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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