(t, x,y)Let A be a partition of E. We define pA(t, x, y) = Pv(A)(t, x-> V)-

176

Heat Kernels

Lemma 5.3.8.

otherwise. Proof. Let H\ be the non-negative self-adjoint operator on L2(K^) associated with the closed form ( £ , . F B ) , where B = V'(A). Then, the same argument as in the proof of Lemma 4.1.9 implies that H\u = Xu if and only \£HD(UOFW) — μwrwX(uoFw) for any w G A. Comparing Definition A.2.12 with (5.1.1), we immediately obtain this lemma. •

For x G K, we define PA,x(t,z,y) = VduA{x){t,z,v)Lemma 5.3.9. / JuA(x)

pA,x(t,x,y)μ{dy)>

mm

inf

WEAX zeFw(Ko)

/

Ph(t,

JK

We will give an analytical proof of this lemma. However, so far, the proof works only when the harmonic structure is regular. Using probabilistic methods, we may prove this lemma even if the harmonic structure is not regular. See A.3 for details.

Proof. Assume that (-D,r) is regular. Let A — U^^^ Fw(dKo) and let B = A U dU\(x). Then for any q G dU\(x), x and q belong to different connected components of K\A. Hence, by Theorem A.2.19, there exists a neighborhood, U, of q G dU\(x) such that ps{t,x,y) = 0 for any y G U. This implies that (dps^x)q = 0 for any q G dU^{x) and any s>0. Since A D dU\{x), by Theorem A.2.16,

1S

for any t > 0 and any z,y £ K. Also note that, for any z G K, Hp^lli monotonically decreasing and is no greater that 1 for any t > 0 . Combining those facts with Theorem A.2.18 and (A.2.10), we have the following

177

5.3 Asymptotic behavior of the heat kernels assertion. JK

/ PB(t,x,y^{dy)+ I (PAA^X^V) ~PB{t,x, JK

JK s

1 - J2 I (-dp f)qds + J2 f J

J

qeB o

qeA *

x

I PA,*(t - s,q,y)μ(dy)(-dp% )qds JK

[

A

x

pA(t -

s,q,y)μ(dy)(-dp% )qds

K

x

- 1) f{-dp% )qds x

> 1 + (min / pA{t,q,y)μ(dy) - 1) V By (A.2.10), 0 < Y,qeASl(-dp%x)qds

/ (-dp% )qds. < 1. Hence

/ PA,x(t,x,y)μ(dy)>mm

JK

^

A

JK

pA(t,q,y)μ(dy).

Since PA, X (^ %, y) = 0 if y £ U\(x), we obtain the desired inequality. P r o p o s i t i o n 5.3.10. There exists c>0

such that

2

t )

(

•

max

c(

μ™)p i v(2£,£,:r)

(5.3.5)

for any x G K and any partition A. Proof. By Lemma 5.3.8, / pA(t,z,y)μ(dy)=

/

JK

JKW

(μw)~1pD(t/(μwrw),Fw~1(z),Fw-1(y))μ(dy)

= / PD(t/(μwrw),Fw~1(z),y)μ(dy)

>

q(t/(μwrw))

JK

for any z G Fw(Ko).

/

Also, by Theorem A.2.16, it follows that

pN(t,x,y)μ(dy)>

JUA(X)

/

p^x(t,x,y)μ(dy).

JUA(X)

Therefore, Lemma 5.3.9 along with Lemma 5.3.7 implies that / pN(t1x,y)μ(dy) JuA(x)

> min q(t/^wrw)) weAx

> q(— mmwej^

). (5.3.6) μ^

178

Heat Kernels

On the other hand, by Schwarz's inequality, it follows that 2 pN(t, x,y)μ^y)) < μ^ix))

/ p(t,x,y)2μ(dy). UK

Now by a similar argument to the one in the proof of Lemma 4.2.3, we can see that there exists c > 0 such that #(ANA(X)) < c f° r a n Y x € K and any partition A. Also, by Proposition 5.1.2-(5), we obtain that fKPN(t, #, y)2μ(dy) = £>TV(2£, x, x). Combining those facts with (5.3.6), we obtain (5.3.5). • Proof of Theorem 5.3.l-(2). Since
2 ) <

for any x G K and any partition A. Let A = A(z/, A) and let t — m i n ^ A μwrw/2. Then we have q(l/2)2

< ch(X : i/, μ)pN(

min μ^r,,;, x, x) K;GA(^A)

for any X G X and any A > 0. Using Proposition 4.5.2, we see that there exists d > 0 such that cf < sds-/2pN(s,x,x) for any x G K and any s>0. •

Notes and references In [20], Barlow & Perkins studied the heat kernel associated with Brownian motion on the Sierpinski gasket, which is the Neumann heat kernel associated with the standard Laplacian in our context. They obtained a uniform off-diagonal (Aronson-type) estimate, exp ( - c2(\x - ^1 < c3t-d^2

exp ( - c4(\x -

where we think that the Sierpinski gasket is embedded in R 2 , ds — log 9/ log 5 is the spectral dimension and dw = log 5/ log 2 is the walk dimension. This result was extended to nested fractals by Kumagai in [99] and then to afrine nested fractals by Fitzsimmons, Hambly & Kumagai in [37] as follows. Let K be an afrine nested fractal. Let (D, r) be the harmonic structure on K obtained in Theorem 3.8.10. Suppose that (D,r) is regular. Also let μ = μ* be the self-similar measure defined in Theorem 4.2.1. Recall that ( r ^ ^ ) d s / 2 = μi and that ds = 2dH/{dH 4-1), where

Notes and references

179

dn = dim H (if, R). Let PN(t,x,y) be the Neumann heat kernel associated with (D, r) and μ. Then there exist constants Q such that

for any (£, x, y) G ( 0 , l ] x X x if, where d^ is the walk dimension with respect to the shortest path metric. See [99] and [37] for the definition of the shortest path metric. The Aronson-type estimate above is obtained by a probabilistic approach. In [6], one can find a good presentation of the probabilistic methods including the proof of the estimate above. In [60], Hambly and Kumagai have shown that a uniform off-diagonal estimate as above fails without the strong symmetry that affine nested fractals possess. Moreover, if μ ^ //*, then \m\tio\ogpN(t,x,x)/\ogt depends sensitively on x and exhibits multi-fractal nature. See [19] and [58]. For example, by those results, it follows that the heat kernel estimates in Theorem 5.3.1 are best possible. Specifically, define no

logt

for any x G K. Replacing the limit infimum by the limit supremum, we also define d(x). Then s6 = inf d(x) < sup d(x) = Note that Proposition 5.3.10 essentially suffices to show that supxeK

d(x) =

5.1 The expression of the heat kernel by using the eigenfunctions of the Laplacian has been known in many places. See, for example, [6, Theorem 3.44]. The proof of the positivity of the heat kernel, Proposition 5.1.10, is essentially based on that in [14]. 5.3 The essential idea in proving the upper estimate of the heat kernel, Theorem 5.3.1-(1), is the use of the Nash inequality. This idea is originally used in [37] in proving the counterpart of Theorem 5.3.1-(1) for the symmetric Laplacian (corresponding to the harmonic structures obtained in Theorem 3.8.10) on an affine nested fractal. See also [6, Theorem 8.3].

Appendix A Additional Facts

A.I Second eigenvalue of A^ Let (D, r) be a harmonic structure on a connected p. c. f. self-similar structure (K, S, {Fi}ieS), where S = {1,2,... ,N} and r = (rur2,... ,r N ). In this section, we study the eigenvalues of Ai, which were defined in (3.2.2). Recall (3.2.3). We have Aiu\Vo

=

u\Fi{Vo)

for any harmonic function u and any i G S. So the matrices {Ai}iGs completely determine the structure of harmonic functions. The following proposition follows easily from (3.2.3). Proposition A.I.I. If k is an eigenvalue of Aiy then \k\ < 1. Moreover, if \k\ = 1 then k = 1 and AiU = u if and only if u is a constant on VbWe will focus on the second eigenvalue of Ai in the rest of this section. It has been suggested in Example 3.2.6 that the second eigenvalue of A^ equals the resistance scaling ratio r^. We will show that this is true under some assumptions. Indeed, Strichartz [175] has proved that if K\VQ is connected, then this is true. In this section, we will give a complete answer in the general case. By the proposition above, if we set Xi = max{|/c| : \k\ < 1, k is an eigenvalue of Ai}, then Xi < 1. For U C K, define m(U) = the number of connected components of K\U. U is said to be a non-focal set if m(U) = 1. If U = {p} for p e K, we use m(p) instead of ra({p}). We say that p is a focal point if m(p) > 2. 180

A.I Second eigenvalue of A^

181

Now let p be the fixed point of Fi for some i. Then, by Proposition 1.6.7, m(p) < #(Vo). Let {Jj}j=i,...,m(p) be the collection of all connected components of K\{p}. Then, again by Proposition 1.6.7, there exists a permutation of {1,... ,m(p)}, r, such that Fi(Jj) = Ki n JT(j)- Define k(p) by k(p) = min{/c : rk = e}, where e is the identity. The following is the main result on the second eigenvalue of Ai, which includes information on corresponding eigenvectors and multiplicity of the eigenvalue as well. Theorem A.I.2. Let p be the fixed point of Fi. (1) If p G Vo, then Xi = Vi and Vi is an eigenvalue of Ai. There exists a non-negative (non-trivial) eigenvector of Ai belonging to the eigenvalue ri. Also, rik^ is the unique eigenvalue of Aik^ whose absolute value equals Xik^ and

where E(k,A) = {u G ^(V^) : Au = ku} for any linear map A : i(Vo) —• t(Vo). (2) Assume p £ Vo. If p is non-focal point, then Xi < r^. If p is a focal point, then X{ = r{ and dimE(rf (p) , A ^ ) = m(p) - 1. Remark. In the case (1) of this theorem, one may obtain that dim Efa, Ai) equals the number of orbits of the action of r. Specifically, let r be the permutation appearing in the definition of k(p). We write i ~ j for z, j G {1,... , m(p)} if and only if there exists / G N such that rl(i) = j . Then ~ is an equivalence relation on {1,... , m(p)}. Each equivalence class of ~ is called an orbit of the action of r. So, what we mean is dimEiruAi) = #({!,...

