Self-similar and Self-affine Sets and Measures

Balázs Bárány, Károly Simon, Boris Solomyak

• Publisher: American Mathematical Society
• Year: 2023
• Topic: Mathematics
• eBook ISBN:
  9781470475505
• Subtopic: Mathematics General
• Index: Mathematics

Although there is no precise definition of a "fractal", it is usually understood to be a set whose smaller parts, when magnified, resemble the whole. Self-similar and self-affine sets are those for which this resemblance is precise and given by a contracting similitude or affine transformation. The present book is devoted to this most basic class of fractal objects. The book contains both introductory material for beginners and more advanced topics, which continue to be the focus of active research. Among the latter are self-similar sets and measures with overlaps, including the much-studied infinite Bernoulli convolutions. Self-affine systems pose additional challenges; their study is often based on ergodic theory and dynamical systems methods. In the last twenty years there have been many breakthroughs in these fields, and our aim is to give introduction to some of them, often in the simplest nontrivial cases. The book is intended for a wide audience of mathematicians interested in fractal geometry, including students. Parts of the book can be used for graduate and even advanced undergraduate courses.

Table of contents

• Contents
• Preface
• Chapter 1. Introduction
• Chapter 2. Elements of Geometric Measure Theory
• Chapter 3. General properties of self-similar sets and measures
• Chapter 4. Separation properties for self-similar IFS
• Chapter 5. Multifractal Analysis for self-similar measures
• Chapter 6. Transversality techniques for self-similar IFS
• Chapter 7. Further properties of self-similar IFS with overlaps
• Chapter 8. Fourier-analytic and number-theoretic methods
• Chapter 9. Elements of Ergodic Theory
• Chapter 10. Self-affine sets and measures
• Chapter 11. Diagonally self-affine IFS
• Chapter 12. Exact dimensionality and dimension conservation
• Chapter 13. Local entropy averages and projections of self-affine sets and measures
• Chapter 14. Nonlinear conformal iterated functions systems
• Appendix A. Some elements of Linear algebra
• Appendix B. Some elements of measure theory
• Appendix C. Some elements of Harmonic Analysis
• Appendix D. Some facts about algebraic numbers
• Bibliography
• Index
• Basic notation