内容

目次

Introduction vii Chapter 1. Measure, Energy, and Metric 1 1.1 GraphApproximations 1 1.2 Self-similar Measures 5 1.3 Graph Energies 10 1.4Energy 18 1.5 Electric Network Interpretation 23 1.6 Effective ResistanceMetric 27 1.7 Notes and References 29 Chapter 2. Laplacian 31 2.1 WeakFormulation 31 2.2 Pointwise Formula 34 2.3 Normal Derivatives 37 2.4Gauss-Green Formula 41 2.5 Gluing 44 2.6 Green's Function 46 2.7 LocalBehavior of Functions 55 2.8 Notes and References 62 Chapter 3.Spectrum of the Laplacian 63 3.1 Fourier Series Revisited 63 3.2 SpectralDecimation 68 3.3 Eigenvalues and Multiplicities 73 3.4 LocalizedEigenfunctions 79 3.5 Spectral Asymptotics 83 3.6 Integrals InvolvingEigenfunctions 86 3.7 Notes and References 89 Chapter 4. PostcriticallyFinite Fractals 91 4.1 Definitions 91 4.2 Energy Restriction andRenormalization 96 4.3 Examples 101 4.4 Laplacians 109 4.5 Geography IsDestiny 114 4.6 Non-self-similar Fractals 116 4.7 Notes and References 119Chapter 5. Further Topics 121 5.1 Polynomials, Splines, and Power Series121 5.2 Local Symmetries 125 5.3 Energy Measures 127 5.4 Fractal Blow-upsand Fractafolds 131 5.5 Singularities 136 5.6 Products of Fractals 140 5.7Solvability of Differential Equations 146 5.8 Heat Kernel Estimates 149 5.9Convergence of Fourier Series 152 5.10 Notes and References 156References 159 Index 167