,m(p)}/~).

Corollary A.1.3. Let p be the fixed point of Fi. If p G VQ and p is a nonfocal point, then Xi = ri andri is the unique eigenvalue of Ai whose absolute value is Xi. Moreover, dimE(7^, A^) = 1 and there exists u G E{r^Ai) that satisfies u(q) > 0 if q ^ p and u(p) = 0. To prove the above theorem, we need several lemmas. In the following lemmas, we assume that p = Fi(p) G VQ. Lemma A. 1.4. For any m>0 and any u G £(Vo), (D(Ai - kl)mu)(p) = (n where I is the identity matrix.

k)m(Du)(p),

182

Additional Facts

Proof. Let u be a harmonic function on K. By the definitions of Hm and Au (Hmu)(p) = ( r , ) - m ( £ ( ^ ) m u ) ( p ) . Also (Hmu)(p) = (Du)(p) = — {du)p. Hence rirn(Du)(p) — (D(Ai)mu)(p). This immediately implies the lemma. • By this lemma, we can immediately obtain L e m m a A.1.5. If A^u — ku for u G ^(Vo) and (Du)(p) / 0, then k = Vi. Lemma A.1.6. Suppose that Aiu = ku for u G £(Vo) and that u(p) ^ 0. Then u is a constant on Vo and k = 1. Proof. As p is a fixed point of F^, we have (Aiu)(p) = u(p). Hence, by Proposition A. 1.1, if u(p) ^ 0 then k = 1 and u is a constant on V^. • Recall the notion of an i/ m -path defined in Definition 3.2.9. Lemma A. 1.7. Define Um}X = {y G Vo : there exists an H^-path between x and y} for x G Vm. Suppose Fw{q) G Vm\Vb for q G Vo and w G Wm< Then {Aw)qq^ > 0 for q* G Vo if and only if q* G UmjFw(qy Using Lemma 3.2.13, we see that if x G Vm\Vo, then Um^x = Vo D C, where C is the connected component of K\Vo containing q. Proof. Let C be the connected component of if \Vo containing Fw(q). Then Um,Fw{q) — CD Vo. Hence, by Theorem 3.2.14, if u is a harmonic function, then u(Fw(q)) is determined by u\umtFw(q)' Since (Awu)(q) = u{Fw(q)), we can verify the claim of this lemma. • We also assume that Fi (p) = p G Vo in the following three lemmas. Lemma A.1.8. Set Up = {q : q G Vo,Dpq > 0}. Then, for sufficiently large m, Um,q QUPL) {p} for any q G F^™(Vo). Proof. Choose m so that Fi m (Vb)nVb = {p}. Thenp G C/m?p. By Theorem 3.2.11 and Lemma 3.2.13, it follows that Um,p C Up. D By Theorem 3.2.11, we obtain the following. Lemma A. 1.9. p is a non-focal point if and only if for any q\,q<2, G Vo\{p}, there exists {p»}t=i,2,...,m C V0\{p} such that pi = qi, p m = q2 and DPiPi+1 ^ 0 for i = 1,2,... , m — 1.

A.I Second eigenvalue of Ai

183

Lemma A.1.10. If p is a non-focal point, then for sufficiently large m, Um,q = UpU {p} for any q G Fr(V0)\{p}. Proof Choose m so that F^™ (Vo) nV0 = {p}. If q G F^ (Vo) and q / p, then q G Vm\Vo. Hence, by Lemma 3.2.13, Um,q = VonC, where C is the connected component of K\VQ containing q. Since there exists an i7 m -path between q and p, Lemma 3.2.13 implies that p G C. Now, by Proposition 1.6.8-(3), #(C(p,V0)) = m(p) = 1. Therefore C(p,V0) = {C}. Hence, by Theorem 3.2.11, we see that Up U {p} = Vo n C. D Proof of Theorem A.1.2. First assume that p G Vo. Let U = {u G f (Vb) : u(p) = 0}. We identify U as ^(F 0 \{p}). Note that A{U C [/. Set ^ = -Ailc/. Then, by Lemma A. 1.6, the maximum of the absolute value of the eigenvalues of Bi equals A^. Now, B{ is a non-negative matrix (i.e., (Bi)pq > 0). Hence, by the Perron-Frobenius Theorem, A^ is an eigenvalue of B{ and there exists a non-negative (non-trivial) eigenvector v belonging to the eigenvalue A^. By Lemma A.1.7 and Lemma A.1.8, we see that (Ai™)^ = 0 for q* G Z p , where Zv = Vo\(Up U {p}). Hence, Aim is expressed by Aim= where Bm : 1{UP) -> 1{UP) and C m : £(UP) ->• ^(Z p ) are non-negative matrices. Hence,

Note that Cmu = ku for u G ^(J7P) and k ^ 0 if and only if Bf1!; = fev, where v G £(V^ U Zp) is defined by v|c/p = u and v\zp = ^Dmu. Now, if v is the non-negative (non-trivial) eigenvector of Bi belonging to the eigenvalue At, then v\Up ^ 0. Hence (Dv)(p) = ^2qeUp Dpqv(q) > 0. By Lemma A.1.5, we see that A; = r^. Next assume that p is a non-focal point. Then, by Lemma A. 1.10, (Aim)qq^ > 0 for q* e UpU {p} and q G V0\{p}. Therefore Cm and jDm in (A.1.1) are strongly positive matrices. Hence the Perron-Frobenius theorem implies that if k is an eigenvalue of Cm and |fc| = r^1, then k = r™ and dim E(rim, Ai171) = 1. Since this holds for sufficiently large ra, we deduce that if k is an eigenvalue of A^ with |fc| = r^, then k = Vi and

dim E(ri,Ai) = 1. Next suppose m(p) > 1. Let {Jj}j=i,2,...,m(p) D e the connected components of K\{p} and define /,- - J, n Vb, C^' = Upn Ij and Z^ - Zp n /j

184

Additional Facts

for any j = 1,2,... ,m(p). By Proposition 1.6.7, Ij ^ 0 for any j = 1,2,... , m(p). In fact, U£ ^ 0. By analogy with Lemma A.I.10, it follows for sufficiently large m. Hence, that £/fc(p)m,9 = U£u{p] for q G Fik^m(Ij) 0 and

Bmj=

0 where Bmj : £(Ij) -+ 1(1 j), Cmj : £(U£) -> £(U>) and L> mJ : £(U£) -> ^(Z^) are strongly positive for any j = 1,2,... , m(p). Applying the same discussion as in the non-focal case to Bmj, we see that if k is an eigenvalue of Bmj with \k\ = nk^m, then k = nk^m and dim E(rf^171, Bmj) = 1 for all j . Hence, if k is an eigenvalue of Bik^m with |fc| = nk^m, then k = rMp)™> a n d d i m £ l ( r ^ / c ^ m , 5 i f c ^ m ) = m(p). Since this holds for sufficiently large m, it follows that if k is an eigenvalue of Aik^ with |fc| = rik(p\ then jfe = r^) and dim^(ri f c ( p ) , Ai fe(p) ) = m(p). Next, suppose p £ yQ. Define VQ = Vb U {p} and V^ = UW£wmFw(Vo). Then V^ C V^+i for any m>0. There exists a Laplacian Hm on V^ such that

for any u G ^(V^). We write D = ^o- Then the Vb-harmonic function with boundary value p G ^(Vb) is the unique u G T with iz|y- = p that attains the following minimum: (w, u) : u G .T7, w|yo = p}. In this case, also, there exists A\ : £(VQ) —•> -f(Vb) such that

where we identify ^(F^(Vb)) with ^(Vb) through the one-to-one mapping F{. Note that a T^o-harmonic function w i s a harmonic function if and only if (Du)(p) = 0. Hence, for any k and n, £(fc, Ai n ) = {W|y0 : u G £7(fc, i , n ) , (Du)(p) = 0}.

(A.1.3)

Therefore, if A^ is the maximum of the second largest absolute value of the eigenvalues of A^ then A^ > A^. We can apply exactly the same arguments for A{ as for A^ and get analogous results. So if p is non-focal, then v\ is the unique eigenvalue of Ai whose absolute value is A^ and dim E(r^, Ai) = 1. Also, we can choose

A.2 General boundary conditions

185

a non-negative generator of E(ri,Ai) that satisfies (Du)(p) > 0. Hence, there is no eigenvalue of Ai whose absolute value equals r^. Hence A^ < r^. k p k Next, if p is a focal point, it follows that dim E(ri ^ \ Ai ^) = m(p). By k k (A.1.3), dimE{ri ^\Ai ^) = m(p)-l. This also implies that \ = r{. •

A.2 General boundary conditions In Chapters 3, 4 and 5, we always think of Vb as the boundary of the self-similar set K. This choice of a boundary is quite natural from the topological point of view; see, for example, Proposition 1.3.5. Prom the analytical point of view, however, we may choose any "small" subset of K as a boundary and wish to extend all the results in Chapters 3, 4 and 5 to this situation. In this section, we choose an arbitrary (finite) subset of V* as a boundary of K and present such extensions. One advantage of such extensions is that we can unify the cases of the Dirichlet boundary condition, which was represented by the symbol "£)", and the Neumann boundary condition, represented by the symbol "JV". More precisely, in the scheme of this section, if we choose Vb as the boundary, then we get the statements for the Dirichlet boundary condition, and if we choose the empty set 0 as the boundary, then we get the ones for the Neumann boundary condition. Let C = (K, 5, {Fi}ies) be a connected, p. c. f. self-similar structure and let (JD, r) be a harmonic structure on C. Also let (S^J7) be the resistance form associated with (D,r). Throughout this section, we assume that μ is a self-similar measure on K with weight (μ^ies and that μ^i < 1 for all i G S. Let d be a distance on K which is compatible with the original topology of K. Note that (K, d) is compact. Let B be a finite subset of V*. We will regard B as the boundary of K hereafter and define the Dirichlet form, the Laplacian and the heat kernel associated with the boundary B. Most of the results in this section are straightforward extensions of the corresponding ones in Chapters 3, 4 and 5. The proofs can be obtained by slight modifications of the original proofs. So we only present a sketch of the proofs in this section and leave the details to the reader. First we introduce the Dirichlet form associated with the boundary B. The following theorem is an extension of Theorem 3.4.6. Theorem A.2.1. Define TΒ = {U € J7 : U\B = 0} and SΒ — Then (£B,3~B) is a local regular Dirichlet form on L2(K\B^).

