分形分析（英文版） (2010 年度畅销榜NO.23463 )

本书讨论了分形分析，这是一个发展很快的数学分支，主要研究分形的动态特性，如分形的热扩散及分形结构材料的振动。本书从基本的自相似集合的几何学原理开始，逐步对新近的研究成果加以讨论，其中包括拉普拉斯算子的特征值和特征函数的性质、自相似集合热核的渐近特性等。
阅读本书只需具备高等分析学、普通拓扑学和测度论的基础知识。本书特别适合分析和概率论专业的研究生和研究人员阅读，同时本书也非常适合作为与分形有关的研究生课程的补充教材。

1 Geometry of Self-Similar Sets 8
1.1 Construction of self-similar sets 8
1.2 Shift space and self-similar sets 12
1.3 Self-similar structure 17
1.4 Self-similar measure 25
1.5 Dimension Of self-similar sets 28
1.6 Connectivity of self-similar sets 33
Notes and references 38
Exercises 39
2 Analysis on Limits of Networks 41
2.1 Dirichlet forms and Laplacians on a finite set 41
2.2 Sequence of discrete Laplacians 51
2.3 Resistance form and resistance metric 55
2.4 Dirichlet forms and Laplacians on limits of networks 63
Notes and references 66
Exercises 66
3 Construction of Laplacians on P. C. F. Self-Similar
Structures 68
3.1 Harmonic structures 69
3.2 Harmonic functions 73
3.3 Topology given by effective resistance 83
3.4 Dirichlet forms on p. c. f. self-similar sets 88
3.5 Green's function 94
3.6 Green's operator 102
3.7 Laplacians 107
3.8 Nested fractals 115
Notes and references 127
Exercises
4 Eigenvalues and Eigenfunctions of Laplacians
4.1 Eigenvalues and eigenfunctions
4.2 Relation between dimensions
4.3 Localized eigenfunctions
4.4 Existence of localized eigenfunctions
4.5 Estimate of eigenfunctions
Notes and references 155
5 Heat Kernels 157
5.1 Construction of heat kernels 158
5.2 Parabolic maximum principle 164
5.3 Asymptotic behavior of the heat kernels 171
Notes and references 178
Appendix 180
A Additional Facts 180
A.1 Second eigenvalue of Ai 180
A.2 General boundary conditions 185
A.3 Probabilistic approach 193
B Mathematical Background 196
B.1 Self-adjoint operators and quadratic forms 196
B.2 Semigroups 199
B.3 Dirichlet forms and the Nash inequality 202
B.4 The renewal theorem 207
Bibliography 212
Index of Notation 221
Index 222

Jun Kigami为日本京都大学应用分析与复杂动力系统系教授。除本书外，他还与人合著有《MathematicsOf Fractals》》一书。