**Contemporary Mathematics**

Volume: 600;
2013;
399 pp;
Softcover

MSC: Primary 28; 11; 37; 58;

Print ISBN: 978-0-8218-9147-6

Product Code: CONM/600

List Price: $123.00

Individual Member Price: $98.40

**Electronic ISBN: 978-1-4704-1082-7
Product Code: CONM/600.E**

List Price: $123.00

Individual Member Price: $98.40

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# Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I: Fractals in Pure Mathematics

Share this page *Edited by *
*David Carfì; Michel L. Lapidus; Erin P. J. Pearse; Machiel van Frankenhuijsen*

This volume contains the proceedings from three conferences: the PISRS 2011 International Conference on Analysis, Fractal Geometry, Dynamical Systems and Economics, held November 8–12, 2011 in Messina, Italy; the AMS Special Session on Fractal Geometry in Pure and Applied Mathematics, in memory of Benoît Mandelbrot, held January 4–7, 2012, in Boston, MA; and the AMS Special Session on Geometry and Analysis on Fractal Spaces, held March 3–4, 2012, in Honolulu, HI.

Articles in this volume cover fractal geometry (and some aspects of dynamical systems) in pure mathematics. Also included are articles discussing a variety of connections of fractal geometry with other fields of mathematics, including probability theory, number theory, geometric measure theory, partial differential equations, global analysis on non-smooth spaces, harmonic analysis and spectral geometry.

The companion volume (Contemporary Mathematics, Volume 601) focuses on applications of fractal geometry and dynamical systems to other sciences, including physics, engineering, computer science, economics, and finance.

#### Table of Contents

# Table of Contents

## Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics I: Fractals in Pure Mathematics

- Preface vii8 free
- Separation Conditions for Iterated Function Systems with Overlaps 110 free
- 𝑘-point Configurations of Discrete Self-Similar Sets 2130
- Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator 5160
- 1. Introduction 5261
- 2. Generalized Fractal Strings and Their Complex Dimensions 5463
- 3. The Spectral Operator 𝔞_{𝔠} and the Infinitesimal Shifts ∂_{𝔠} 6372
- 4. Inverse and Direct Spectral Problems for Fractal Strings 7079
- 5. Quasi-Invertibility and Almost Invertibility of the Spectral Operator 7382
- 6. Spectral Reformulations of the Riemann Hypothesis and of Almost RH 7685
- 7. Concluding Comments 7988
- 8. Appendix A:Riemann’s Explicit Formula 8089
- 9. Appendix B:The Momentum Operator and Normality of ∂_{𝑐} 8291
- References 8493

- Analysis and Geometry of the Measurable Riemannian Structure on the Sierpiński Gasket 91100
- 1. Introduction 92101
- 2. Sierpiński gasket and its standard Dirichlet form 94103
- 3. Measurable Riemannian structure on the Sierpiński gasket 97106
- 4. Geometry under the measurable Riemannian structure 100109
- 5. Short time asymptotics of the heat kernels 106115
- 6. Ahlfors regularity and singularity of Hausdorff measure 109118
- 7. Weyl’s Laplacian eigenvalue asymptotics 112121
- 8. Connections to general theories on metric measure spaces 116125
- 9. Possible generalizations to other self-similar fractals 123132
- Appendix A. Case of the standard Laplacian on the Sierpiński gasket 126135
- References 128137

- A Survey on Minkowski Measurability of Self-Similar and Self-Conformal Fractals in ℝ^{𝕕} 135144
- Minkowski Measurability and Exact Fractal Tube Formulas for 𝑝-Adic Self-Similar Strings 161170
- 1. Introduction 162171
- 2. 𝑝-Adic Numbers 162171
- 3. 𝑝-Adic Fractal Strings 163172
- 4. Volume of Inner Tubes 166175
- 5. Explicit Tube Formulas for 𝑝-Adic Fractal Strings 168177
- 6. Nonarchimedean Self-Similar Strings 169178
- 7. Geometric Zeta Function of 𝑝-Adic Self-Similar Strings 173182
- 8. Rationality of the Geometric Zeta Function 174183
- 9. Exact Tube Formulas for 𝑝-Adic Self-Similar Strings 177186
- 10. The Average Minkowski Content 178187
- References 182191

- Minkowski Measurability Results for Self-Similar Tilings and Fractals with Monophase Generators 185194
- Multifractal Analysis via Scaling Zeta Functions and Recursive Structure of Lattice Strings 205214
- 1. Introduction and summary 206215
- 2. Multifractal analysis of self-similar systems 207216
- 3. Fractal strings and complex dimensions 213222
- 4. Generalized lattice strings and linear recurrence relations 219228
- 5. Multifractal analysis via scaling regularity and scaling zeta functions 225234
- 6. Further results and future work 232241
- References 236245

- Box-Counting Fractal Strings, Zeta Functions, and Equivalent Forms of Minkowski Dimension 239248
- Hausdorff Dimension of the Limit Set of Countable Conformal Iterated Function Systems with Overlaps 273282
- Multifractal Tubes: Multifractal Zeta-Functions, Multifractal Steiner Formulas and Explicit Formulas 291300
- Laplacians on Julia Sets III: Cubic Julia Sets and Formal Matings 327336
- Lipschitz Equivalence of Self-Similar Sets: Algebraic and Geometric Properties 349358
- 1. Introduction 349358
- 2. Techniques for Lipschitz Equivalence of Dust-Like Cantor Sets 352361
- 3. Recent Results on dust-like self-similar sets 356365
- 4. Touching IFS and Lipschitz equivalence: One dimensional case 358367
- 5. Touching IFS and Lipschitz equivalence: Higher dimensional case 362371
- Acknowledgements 363372
- References 363372

- Riemann Zeros in Arithmetic Progression 365374
- Curvature Measures of Fractal Sets 381390

#### Readership

Graduate students and researchers interested in fractal geometry and dynamical systems.