£\TB*FB-

The cor-

186

Additional Facts 2

responding non-negative self-adjoint operator Hβ on L (K\B^) pact resolvent.

has com-

2

Remark. Since μ(B) = 0, we may naturally identify L (K\B^) with 2 L (K, μ). In this manner, we think of Hβ as a self-adjoint operator on The proof of this theorem is similar to that of Theorem 3.4.6. If B = 0, then this theorem coincides with the Neumann boundary case, Theorem 3.4.6, while if B = Vo, then this theorem coincides with the Dirichlet boundary case, Corollary 3.4.7. Note that if B ^ 0, then Hβ is invertible and the inverse ( i J ^ ) " 1 is a compact operator on L2{K, μ). We write Gβ = (Hβ)"1. Next we introduce the Laplacian associated with the boundary B. Definition A.2.2. Define V^ = {u G C(K, d) : there exsits

max ^^^(HmU^p)

- (p(p)\ = 0}.

Furthermore, we define a linear operator Aβ^ : V^ —> C(K, d) by if, where y> G C(K,d) is the function appearing in the definition of V^. AB ? / 1 is called the Laplacian associated with B. Lemma A.2.3. Let p G Fw(Vo) for some w G Wm. Define (Hn,wf){p) = (irw)~1(Hn-rn(foFw))((Fw)~'1(p)) for any n > m. Assume that BC\KW C Fw(Vo). Then, for any feV*,

n

lim -(Hn,wf)(p) = -(HmtWf)(p)

/

^°° Note that, for any p G Kn,

(Hmf)(p) =

JKW

Y^

(Hm,wf)(p).

(A.2.2)

Hence, the quantity limn^oo —(iJ n>t/; /)(p), which is denoted by (df)WiP, is a kind of directional (Neumann) derivative of / at p. 1

Proof. Write h = f o Fw and q = (Fw)~ (p). that h G Vμ. Hence, by Lemma 3.7.5,

Then since / G D^, we see

lim -(Hkh)(q) = -(Dh)(q) / k+oo JK Since (Hkh)(q) = rw(Hk+m,wf)(p) for any k > 0, the above equation immediately implies (A.2.1). •

187

A.2 General boundary conditions

Choose m so that B C Vm. Then B Pi Kw C FW(VO) for any w G W m . Recall (A.2.2). Then, for any p G F* and any / G £>*,

lim -(Hnf)(p) =

]T

Definition A.2.4. For any / G X^ and any p G K , define

tof)p = lim -(Hnf)(p). n—>-oo

(df)p is called the Neumann derivative of / at p. By the above argument,

(df)p =

E

Also, we immediately obtain the following proposition. Proposition A.2.5. Assume that B C Vm and that p G Vm. Then, for any f G V*,

/

(df)p = -(Hmf)(p)

JK

It is easy to see that (df)p = 0 if p £ B. Therefore, for any p G V*\B, if

(A.2.3)

(Hmf)(p) = f rl>?ABi^. JK

1

Conversely, if (df)p = 0, (A.2.3) implies that lim^ooO^p)- (H^f)(p) )- Hence we have the following fact. Proposition A.2.6. If Bi CB2Q

=

Vm, then V^1 C V^2. More precisely,

Vμ1 ={ue V^ : (du)p = Ofor all p G B2\B1}. Ifue

V^1, then ABl^u

=

Next we present an extension of Theorem 3.7.8, the Gauss-Green's formula. Proposition A.2.7. V^ c T. Moreover, for any u G T and any v G V^, £(u,v) = V u(p)(dv)p - J UAB^μ.

T

(A.2.4)

J

The proof of this proposition is analogous to that of Theorem 3.7.8. We also obtain the counterpart of Theorem 3.7.9.

188

Additional Facts

Theorem A.2.8. Define VB^ by VB^ = {u e V^ : u\B = 0}. Suppose that B^®. Then VB^ = GB(C(K,d)) C Dom(HB) and AB^\vB^ = —HB\x>BtJi' Furthermore, HB is the Friedrichs extension of — AB^\x>B »Remark. If B = 0, then VB^ = V^N and HB = HN. So, by Theo= rem 3.7.9, we still see that AB^\vB^ ~HB\VB^ and that HB is the Priedrichs extension of —AB^\vB^The key fact in proving the above theorem is the following lemma corresponding to Lemma 3.7.10. Lemma A.2.9. Assume that B ^ 0. For any ip G C(K,d), GB(p G VB^ and AB^(GB(f) = - u) = (f,(f). If m > k and p G Vm\B, then ty™ G TB. Therefore we obtain — (Hmu)(p).

I

Hence it follows that

(Hmu)(p) = - [ V£V^.

(A.2.5)

JK

(This equation is analogous to that of Lemma 3.6.4.) Using a similar argument to that in the proof of Lemma 3.7.10, we see that

lim

max {(μ^p)-1 (H^u)^) +
m-+oopeVm\B

Let w G Wfc. Define uw = u o Fw and (pw =

,JK/ K

r m

for any m > 1 and any p G V \Vb. Therefore, using Theorem 3.7.16, we obtain • rwμwGμ(pw and that uw G Vμ C C(K,d). and AB^(GB(f) = -(p.

(A.2.6)

Hence -u G C(K,d) and therefore w G £>B,M •

Given this lemma, a similar argument as in the proof of Theorem 3.7.9 suffices to show Theorem A.2.8. Using (A.2.6), we may define the Green's function gB(x,y) as follows.

189

A.2 General boundary conditions P r o p o s i t i o n A . 2 . 1 0 . Assume into four

parts,

and

that

B ^ β

and that

B C Vk.

Divide

Hk

write TTk

H

=

/rp 1 (B \\JBT

tJ \ JB\

v XΒ) >

where TB : 1(B) -> 1(B), JB : £(B) -> i(Vk\B) 1 £(Vk\B). Set GB = —(Xβ)" and define

and XB

: l(Vk\B) -+

gB(x,y)= p,qeVk\B

where jgdF^-^x), 9w{x,y) = < [0

(F^'1

(y))

if both x and y e

K w,

otherwise,

where g is the Green's function defined in 3.5. Then, for if G C(Kyd), following four conditions are equivalent: (1) u e T>B^ and AB^u = tp, (2) u = -GBV, (3) u(x) = -JK 9B(X, y)v(y^ for any xeK, (4) u e TΒ and, for any m> k and any p e Vm\B,

the

(Hmu)(p) = [

JK

gs is called the Green's function associated with the boundary B. Remark. The original Green's function g equals gv0 • Using Lemma A.2.9 and Proposition A.2.10, we can prove Theorem A.2.8 by arguments similar to those in the proof of Theorem 3.7.9. Next, we discuss the eigenvalues and the eigenfunctions of A^μ. By Proposition A.2.10, we can verify the counterpart of Corollary 3.6.6, which was the key fact in proving Proposition 4.1.2. So, in the same manner, we may obtain the following statement corresponding to Proposition 4.1.2. Proposition A.2.11. For A € R, define EB(X) = {

(f e DOUI(HB)

: &B^U =

and HB¥> = Xip,

(2) (f e TΒ and £(ip,u) = -X(^p,u)μ for any u e TΒ(3) Combining this proposition with the fact that Hβ has compact resolvent, 2 we see that there exist a complete orthonormal system {iff }i>\ of L (K, μ)

190

Additional Facts B

and a sequence of non-negative numbers, {A^}i>i, such that \f < X +1 for every i > 1 , limn-^oo A^ = oo and ipf G Es(Xf) for every z > 1 . So, if we define the eigenvalue counting function of — A# ) M , pB (#,AO, by pB{x,p) = J ^ ^ dim ^ ( A ) , then pB(x^) = # { n : A^ < x}. Note that p0(x,/i) = PN(X^) and pVo(x,v) = pD(x^). Since TΒ £ ^ , Corollary 4.1.8 implies that PN(X^)

- #(B) < PB{X^)

<

PN(X^)'

Therefore, the asymptotic behavior of PB(X,/J>) as x —» oo turns out to be exactly the same as that of p^-(x,/x). More precisely, replacing pb(x,/i) by PB{X,P), we can immediately verify all the statements of Theorem 4.1.5. Finally, we discuss the heat kernel associated with the Laplacian A^μ. Definition A.2.12. We define the heat kernel associated with the Laplacian Aβ^μ by pB(t,X,y)

= n>1

Note that the Neumann heat kernel PN(t, x, y) equals p$(t, x, y) and the Dirichlet heat kernel PD{1, X, y) equals py0(£, x, y). Since we have the counterpart of Lemma 5.1.3 for Aβ j M , which is shown by Theorem 4.1.5, it follows that pB(t)X,y) is a non-negative, continuous function on (0, oo) x K x K, like the Neumann and Dirichlet heat kernels. Moreover, replacing Pb(t,x,y), Aμ and 2 \ M by pB(t,x,y), AB^ and VB^ respectively, we can verify Propositions 5.1.2 and 5.1.5. Also define the heat semigroup {TtB}t>o by

(Tfu)(x)=

/ JK

pB(t,x,y)u(yMdy)

for u G Ll{K, μ). Then we immediately have the counterpart of Theorem 5.1.7 by using exactly the same arguments as in the proof of the original theorem. Definition A.2.13. (1) Let u : (0, oo) x K -> R. For any t > 0, define ut : K -> R by ixt(x) = u(£,x) for any x e K. li ut e VB for all t > 0 , w(-,x) G C f l ((0, oo)) for all x G K and — - 7 — = (AB^ut){x) (A.2.7) at holds for any (£, x) G (0, oo) x if, then u is called a solution of the heat equation (A.2.7) (associated with the Laplacian Aβ^) on (0, oo).

