Horizons of fractal geometry and complex dimensions : 2016 Summer School on Fractal Geometry and Complex Dimensions, June 21-29, 2016, California Polytechnic State University, San Luis Obispo, California / Robert G. Niemeyer [and three others], editors.

Format:

Book

Author/Creator:

Niemeyer, Robert G., 1983- editor.

Series:

Contemporary mathematics (American Mathematical Society). 0271-4132 731

Contemporary mathematics, 731 0271-4132

Language:

English

Subjects (All):

Fractals--Congresses.

Fractals.

Physical Description:

1 online resource (320 pages).

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2019]

Language Note:

English

Summary:

This volume contains the proceedings of the 2016 Summer School on Fractal Geometry and Complex Dimensions, in celebration of Michel L. Lapidus's 60th birthday, held from June 21-29, 2016, at California Polytechnic State University, San Luis Obispo, California. The theme of the contributions is fractals and dynamics and content is split into four parts, centered around the following themes: Dimension gaps and the mass transfer principle, fractal strings and complex dimensions, Laplacians on fractal domains and SDEs with fractal noise, and aperiodic order (Delone sets and tilings).

Contents:

Cover

Title page

Contents

Preface: Horizons of Fractal Geometry and Complex Dimensions

The Mass Transference Principle: Ten years on

1. Introduction

2. The Mass Transference Principle

3. Extension to systems of linear forms

4. Extension to rectangles

5. Random Mass Transference Principles

References

A measure-theoretic result for approximation by Delone sets

2. Proof of main theorem

Self-similar tilings of fractal blow-ups

2. Tilings, Similitudes and Tiling Spaces

3. Definition and Properties of IFS Tilings

4. Structure of {Ω_{ }} and Symbolic IFS Tilings

5. A Canonical Sequence of Self-similar Tilings

6. Theorem 3.2: Existence and Continuity of Tilings

7. Theorem 3.3: When Do all Tilings Repeat the Same Patterns?

8. Relative and Absolute Addresses

9. Strong Rigidity, Definition of "Amalgamation and Shrinking" Operation on Tilings, and Proof of Theorem 3.4.

10. Theorem 3.5: When is a Tiling Non-Periodic?

11. When is :[ ]*∪\lbrack ]^{∞}→ invertible?

12. Examples

Acknowledgement

Regularly varying functions, generalized contents, and the spectrum of fractal strings

2. Regularly varying functions and statement of the main results

3. Some preliminaries on asymptotic equivalence and similarity

4. Regularly varying functions and Karamata theory

5. The geometric part of the proof

6. The 'spectral' part of the proof of Theorem 2.5

Dimensions of limit sets of Kleinian groups

1. Dimensions and invariants in conformal dynamics

2. Brooks' Theorem

the role of amenability

3. Group extensions of topological Markov chains

4. The convex core entropy

5. An outlook

open problems

The spectral operator and resonances

1. Introduction.

2. Fractal strings

3. The spectrum of a fractal string

4. The Riemann zeta function

5. The spectral operator

6. Zeros in arithmetic progression

7. Resonances

Measure-geometric Laplacians for discrete distributions

An overview of complex fractal dimensions: from fractal strings to fractal drums, and back

2. Fractal Strings and Their Complex Dimensions

2.1. Fractal tube formulas

2.2. Other examples of fractal explicit formulas

2.3. Analogy with Riemann's explicit formula

2.4. The meaning of complex dimensions

2.5. Fractality, complex dimensions and irreality

2.6. Inverse spectral problems and the Riemann hypothesis

3. A Taste of the Higher-Dimensional Theory: Complex Dimensions and Relative Fractal Drums (RFDs)

3.1. Brief history

3.2. Fractal zeta functions and relative fractal drums (FZFs and RFDs)

3.3. A few key properties of fractal zeta functions

3.4. Examples of fractal zeta functions and complex dimensions

3.5. Fractal tube formulas and Minkowski measurability criteria: Theory and examples

3.6. Fractality, hyperfractality and unreality, revisited

4. Epilogue: From Complex Fractal Dimensions to Quantized Number Theory and Fractal Cohomology

4.1. Analogy between self-similar geometries and varieties over finite fields

4.2. Quantized number theory: The real case

4.3. Quantized number theory: The complex case

4.4. Towards a fractal cohomology

Acknowledgements

Glossary

Eigenvalues of the Laplacian on domains with fractal boundary

2. Our domains

3. Our spectral zeta-functions

4. Are they equal?

5. Acknowledgments

Forward integrals and SDE with fractal noise

0. Introduction

1. Fractional integrals and Sobolev type spaces.

2. An integral operator and its contraction property

3. Stochastic forward integrals, quadratic variation and Itô formula

4. Differential equations driven by functions with fractional smoothness of order greater than 1/2

5. Stochastic differential equations with mixed fractal noise

6. Hölder continuity of the solution

7. Appendix: Stochastic integrals with respect to random Weierstrass-type functions

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-5315-0