Preface |
|
ix | (2) |
Introduction |
|
xi | (5) |
Notes and references |
|
xvi | |
|
Chapter 1 Mathematical background |
|
|
1 | (18) |
|
|
1 | (2) |
|
1.2 Some useful inequalities |
|
|
3 | (3) |
|
|
6 | (7) |
|
1.4 Weak convergence of measures |
|
|
13 | (3) |
|
|
16 | (1) |
|
|
16 | (3) |
|
Chapter 2 Review of fractal geometry |
|
|
19 | (22) |
|
|
19 | (10) |
|
2.2 Review of iterated function systems |
|
|
29 | (10) |
|
|
39 | (1) |
|
|
39 | (2) |
|
Chapter 3 Some techniques for studying dimension |
|
|
41 | (18) |
|
|
41 | (10) |
|
3.2 Box-counting dimensions of cut-out sets |
|
|
51 | (5) |
|
|
56 | (1) |
|
|
57 | (2) |
|
Chapter 4 Cookie-cutters and bounded distortion |
|
|
59 | (12) |
|
|
59 | (3) |
|
4.2 Bounded distortion for cookie-cutters |
|
|
62 | (7) |
|
|
69 | (1) |
|
|
69 | (2) |
|
Chapter 5 The thermodynamic formalism |
|
|
71 | (26) |
|
5.1 Pressure and Gibbs measures |
|
|
71 | (4) |
|
5.2 The dimension formula |
|
|
75 | (4) |
|
5.3 Invariant measures and the transfer operator |
|
|
79 | (5) |
|
5.4 Entropy and the variational principle |
|
|
84 | (4) |
|
|
88 | (4) |
|
5.6 Why `thermodynamic' formalism? |
|
|
92 | (2) |
|
|
94 | (1) |
|
|
95 | (2) |
|
Chapter 6 The ergodic theorem and fractals |
|
|
97 | (16) |
|
|
97 | (5) |
|
6.2 Densities and average densities |
|
|
102 | (9) |
|
|
111 | (1) |
|
|
112 | (1) |
|
Chapter 7 The renewal theorem and fractals |
|
|
113 | (16) |
|
|
113 | (10) |
|
7.2 Applications to fractals |
|
|
123 | (5) |
|
|
128 | (1) |
|
|
128 | (1) |
|
Chapter 8 Martingales and fractals |
|
|
129 | (20) |
|
8.1 Martingales and the convergence theorem |
|
|
129 | (7) |
|
|
136 | (7) |
|
8.3 Bi-Lipschitz equivalence of fractals |
|
|
143 | (3) |
|
|
146 | (1) |
|
|
146 | (3) |
|
Chapter 9 Tangent measures |
|
|
149 | (20) |
|
9.1 Definitions and basic properties |
|
|
149 | (6) |
|
9.2 Tangent measures and densities |
|
|
155 | (8) |
|
|
163 | (4) |
|
|
167 | (1) |
|
|
167 | (2) |
|
Chapter 10 Dimensions of measures |
|
|
169 | (16) |
|
10.1 Local dimensions and dimensions of measures |
|
|
169 | (8) |
|
10.2 Dimension decomposition of measures |
|
|
177 | (7) |
|
10.3 Notes and references |
|
|
184 | (1) |
|
|
184 | (1) |
|
Chapter 11 Some multifractal analysis |
|
|
185 | (22) |
|
11.1 Fine and coarse multifractal theories |
|
|
186 | (6) |
|
11.2 Multifractal analysis of self-similar measures |
|
|
192 | (9) |
|
11.3 Multifractal analysis of Gibbs measures on cookie-cutter sets |
|
|
201 | (3) |
|
11.4 Notes and references |
|
|
204 | (1) |
|
|
205 | (2) |
|
Chapter 12 Fractals and differential equations |
|
|
207 | (40) |
|
12.1 The dimension of attractors |
|
|
207 | (16) |
|
12.2 Eigenvalues of the Laplacian on regions with fractal boundary |
|
|
223 | (7) |
|
12.3 The heat equation on regions with fractal boundary |
|
|
230 | (6) |
|
12.4 Differential equations on fractal domains |
|
|
236 | (8) |
|
12.5 Notes and references |
|
|
244 | (1) |
|
|
245 | (2) |
References |
|
247 | (6) |
Index |
|
253 | |