A.2 General boundary conditions

191

(2) Let u : [0, oo) x K. For t > 0 , define Ut exactly the same as above, u is called an L°°-solution of the heat equation (A.2.7) (associated with the Laplacian A# )/z ) on [0, oo) if it is a solution of the heat equation (A.2.7) on (0, oo) and u is continuous on [0, oo) x K. Remark. Suppose that it is a solution of the heat equation (A.2.7) on (0, oo). Then u is an L°°-solution of the heat equation (A.2.7) if and only if \\ut - ^olloo —> 0 as t i 0. The next theorem is the counterpart of Corollary 5.1.9 and Theorem 5.2.5 describing the relation between the heat semigroup and solutions of the heat equation. Theorem A.2.14. (1) Let

and ut\B = 0 for any

t>0.

(2) Suppose (D, r) is a regular harmonic structure. Let

0 and uo = (f. From this theorem, the heat kernel pB(t,x,y) may be thought of as the fundamental solution of the heat equation (A.2.7). We can prove Theorem A.2.14-(l) by using the counterpart of Theorem 5.1.7. We need the following version of parabolic maximum principle to show Theorem A.2.14-(2). Theorem A.2.15. Let u : [0, oo) x K —> E be an L°°-solution of the heat equation (A.2.7) on [0, oo). Then, for any T > 0, max u(t, x) = max u(t,x) and min u(t,x) = min u(t, x), (t,x)euT {t,x)eduB (t,x)euT (t,x)edu* where UT = [0,T] x K and dU$ = {0} x KU [0,T] x B. The proof of this theorem is analogous to that of Theorem 5.2.2. The following comparison theorem for heat kernels is the counterpart of Theorem 5.2.9 and Theorem 5.2.13-(1). Theorem A.2.16. Let B\ and B2 be finite subsets ofV*. If B\ C B2, then 0 < PB2 (t, x, y) < pBl (t, x, y)

(A.2.8)

192

Additional Facts

for any (t,x,y) G (0, oo) x K x K. Moreover, if (D,r) is regular, then pBl(t,x,y)-pB2(t1x,y)

]T

/

qeB2\B1

J 0

=-

pB1(t-s,x, (A.2.9)

for any t>0

and any x, y G K\B2.

Remark. As in Theorem 5.2.13, the integral J0PB1(t — s,x,Q)(dpSB2J)qds actually means lim /

PBiit -

s,x,q)(dp8jg)qds.

If (D, r) is regular, then this theorem can be proven in an analogous way to 5.2 by using the parabolic maximum principle, Theorem A.2.15. Even if (D, r) is not regular, (A.2.8) is still true. However, to prove it, we must employ probabilistic arguments on the diffusion processes whose transition densities are psx and pB2 respectively. See Corollary A.3.4. Also, we can verify the following theorem, which is an extension of Corollary 5.2.14. Theorem A.2.17. Let B be a finite subset ofV*. Suppose that (D,T) is regular. Then, for any x G K\B,

\\p%x\\i = I pB(t,x,y^(dy) JK

= 1 - ( - J2 I (dpSBX)qds). Jo

(A.2.10)

qeB

Moreover, if B ^ 0, then /•o

J 0

Proof The arguments are essentially the same as in the proof of Theorem 5.2.13. To obtain (A.2.11), let B1 = 0 and B2 = B in (A.2.9). Then integrate it on K with respect to the measure μ. Using Lemma 5.2.10, we have (A.2.11). Next, if B ^ 0, then k e r # # = {0}. Therefore all the eigenvalues of Hβ are positive. Recalling Definition A.2.12, we see that limt^oo fK PΒ (t)X,y)fi>(dy) = 0. Then (A.2.11) is immediate from (A.2.10). • Next, we present an upper estimate for the asymptotic behavior of the heat kernel ps (t, x,y) as 11 0.

A.3 Probabilistic approach T h e o r e m A.2.18. There exists c>0 sup

for any

193

such that

pB(t,x,y)
t>0.

To prove this theorem, we first establish the counterpart of the Nash inequality, Theorem 5.3.3, and apply Theorem B.3.7 as in the proof of Theorem 5.3.1-(1). Theorem A.2.19. Let x,y G K\B and let t>0. Then pB(t,x,y) > 0 if and only if x and y belong to the same connected component of K\B. This theorem is the counterpart of Proposition 5.1.10. To prove this theorem, first establish the counterpart of Lemma 5.3.5 by using Theorem A.2.18. Then note the following lemma and proceed with the same argument as in 5.3. Lemma A.2.20. Let x,y e K\B. Then gB(x,y) > 0 if and only if x and y belong to the same connected component of K\B. Proof. We present only a brief sketch of the proof. It is enough to prove that J2p qevk\B(GB )pqipp(x)i}q{y) > 0 ii x and y belong to the same connected component of K\B. This can be shown by an argument similar to that in the proof of Proposition 3.5.5. • A.3 Probabilistic approach In this section, we will briefly present the probabilistic approach to the heat kernels, pB(t,x,y). In particular, we will give probabilistic proofs of Theorem 5.2.9 and Lemma 5.3.9. Recall that our analytical proofs only work when the harmonic structure is regular. Let C = (K, S, {Fi}ieS) be a p. c. f. self-similar set with S = {1,2,... , N} and let (D, r) be a harmonic structure on K. Also let μ be & self-similar measure on K with weight (/x^)^^. We assume that μ^i < 1 for all i G S. 7 Then, by Theorem 3.4.6, the resistance form (S^J ) derived from (D, r) is 2 a local regular Dirichlet form on L (K, μ). Set Q, = {u : u is a continuous function from [0, oo) to K}. Also, for t > 0 , define Xt : f£ —> K by Xt(u) = u(t). By the general theory of Dirichlet forms (see [43, Theorem 7.2.1] for example), there exists a diffusion process associated with the local regular Dirichlet form (£,.F). One may also find a concise exposition on Dirichlet forms and diffusion

194

Additional Facts

processes in [6, Section 4], in particular, [6, Theorem 4.8]. See also [6, Section 7]. Using those discussions, we can prove the following theorem. Theorem A.3.1. There exists a diffusion process (£2, {Xt} t > 0 , such that, for any x G K and any f £ C(K),

{PX}XEK)

Ex(f(Xt))= I PN(t,x,y)f(yWdy),

JK where p^ is the Neumann heat kernel defined in Definition 5.1.1. Ex{-) is the expectation with respect to the measure P^, where x is the starting point of the process. Remark. By [43, Theorem 7.2.1] and [43, Theorem 7.2.2], there exists a diffusion process associated with (E^J7). This diffusion process is, in general, defined only on q. e. starting points. However, since the heat kernel Piv(t, x, y) is continuous on (0, oo) x K x K, we can obtain the above theorem. Definition A.3.2. For any A C if, define the hitting time of A, GA>, by <jA(u) = inf{* >0:Xt{u)e

A}

for any u € Q. Theorem A.3.3. Let B be a finite subset ofV*. Then, for any Borel set A ofK, //

= PX(<JB

JΑ Α

>t,XteA),

where PB is the heat kernel associated with the boundary B defined in Definition A. 2.12. By this theorem, we immediately obtain the following fact, which is (A.2.8). Corollary A.3.4. Let B\ and B2 be finite subsets of V* with Bi C B2. Then, for any t>0 and any x,y G K, 0 < PB (t, x, y) < pBl (t, x, y). 2

If Bi = 0 and B2 = Vo, then the theorem above is Theorem 5.2.9. Proof. Since GBX > O~B2I & follows that PxiPBi >t,XteA)>

PX((JB2 >t,Xte

A)

for any £ > 0, any A and any x £ K. Hence Theorem A.3.3 implies the desired inequality. •

195

A. 3 Probabilistic approach Next we give a probabilistic proof of Lemma 5.3.9. Proof of Lemma 5.3.9. Let A = Uwe^xFw(dK0). t

lim / pA{t,x,y)μ(dy)

By Definition A.2.12, = 0.

^ooJK

This implies that Y^

x(XaA =q) = Px(aA < oo) = 1.

(A.3.1)

Recall that for any p G dU^x), x and p belong to different connected component of K\A. Therefore, any continuous path between p and x must intersect with A. Hence (A.3.2)

(TΑ
Y^

x{X*A = q> o-duA(x) < t)

PX{X«A

= q)Pq(adUA{x)

< t)

Px{XOA

= q)Pq(
t)

qeA

The equality in the fourth line is deduced by the Markov property of the process and (A.3.2). Again by (A.3.1), - / PA,x(t)X,y)μ(dy) < maxP q (a V ( A ) < t) JK

^

A

= max(l — Since fKPA,x(t,x,y)μ(dy) plies Lemma 5.3.9.

= fUA^x)PA1x(t,x,y)μ(dy),

this immediately imD

Comparing the proof above with the one in 5.3, we have the following. Theorem A.3.5. Suppose (D,r) is regular. Let B be a finite subset ofV*. Then, for any x € K, any q e B and any t>0, Px(XaB =q,aB
[

Jo

-{dp%x)qds.

