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