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Tables of Contents for Analysis on Fractals

Chapter/Section Title

Page #

Page Count

Introduction

1

7

Geometry of Self-Similar Sets

8

33

Construction of self-similar sets

8

4

Shift space and self-similar sets

12

5

Self-similar structure

17

8

Self-similar measure

25

3

Dimension of self-similar sets

28

5

Connectivity of self-similar sets

33

8

Notes and references

38

1

Exercises

39

2

Analysis on Limits of Networks

41

27

Dirichlet forms and Laplacians on a finite set

41

10

Sequence of discrete Laplacians

51

4

Resistance form and resistance metric

55

8

Dirichlet forms and Laplacians on limits of networks

63

5

Notes and references

66

1

Exercises

66

2

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

68

63

Harmonic structures

69

4

Harmonic functions

73

10

Topology given by effective resistance

83

5

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

88

6

Green's function

94

8

Green's operator

102

5

Laplacians

107

8

Nested fractals

115

16

Notes and references

127

1

Exercises

128

3

Eigenvalues and Eigenfunctions of Laplacians

131

26

Eigenvalues and eigenfunctions

132

5

Relation between dimensions

137

4

Localized eigenfunctions

141

5

Existence of localized eigenfunctions

146

6

Estimate of eigenfunctions

152

5

Notes and references

155

2

Heat Kernels

157

23

Construction of heat kernels

158

6

Parabolic maximum principle

164

7

Asymptotic behavior of the heat kernels

171

9

Notes and references

178

2

Appendix

180

32

A Additional Facts

180

16

A.1 Second eigenvalue of Ai

180

5

A.2 General boundary conditions

185

8

A.3 Probabilistic approach

193

3

B Mathematical Background

196

16

B.1 Self-adjoint operators and quadratic forms

196

3

B.2 Semigroups

199

3

B.3 Dirichlet forms and the Nash inequality

202

5

B.4 The renewal theorem

207

5

Bibliography

212

9

Index of Notation

221

1

Index

222