Appendix B Mathematical Background

B.I Self-adjoint operators and quadratic forms In this section, we will introduce basic concepts and results on self-adjoint operators and non-negative quadratic forms on a Hilbert space. One can find detailed accounts and proofs of those subjects in Davies [26]. Also one can refer to classical textbooks, for example, Kato [81], Reed & Simon [156]. Let H be a (real or complex) separable Hilbert space with an inner product (•,•). A linear map H is called a linear operator on 'H if and only if the domain of if, denoted by Dom(if), is a dense subspace of TL and H : Dom(H) —> H. In this appendix, we assume that H is a real Hilbert space for simplicity. Definition B.I.I. Let if be a linear operator on H. (1) H is called symmetric if and only if (Hf,g) = (f,Hg) for any /, g G Dom(if). (2) if is said to be a self-adjoint operator if and only if if is symmetric and Dom(if) = {g E H : there exists h GH such that (Hf,g) = (/, A) for all / G Dom(if)}. (3) A symmetric operator H is called non-negative if (Hf, / ) > 0 for all / G Dom(if). Proposition B.1.2. Let H be a non-negative self-adjoint operator. Then there exists a unique non-negative self-adjoint operator G on H such that Dom(if) C Dorn(G), Dom(if) = {/ : / G Dom(G) and Gf G Dom(G)} andG2 = H. We write G = Definition B.1.3. Let if be a non-negative self-adjoint operator on H. 196

B.I Self-adjoint operators and quadratic forms

197

The associated quadratic form QH with domain Dom(i71/2) x Dom(H*/2) is defined by QH(f,g) = (H^f.H^g) for f,g G The following lemma is an immediate consequence of the definition of a self-adjoint operator and the definition of if 1 / 2 . Lemma B . I . 4 . Let H be a non-negative self-adjoint operator on 7i. Then Dom(#) = {g G Dom(H1/2) QHU,9)

: there exists heH

such that

= (/, h) for all f G Dom(if 1 / 2 )}.

In particular, in the above situation, h = Hg. Definition B.I.5. Q(-, •) is called a non-negative quadratic form on H if (QFl) Q : Dom(Q) x Dom(Q) -> R, where the domain of Q, Dom(Q) is a dense subspace of H. (QF2) Q is bilinear and symmetric: Q(af + bg, h) = aQ(/, h) + bQ(g, h) = aQ(h,

f) + bQ(h, g) for a n y f,g,heV

a n d a, b G R .

(QF3) Q is non-negative definite: Q(f, / ) > 0 for any / G V. Theorem B.1.6. Let Q be a non-negative quadratic form on 7i with dense domain Dom(Q). Then the following conditions (1) and (2) are equivalent. (1) Dom(Q) = Dom(i71/2) andQ = QH for some non-negative self-adjoint operator H onTi. (2) Define Q*(f,g) = Q(f,g) + (f,g) for any f,g G Dom(Q). Then (Dom((2),<2*) is a Hilbert space. A non-negative quadratic form Q on H is said to be a closed form if Q satisfies the conditions in Theorem B.1.6. Definition B.1.7. (1) Let Q and Qr be non-negative quadratic forms on H. Q is called an extension of Qf if and only if Dom(Q) D Dom(Q/) and Q\Dom(Q')xDom(Q') = Q'•

(2) A non-negative quadratic form on Ji is said to be closable if there exists a closed extension of Q. If a non-negative quadratic form Q is closable, then there exists a minimal closed extension of Q. Precisely, let ([/, Q*) be the completion of (Dom(<2),Q*). Then define Q(u,v) = Q*(u,v) — (u,v) for any u,v G U. Then Q, whose domain is C/, is the minimal closed extension of Q. This Q is called the closure of Q. Theorem B.1.8 (Friedrichs). Let H be a non-negative symmetric operator on H. Define a symmetric bilinear form Q : Dom(if) x Dom(if) —• R by Q(f,g) = (Hf,g) for f,g G Dom(F). Then Q is closable.

198

Mathematical Background

Let Q be the closure of the symmetric form Q in Theorem B.1.8. Then, by Theorem B.1.6, there exists a non-negative self-adjoint operator H on H such that Q = Qjj. It is known that H is an extension of H: Dom(iJ) C Dom(iJ) and H\Dom(H) = H- This if is called the Priedrichs extension of H. In fact, Dom(<3) is the completion of Dom(H) with respect to the inner product (J*. By this fact, it follows that H is the minimal self-adjoint extension of H: if H is a self-adjoint extension of if, then Dom(if) C Dom(if) and H\Dom(n) = H. Next we will introduce a non-negative self-adjoint operator with compact resolvent. For this purpose, we need the notion of compact operators. Definition B.1.9. Let (Bu || •||i) and (B 2 , II •II2) be Banach spaces. (1) A linear operator A : B\ —* B^ is called a compact operator if and only if A(U) is relatively compact in B2 for any bounded subset U C B\. (2) A linear operator A : B\ —> B2 is called a finite rank operator if A{B\) is a finite dimensional vector space. Remark. A compact operator is bounded and hence it is continuous. The following are important facts about compact operators. Proposition B.I. 10. A finite rank operator is a compact operator. Theorem B.I. 11. Let A : B\ —» B2 be a bounded operator. If there exists a sequence of compact operators {A n } m >i such that \\An — A\\ —• 0 as n —• 00, where 11 •11 is the operator norm, then A is a compact operator. Definition B.I. 12. A non-negative self-adjoint operator if on a Hilbert space Ti is said to have compact resolvent if the resolvent (H + I)~ x is a compact operator, where / is the identity map. Theorem B.I. 13. Let H be a non-negative self-adjoint operator on H. Then the following three conditions are equivalent. (1) H has compact resolvent. (2) There exists a complete orthonormal basis of 7i, {(pn}n>i such that = Xn(fn for all n > 0, 0 < Xi < •" < Xn < Xn+i < • • • and oo A n = 00.

(3) Define id : Dom(if 1 / 2 ) -> H by id(x) = x for all x e Dom(H^2). x 2 Then id is a compact operator from (Dom(if / ), (QH)*) —• (W, (•,•))> where (QH)* is the inner product defined in Theorem B.1.6. Remark. By Theorem B.1.6, (Dom(if 1/2 ),

(QH)*)

is a Hilbert space.

B.2 Semigroups

199

Theorem B.I.14 (Variational formula). Let H be a self-adjoint operator on H. Suppose that H is non-negative and has compact resolvent. Let L be any finite-dimensional sub space o / D o n ^ i / 1 / 2 ) . Define

where \\f\\ is the H-norm of f £ 7Y. Also let {A n } n >i be the set of eigenvalues of H appearing in Theorem B.I. 13. Then

Xn = inf{A(L) : L C Dom(iJ 1/2 ),dimL = n} = inf{A(L) : L C Dom(#), dimL = n}. B.2 Semigroups In this section, we will briefly introduce basic notions and results on one parameter semigroup of symmetric bounded operators on a Hilbert space. See Fukushima, Oshima and Takeda [43] for details and proofs. In the following, TL is a Hilbert space with the inner product (•,•). Definition B.2.1 (Semigroup). A family of bounded symmetric operators {Tt}t>o from Ti to itself is called a semigroup (of symmetric operators) on H if it satisfies the following two properties. Semigroup property: For any t, s > 0, T t + S = TtTs. Contraction property: (Ttu, Ttu) < (u, u) for any u G W and any t > 0 . Moreover, if a semigroup {Tt}t>o satisfies

lim(Ttu - u, Ttu - u) = 0 for any u G 7i, then {Tt}t>o is called a strongly continuous semigroup. Remark. In [186], the family {Tt}t>o, where the Tt are not necessarily symmetric, but with the semigroup property and strong continuity, is called a semigroup of class (Co). In addition, if {Tt}t>o has the contraction property, it is called a contraction semigroup of class (Co). Theorem B.2.2. Let {Tt}t>o be a strongly continuous semigroup on H. Set Dom(A) = {u e n : there exists f eH such that lim | | /

— — - | | = 0}

and define Au = limtio(Ttu — u)/t for any u G Dom(A). Then A is a densely defined non-positive self-adjoint operator on H and lmTt C Dom(j4) for any t > 0 . A is called the generator of the strongly continuous semigroup {Tt}t>o- Moreover, for a non-positive self-adjoint operator A

200

Mathematical Background

on H, there exists a unique strongly continuous semigroup {Tt}t>o whose generator is A. By the above theorem, we see that there is a one-to-one correspondence between strongly continuous semigroups and non-positive self-adjoint operators. Also, by Theorems B.I.6 and B.I.8, there is a one-to-one correspondence between non-negative self-adjoint operators and closed forms. Combining those two correspondences, we obtain a one-to-one correspondence between closed forms and strongly continuous semigroups as follows. {(£, F) : {£,T) is a closed form on H} \£

= QH

{H : H is a non-negative self-adjoint operator on W} I — H is a generator of {Tt}t>o{{Tt}t>o : {^}t>o is a strongly continuous semigroup on 7i}. Now let (5, T) be a closed form on a Hilbert space H. Let H be the self-adjoint operator associated with (£,T). Also let {Tt}t>o be the strongly continuous semigroup associated with (5,^*). We will assume these throughout the rest of this section. Lemma B.2.3. Let f and g belong to T)om(H). (1) -HTtf = -TtHf = \imh^o(Tt+hf - Ttf)/h for any t > 0, where the limit is the strong limit in Ji. (2) For any t > 0, £{Ttf,g) = £(f,Ttg). (3) £(Ttf,Ttf) is monotonically decreasing on (0, oo), and £(Ttf,Ttf) f £(/,/) as 110. Proof. (1) Set ft = Ttf. Then

(Tt+hf - Ttf)/h = Tt(Thf - f)/h = (Thft -

ft)/h.

Note that ft = Ttf €Dom(H) by Theorem B.2.2 and -H is the generator of {Tt}t>o- Letting t —> 0, the equality follows immediately. (2) Let (•,•)be the inner product of H. Then £(Ttf,g) = (Ttf,Hg) = (HTtf,g) = (TtHf,g) = (Hf,Ttg) = £(f,Ttg). (3) Let u(t) - £(Tt/2f,Tt/2f). Then u(t) - £{f,Ttf) = (Hf,Ttf). Hence u'(t) = -(Hf,HTtf) = ~{Hf,TtHf) = -(Tt/2Hf,Tt/2Hf) < 0. Therefore u(t) is monotonically decreasing. Since Tt is strongly continuous,

u(t) = (Hf, Ttf) - (Hf, f) = £(f, /) as U 0.

•

Let £*{u,v) = £(u,v) + (u,v). Then, recalling Theorem B.1.6, we see that (F, £*) is a Hilbert space.

B.2 Semigroups Proposition B.2.4. {Tt}t>o

is a strongly continuous

201 semigroup on

Proof. The semigroup property is obvious. Note that Dom(iJ) is dense in T with respect to £*. Let {/ n } n >i C Dom(iif). If fn —• f as n —• oo in £* for some / G T, then, by Lemma B.2.3-(3), {T t / n } n >i is an £*Cauchy sequence. Since {Ti/ n } n >i is an 7Y-Cauchy sequence as well and Ttfn —> Ttf as n —> oo in 7i, it follows that Ttfn —> Ttf as n —> oo in £*. Hence £(T t /, T t / ) < £ ( / , / ) for any / € .F.

If / € Dom(#), £ (Ttf - f,Ttf - f) = £(Ttf,Ttf) - 2E(Tt/2f9Tt/2f) +

£ ( / , / ) . Again by Lemma B.2.3-(3), we have £(Ttf - f,Ttf - f) - • 0 as t —> 0. If / G J7, an ordinary approximation argument by a sequence in Dom(i7) gives strong continuity. • Next we give a characterization of a strongly continuous semigroup {Tt}t>o as a solution of a formal differential equation on a Hilbert space H. Definition B.2.5. A map u : [0, oo) —•> *H is called a solution of

if and only if w : [0, oo) —> TL is strongly continuous (i.e., ||w(£ + h) — ix(t)|| -> 0 as ft -> 0 for any t > 0), u(t) G Dom(iJ) for any t > 0 and (w(t + ft) — u{t))/h converges to —Hu{t) as ft —> 0 with respect to the 7i-norm for any t > 0 . Theorem B.2.6. For any y? G H, there exists a unique solution u : [0, oo) —>H of (B.2.1) that satisfies u(0) = (p. Moreover, the unique solution u(t) is given by u(t) =Ttip. Proof. If u(t) — Tfip, it immediately follows that u(t) is a solution of (B.2.1) with u(0) — (p. Now suppose u(t) and v(t) are solutions of (B.2.1) with

u(p) = v(p) = p. Let f(t) = (u(t) - v(i),u(t) - v(t)). Then we see that f'(t) = -2£(u(t) - v(i),u(t) - v{t)) < 0 for t > 0. Hence f(t) is a non-negative monotone non-increasing function for t > 0 . Since /(O) = 0, f(t) = 0 for any t > 0 . Therefore u(t) = v(t) for any t > 0 . D

Corollary B.2.7. IftpE Dom(H) is an eigenfunction of H belonging to an eigenvalue A (i.e., Hip = \
Tt(p = e" A V

•

202

Mathematical Background

B.3 Dirichlet forms and the Nash inequality In this section, we introduce the notion of Dirichlet forms. One can find a more detailed and well organized account in [43]. We will also introduce the Nash inequality to study the asymptotic behavior of a strongly continuous semigroup associated with a Dirichlet form as t j 0. Let X be a locally compact metric space. Also let ^ b e a cr-finite Borel measure on X that satisfies μ(A) < oo for any compact set A and μ(O) > 0 for any non-empty open set O. Define C$(X) by Co(X) = {/ : X —»R, / is continuous and supp(/) is compact}. Definition B.3.1. Let £ be a closed form on L2{X^). Set T = Dom(£). (1) Markov property: We say that the unit contraction operates on £ if and only if u G T and £(u, u) < £(u, u) for any u G T, where u is defined in the same way as (DF3) in Definition 2.1.1. {£,T} is said to have the Markov property if and only if the unit contraction operates on £. (2) Core: A subspace C of T D CQ(X) is called a core of (£, J7) if and only if C is dense in T with respect to the £*-norm and in C$(X) with respect to the supremum norm. (3) Local property: (£,T) is said to have the local property if £(u,v) = 0 whenever u, v G T, supp(w) and supp(f) are compact and supp(it) fl supp(^) = 0. Definition B.3.2 (Dirichlet form). Let £ be a closed form on L2(X^μ). Set T = Dom(£). (£, T) is called a Dirichlet form on L2(X, μ) if and only if it has the Markov property. Moreover, a Dirichlet form is called regular if and only if it possesses a core. Also a Dirichlet form which has the local property is called a local Dirichlet form. If (£,T) possesses a core, TO Co(X) is also a core. Hence, a Dirichlet form (£, T) is regular if and only if T H Co(X) is a core. In the rest of this section, we assume that (£,F) is a Dirichlet form on L2(X^) and H is the associated non-negative self-adjoint operator. Then Theorem B.2.2 implies that there exists a corresponding strongly continuous semigroup {Tt}t>o whose generator is equal to —H. Since (£, T) satisfies the Markov property in addition to being a closed form, {Tt}t>o also has the following additional property. Definition B.3.3. A strongly continuous semigroup {X^} on L2(X, μ) is said to have the Markov property if and only if it satisfies the following condition: if u G L2(X, μ) and 0 < u(x) < 1 for μ-a. e.x G X, then 0 < (Ttu)(x) < 1 for μ-a.e.x e X and any t > 0 .

B.3

Dirichlet forms and the Nash inequality

203

Theorem B.3.4. There is a one-to-one correspondence between Dirichlet forms on L2(X, μ) and strongly continuous semigroups on L2(X, μ) with the Markov property. The correspondence is given by the natural one between closed forms and strongly continuous semigroups described in B.2. One can also find details on semigroups associated with Dirichlet forms (including a proof of the above theorem) in Davies [25]. Next we show that Tt can be extended to a bounded operator on LP(X, μ) for p = 1, oo. For //-measurable functions / and ^ on I , we write f > g if and only if f(x) > g(x) for μ-almost every x £ X. Also, if no confusion can occur, we use LP in place of LP(X, μ). Lemma B.3.5. Let f G L2. (1) Lf f>0, then Tt f>0. (2) Tt\f\>\Ttf\. Proof. (1) Let Xn = /" 1 ([0,n]) and let / n = / • Xxn, where \xn is the characteristic function of Xn. Then fn —» / as n —> oo in L 2 . Since 0 < fn < ni the Markov property of {Tt}t>o implies that 0 < Ttfn < n. Letting n —> oo, we obtain 0 < Tt f. (2) Since - | / | < / < |/|, this follows immediately from (1). • Proposition B.3.6. For p = l,oo, Tt(L2 D Lp) C Lp for any t > 0 and \\Ttf\\p < WfWp for any f G L2 n Lp and for any t > 0, where || • | | p is the Lp-norm. Proof. First, we will show t h e case p=1. {En}n>i

C B(X)

As /i is α-finite, there exists

such t h a t U n > i £ n = X , μ^n)

< oo a n d En C En+X

for

2

anyn > 1. Let / € L nL\ By Lemma B.3.5, (\Ttf\,XEn) (Tt\f\,XEn) = (\f\,TtXEn) ||/||i> where (•, •) is the L2-inner product. (The last inequality comes from the fact that 0 < TtXEn < 1.) Hence, we see that Ttf G L1

and llH/ll! < ll/lli.

Next let p = oo. By the Markov property, 0 < Tt\f\ < \\f\\oo for / G L2 n L°°. Hence Lemma B.3.5 implies that 0 < \Ttf\ < Tt\f\ < \\f\\ooTherefore Ttf G L°° and \\Ttf \\oo < ||/||oo• By the above proposition, for p = l,oo, Tt : L2 (1 Lp -> Lp naturally extends to a bounded operator T t (p) : Lp -> Lp with \\T^p)\\p < 1, where is the operator norm of a bounded operator A : L p —> L p . (If = oo, then T^°° is a bounded operator from L°° to L°°, where L 00 is the closure of L°°r\L2 in L°°.) Moreover, by the Riesz-Thorin interpolation theorem ([155, Theorem IX. 17], [25]), we can extend this to any p e [1, oo]. If no confusion can occur, we write Tt in place of T t . If A : LP —> Lq

204

Mathematical Background

is a bounded operator, we use H^H^-^ to denote the operator norm of A : D> - + L<*.

Now we consider the asymptotic behavior of Tt as t j 0 by using an idea originally due to Nash. Theorem B.3.7. The following four conditions are equivalent. (1) There exist positive constants c\ and 6 such that, for any f £ J7 (1 L1,

||/|| 2 +4/ * < ci(£(/,/) + ||/|| 2 2 )||/|ir. 1

2

1

(B.3.1)

2

(2) For any t>0, T^L ) C L and Tt : L -+ L is a bounded operator. Moreover, there exists c2 > 0 such that for any te (0,1]. (3) For any t > 0 , Tt(L2) c L°° and Tt : L2 Moreover, there exists c^ > 0 swc/i /or any t G (0,1]. (4) For any t > 0, T^L1) C L°° and Tt : L1 Moreover, there exists c± > 0 such that oo

(B.3.2) L°° Z5 a bounded operator. (B.3.3) L°° is a bounded operator. (B.3.4)

for any t G (0,1]. Remark. As ||T t || p < 1 for p = 2,oo, it follows that ||T t ||i_ p < ||Ti||i_ p for any t > 1 and that ||Tt||2-oo < H^ilb^oo for any t > 1. Hence (B.3.2), (B.3.3) and (B.3.4) are equivalent, respectively, to

and

for any t > 0 . (B.3.1) is called the Nash inequality. In [144], Nash used essentially the same argument as (1) => (4) to study the asymptotic behavior of the fundamental solution of a parabolic partial differential equation as t j 0. The present form of the theorem was that essentially obtained by Carlen, Kusuoka and Stroock [23]. See also Davies [25].

B.3 Dirichlet forms and the Nash inequality

205

To prove the theorem, we need one lemma about a dual operator. Lemma B.3.8. Let A : L2 —> L2 be a bounded self-adjoint operator. Then the following two conditions are equivalent. (1) There exists a bounded linear operator B from L1 to L2 such that 2 ML^HL2 = ^li^ni, 2 (2) A(L ) C L°° and A is a bounded operator from L2 to L°°. Moreover, if one of the above conditions holds, then |A|2—>oo = ||#||i-»2Proof. (1) => (2) : Since the dual spaces of L 1 and L2 are L°° and L2 respectively, the dual operator of B, B*, is a bounded operator from L2 to L°°. Hue L2 and v e L1 (1L2, then (v,Au) = (Av,u) = (Bv,u) = {v,B*u). Hence Au = B*u for any u G L2. Since ||i?||i_>2 = ||i?*||2->oo (see? f° r example, [186, VII. 1, Theorem 2']), we have ||2?||i—2 = H^lb^oo(2) => (1) Let A* be the dual operator of A : L2 -> L°°. As L1 C (L°°)*, we see that A*u G L2 for any u G L1. Set B = A*\Li. If u G L2 and ^ G L1 flL 2 then (u,Av) = (Au,v) = (u,Bv). Therefore, A| L i nL 2 = B\LinL2. Hence we obtain (1). • Proof of Theorem B.3.7. (1) => (2): Let f G L2 n L1 with \\f\\i = 1. Set u(t) = (Ttf,Ttf). I/ft + ft) - ?/(*)

^

+

Tt/, (T^ -

Then,

I)Ttf/h)

-+ -2(Ttf,HTtf) = -2£(Ttf,Ttf) as ft -> 0. Hence w;(t) = -2£(Ttf,Ttf). 2u(*)

1+2

Now, by the Nash inequality,

/* < ciC-fi'W + 2ti(t))||Tt/||J / ' < ai-u'tf)

+2

where we used the fact that | | T t / | | i < | | / | | i = l. This implies Set v(t) = (e- 2t it(t))- 2 /^. Then we obtain v'(t) > 4/(ci<9). Since v(t) w (0)-2/^ > 0 as 11 0, it follows that v(t) > 4t/(ci0). Therefore where c = (ci^/4)^/ 2. Hence for any f £ L2 HL1. This immediately implies (2). (2) <^> (3): Apply Lemma B.3.8.

206

Mathematical Background

(2) =» (4) As (2) & (3), we see that Tt/2 : L1 -+ L2 and Tt/2 : L2 -> L°° are bounded operators. Hence Tt = Ttj2 o Tt/2 : L1 —• L°° is a bounded operator and (4) => (1): Assume that T* : L 1 —• L°° is a bounded operator and (B.3.4) holds for any t G (0,1]. By the remark immediately after Theorem B.3.7, it follows that ||T t ||i_oo < cAeH~ef2 for any t > 0 . Let / G F C\ L1. Then, if t > e > 0, we have

{e-*Ttf,f) = (e-'Tef,f) + I = (e-*TJ, f)-J

(-(e-sTsf),f)ds e-s((I + H)TJ, f)ds.

Now, using Lemma B.2.3 and Proposition B.2.4, e- s ((7 + H)T.f, / ) = e-°(Ts/2f, Ts/2f) + e~s{Ts/2HTs/2f, f) = e-s(Ts/2f,Ts/2f) + e-s£(Ts/2f,Ts/2f) This, along with the fact that (Ttf,f)

< HTtHi^ooH/Hf, implies

c 4 | | / | | ^ / 2 > {e-'Tef, / ) - (t - e)(||/||l + £(f, /)). Letting e —> 0, we obtain It is easy to calculate the value of t* > 0 which gives the minimum value in the left-hand side of the above inequality. Then, substituting t* for t in the above inequality, we immediately obtain (B.3.1). • Prom the arguments in the proof, in particular, for (1) => (2), we see that the condition (1) can be replaced by the weaker condition (1)7 There exits a positive constant c\ such that (B.3.1) holds for any / G Dom(H) n L1 with / > 0. In practice, however, this condition may be no easier to verify than the original one. Corollary B.3.9. Suppose that the Nash inequality (B.3.1) is satisfied. Let

1. Then (B.3.5)

where c> 0 is a constant which is independent of tp and A.

B.4 The renewal theorem

207

Remark. Under the assumptions of the above corollary, if

IM| 2
If t = I/A and c = C2e, then we get (B.3.5).

•

B.4 The renewal theorem In this section, we will introduce the renewal theorem. One can find the renewal theorems (in various versions) in many places in the literature, for example, Feller [36], Rudin [159]. Definition B.4.1. A function u : R —> R is directly Riemann integrable on R if and only if oo

a(h) = h ^2

oo

SU

P

W

W

an

d
inf

u{t)

are finite for any h>0 and tend to the same limit as h —> 0. The following is essentially the renewal theorem (alternative form) in section XI. 1 p. 363 of [36]. We only present the case where the distribution is atomic.

Theorem B.4.2 (Feller's renewal theorem). Let £* > 0. Let f be a measurable function on R such that f(t) — 0 for t < t*. Suppose f satisfies a renewal equation N

where a i , # 2 , . . . , OLN are positive numbers, X)j=i Pj ~ ^ and Pj ^ ^ for a^ j and u is a non-negative directly Riemann integrable function on R with u(t) =0 fort 0 such that OLi = rriiT for all i, where rai,m2,... ,WIAT are positive integers whose

208

Mathematical Background

greatest common divider is 1. Then \f(t) — G(t)\ —> 0 as t —> oo, where G(t) is a T-periodic function given by N

oo

j=1

j=-oo

(2) Non-arithmetic case (Non-lattice case): Suppose that J2iLi^ai ^ dense additive subgroup ofR. Then f(t) is convergent as t —> oo and

a

N

u(t)dt. Feller only stated the case where t* = 0. Using a parallel translation, we can immediately obtain the present version. Also, in Feller's original version, the conclusion for the arithmetic case is that

lim f(t + nT) = G{t) 71—KX)

for any t. However, in equation (1.18), p.362 of [36], Feller gives the following expression for f(t) :

k>0

where Vk —> (X)j=i rn3lPj)~1 as k ~> °°- Since u is directly Riemann integrable, a standard argument in calculus implies that \f(t) — G(t)\ —• 0 as t —> oo. This version of the renewal theorem suffices to prove Theorem 4.1.5 on the asymptotic behavior of eigenvalue counting functions of Laplacians on p. c. f. self-similar sets. (The conditions of the renewal theorem in [93] were unfortunately incorrect. The author thanks D. Vassiliev for having pointed this out.) Since Feller's version of the renewal theorem, much effort has been made to weaken the conditions, in particular, the condition that f(t) = u(t) = 0 for t oo, instead of assuming that u(t) = 0 for t < t* and that u is directly Riemann integrable. The following renewal theorem for the arithmetic case contains information on the order of convergence of \f(t) — G(t)\ as t —> 0. This kind of error estimate allows us to deduce detailed information about the behavior of the eigenvalue counting function, (4.1.3).

209

B.4 The renewal theorem

Theorem B.4.3. Let f be a measurable function on R with f(t) —• 0 as t —» —oo. Suppose f satisfies a renewal equation (B.4.1), where there exists T > 0

such that oci = rn^T /or a// i, where m i , r a 2 , . . . , wijv are

positive integers whose greatest common divisor is 1. 5e£ Wj(t) = ix(£ + jT) /or* G [0,T]. //Zlj^ool^WI converges uniformly on [0,T], then \f(t) — G(t)\ —• 0 as £ —> oo, wfoere G(t) zs a T-periodic function given 1 by (B.4.2). Moreover, set Q(z) = (1 - E ^ L i f t ^ O A - *)• ^fao de/me /? = min{|2:| : Q(z) = 0} and m = mdiK{multiplicity of Q(z) = 0 at w : |w| = /3? Q(iu) = 0}. If there exist C > 0 and a > 1 s^c/i tfta* |it(t)| < C a ~ f /or a//1, then, as t —> oo,

{ )

ifaT>p,

\G(t)-f(t)\ = { Remark. If Q(;z) = 1, then we set P = +oo. Since J2f=i Pj = 1? Q( z ) i s a polynomial. Furthermore, | ^f-iPjZ™^ < 1 for {z : \z\ < l,z ^ 1}. Hence we see that /? > 1. In the rest of this section, we will give a proof of Theorem B.4.3. Lemma B.4.4. Set F(z) = H* = 1 (l - e^z)'1, where 0 < 6j < 2TT for a n 1 j = 1,2,... ,fc. J/F(z) = E r = o n ^ . ^en |a n | = G(71^ ) as n ^> 00, where m = max{#{j : 0 = Oj} : 0 < 6 < 2n}. Proof We use induction on k. The conclusion is obvious when k = 1. Assume that the conclusion holds for k. For F(z) = Iljiii U ~ e^z)'1 = E^Lo a n^ n 5 if ^' = ^ f° r ^1 i then it is easy to see that \an\ = O(nk). If 0p 7^ fl? for some p + q, then (1 - eie*z)-\l - eie*z)"1 = a(l - e ^ z ) ' 1 + b^ — e^vz)'1 for some a, 6. Hence the statement follows from the induction hypothesis. • Lemma B.4.5. For w = W\W2 • • .Wk G Wk, set m(w) = X^7=i rnwj Pw =pWl"

'Pwk. Define M(k) = T,weW^m(w)=kPw

Then

N 1

- M(n)\ = 0(H^

1

n

/T )

as n —+ 00; where β and m are defined in the statement of Theorem B.4..3. Remark. If {w G W* : m(w) = k} = 0, then we set M(k) = 0. Hence = 0 for any k<0. Also we set M(0) = 1.

210

Mathematical Background

Proof. Using the fact that M(k) = X^li M(k — rrij)pj, we see that oo

N

n=0

j=1

Note that ( E ^ m^) = Q(l). Then we have that E ^ o ^ - ^ W ) * 7 1 = R(z)/Q(z), where M = Q(l)" 1 and R(z) is a polynomial defined by R(z) = (Q(z)/Q(l) - 1)/(1 - 2). If Q(s) = a\[f^Q\z - Zj), it follows that

f^(M-M(n))zn = R(z)/Q(z) = (f^cnzn) J] (z-^)" 1 , n=0

j=0

j:M=/3

where the radius of convergence of ^nt=o c^zn ls greater than p. Hence, applying Lemma B.4.4 to Ylj^.^piz — Zj^1, we obtain the required estimate 3 D of|M-M(n)|. Proof of Theorem B.J^.S. By the renewal equation (B.4.1), we have u(t-m{w)T)pw. k=owewk

Since lim^-^-cx) f{t) = 0 and X ^ o u (^ ~ n ^ ) ^s absolutely convergent, we have

n=0

Hence we obtain oo

G(t) - f{t) = MY^u{t + kT) + Y^u{t - kT)(M - M(k)).

(B.4.3)

fe=0

As M < 1 and M(k) < 1,

\G(t) - fn(t)\ < 2 Y^ \uk(t)\ + f ; K-fcWIlMW - M\, k>n—m

k=m

where fn(t) = f(t+nT) on [0, T]. For e > 0 , choose m so that |M(fe)-M| < e for k > m. Then for sufficiently large n, we have E f c > n _ m |^fc(^)| < e. Hence \fn(t)-G(t)\ < (2+A)e, where A = supo < t < T E ^ - o o l^fcWI- H e n c e fn is uniformly convergent t o G a s n ^ o o o n [0,T]. So /(£) — G(t) —> 0 as t —> oo.

B.4 The renewal theorem

211

Now suppose \u(t)\ < Ca~l. Then, by (B.4.3) and Lemma B.4.5, it follows that

\G(t) - f(t)\ < da'1 + c2arl Y^ km-\aTIP)k + c3 J^ ^ ^ ^ O
<

' 1 + c2a~t

Y^

k>t/T

krn-1(aT/p)k

+

where ci,C2,C3 and C4 are positive constants. Prom this inequality, it is easy to deduce the required estimate. In particular, if aT > /?, we use the fact that there exists c > 0 such that a~nT J2 km-1(aT/P)k

0
for any n > 1 .

•

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221

Index of Notation

α, 103

Km,*, 20

Br(x), 8 X^, see characteristic function X(7, see characteristic function C(K), 69

), 42 ), 42 £(A), 23

Cc, 19

m(p), 36 ro(p, Vb), 36 Vp, 82

), 53 108 M), 152 /i), 152 ), 42 ), 42 diam(-), 14 diniH, see Hausdorff dimension ds, 134 ds(K,R), 138 , 138 3 2

I A), 181 ) , 132 , 94 Gμ, 102 <7X, 99 ) , 152 ), 152 Ws, see Hausdorff measure [H]u, 45 £(F), 41 J(p), 36 ,Vo), 36 ), 23 ).

M{K), 89

,i/, 44

-R£(X), 56 7 2 ^ ( X ) , 56 p6(x,/L*), 133 •RM(X), 56 ), 56 ), 90 i?^(M), 102 ^ r , 70 S w , 13 K,22 ), 19 V(A), 22 y(A,£), 22 F m , 22

W., 18 W m , 18

Index

222

Index

affine nested fractal, 116,117,150 arcwise connected, 33 arithmetic case, 207

contraction ratio, 9 core, 202 critical set, 19

Bernoulli, 25 Borel measure, 25 regular measure, 25, 26 set, 25 α-algebra, 25 boundary condition, 92, 93 Dirichlet, see Dirichlet boundary condition Neumann, see Neumann boundary condition boundary of self-similar set, 19

D-eigenfunction, see Dirichlet eigenfunction D-eigenvalue, see Dirichlet eigenvalue A-Y transform, 48, 67, 129 diameter, 14 diffusion process, 66, 193 directly Riemann integrable, 207 Dirichlet boundary condition, 93, 158 Dirichlet eigenfunction, 132 Dirichlet eigenvalue, 132 Dirichlet form, 65, 88, 92, 185, 202 local, see local Dirichlet form on a finite set, 42 regular, see regular Dirichlet form Dirichlet Laplacian, 93, 95, 111 Dirichlet problem for Poisson's equation, 114

Cantor set, 15, 39, 66 topological, 13, 19 characteristic function, 42, 203 closable, 197 closed form, 197 compact operator, 92, 94, 198 compact resolvent, 64, 92, 93, 133, 198 compatible sequence, 51, 65, 69 complete measure, 25 connected; 33 arcwise, see arcwise connected locally, see locally connected contraction, 9 contraction principle, 9, 10 contraction property, 199

effective resistance, 46, 51, 65 effective resistance metric, 48, 66, 83, 138 eigenfunction Dirichlet, see Dirichlet eigenfunction

Index localized, see localized eigenfunction Neumann, see Neumann eigenfunction fire-localized, see pre-localized eigenfunction eigenvalue cdunting function, 133, 190 decimation method, 135,156 Dirichlet, see Dirichlet eigenvalue Neumann, see Neumann eigenvalue equilibrium point, 9 extension Friedrichs, see Priedrichs extension of a form, 197 finite rank operator, 92, 198 finitely ramified, 23 fixed point, 9, 14, 70 focal point, 180 Friedrichs extension, 111, 188,198 Frostman's lemma, 29 fundamental solution, 162 7-elliptic measure, 89 Gauss-Green's formula, 110,145, 187 generator, 199 Green's function, 95, 98,102,188, 189 Green's operator, 94, 95,102,103, 145 extended, 102 harmonic function, 45, 52, 73, 75, 143 harmonic structure, 69, 73, 138

223 regular, see regular harmonic structure Harnack inequality, 46, 78, 100 Hata's tree-like set, 16, 24, 71, 129 Hausdorff dimension, 28, 29 measure, 28, 138 metric, 10 heat equation, 162, 164, 190 kernel, 158, 171, 190 semigroup, 190 heat kernel, 152 hitting time, 194 inequality Harnack, see Harnack inequality Nash, see Nash inequality integrated density of states, 135, 155 isomorphism between self-similar structures, 18 Koch curve, 15, 17 A-harmonic function, 139 Laplacian, 65, 66, 92, 107, 108, 186 Dirichlet, see Dirichlet Laplacian Neumann, see Neumann Laplacian on a finite set, 42 standard, see standard Laplacian lattice case, 134, 142, 207 L°°-solution of the heat equation on [0,oo), 164

224

Index

Lipschitz, 9 Lipschitz constant, 9 local Dirichlet form, 65, 88, 202 local property, 93, 202 localized eigenfunction, 141 locally connected, 33 ra-harmonic, 81 Markov property, 42, 56, 93, 162, 202 of a semigroup, 202 mass distribution principle, 29 maximum principle, 45, 52, 76, 77, 81 parabolic, see parabolic maximum principle strong version, 81 weak version, 77 measure Bernoulli, 25 Borel, see Borel measure Borel regular, see Borel regular measure complete, see complete measure Hausdorff, see Hausdorff measure probability, see probability measure self-similar, see self-similar measure minimal, 21 modified Sierpinski gasket, 129, 129 Moran's theorem, 32 multi-harmonic function, 143 N-eigenfunction, see Neumann eigenfunction

N-eigenvalue, see Neumann eigenvalue Nash inequality, 153, 172, 202, 204 nested fractal, 117, 146 affine, see affine nested fractal nesting condition, 117 net, 10 Neumann boundary condition, 92, 158 Neumann derivative, 110 Neumann eigenfunction, 132 Neumann eigenvalue, 132 Neumann Laplacian, 92, 111 Neumann problem for Poissons equation, 129 non-arithmetic case, 208 non-lattice case, 134, 208 open set condition, 30, 32 operator compact, see compact operator finite rank, seefiniterank operator Green's, see Green's operator renormalization, see renormalization operator self-adjoint, see self-adjoint operator symmetric, see symmetric operator p. c. f., see post critically finite parabolic maximum principle, 164 partition, 22, 23, 83, 139, 173 pentakun, 119 piecewise harmonic function, 81 Poisson's equation, 114, 129

Index post critical set, 19, 72 post critically finite, 23, 34 pre-localized eigenfunction, 141, 146, 153 probability measure, 25 quadratic form associated with a self-adjoint operator, 197 non-negative, 197 r-network, see resistance network random walk, 54 refinement of partition, 22 regular Dirichlet form, 55, 65, 88, 202 regular harmonic structure, 69, 75, 167 renewal equation, 137, 207 theorem, 135, 137, 207 renormalization operator, 70 resistance form, 55, 61 metric, 56, 61, 70, 75 network, 43, 69 self-adjoint operator, 196 self-affine set, 10 self-similar measure, 25, 26, 105 set, 10, 13 structure, 18, 25, 26 self-similarity of a form, 83 semigroup, 160, 199 of symmetric operators, 199 property, 199 strong continuous, see strong continuous semigroup shift

225

map, 13 space, 12, 18 Sierpinski carpet, 24 gasket, 15, 24, 71, 108, 128, 135, 137, 149, 155 similarity dimension of harmonic structure, 138 similitude, 9, 13 snowflake, 120 solution of the heat equation L°°-, see L°°-solution of the heat equation on (0,oo), 162 spectral dimension, 135, 137, 138, 155 exponent, 134, 137 standard harmonic structure, 71, 77 standard Laplacian, 109, 135, 149 strong continuous semigroup, 199 strongly symmetric, 116 symbol, 12, 18, 20 symmetric operator, 196 strongly, see strongly symmetric weakly, see weakly symmetric symmetry group, 146 of a harmonic structre, 115 theorem renewal, see renewal theorem Weyl's, see Weyl's theorem totally bounded, 10 unit contraction, 202 variational formula, 136, 140,198

226 Vicsek set, 147 weakly symmetric, 37 Weyl's theorem, 133 word, 12 empty, 12 length of, 12

Index

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