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.
Bibliographic references
Includes bibliographical references and index.
Source of description
Description based on print version record.
Contents
Intro
Contents
Preface
Chapter 1. Introduction
1.1. Self-similar sets
1.2. Self-affine sets
1.3. Iterated function systems and the symbolic space
1.4. Preliminaries on fractal geometry
1.5. Similarity dimension
the dimension of self-similar sets with well separated cylinders
1.6. Self-similar sets with overlaps
1.7. Self-similar graph directed IFS on the line\textsuperscript{*}
1.8. Stationary measures for iterated function systems
1.9. Dimension of measures
1.10. Dimension of self-affine sets
1.11. Self-affine sets with aligned cylinders
1.12. Self-affine sets without alignment
Chapter 2. Elements of Geometric Measure Theory
2.1. Densities for Radon measures
2.2. Product theorems
2.3. Projection theorems
2.4. Slicing theorems
2.5. Besicovitch projection theorem and generalizations
2.6. ^{ } dimensions and entropy dimension
2.7. Generalized Hausdorff and packing measures
Chapter 3. General properties of self-similar sets and measures
3.1. Consequences of stationarity
3.2. On the dimension of self-similar measures.
3.3. Equality of box and Hausdorff dimension
3.4. Hausdorff measure of self-similar sets
3.5. Lower semicontinuity of the Hausdorff dimension of self-similar sets.
3.6. Existence of ^{ } dimensions
3.7. Existence of entropy dimension and lower semicontinuity of the entropy and Hausdorff dimensions of self-similar measures
Chapter 4. Separation properties for self-similar IFS
4.1. Conditions that are equivalent to the OSC
4.2. Weak Separation Property
4.3. Example: The {0,1,3}-IFS with contraction ratio =1/3
4.4. Finite type condition
4.5. Special family of finite type\textsuperscript{*}
Chapter 5. Multifractal Analysis for self-similar measures
5.1. Multifractal spectrum and multifractal formalism.
5.2. Multifractal analysis for IFS satisfying the weak separation or finite type condition
5.3. Divergence points of self-similar measures
Chapter 6. Transversality techniques for self-similar IFS
6.1. Self-similar sets on the line with overlaps (linear dependence)
6.2. Self-similar measures on the line with overlaps (linear dependence)
6.3. Self-similar sets and measures on the line (nonlinear dependence)
6.4. How to check transversality?
6.5. First applications
6.6. Generalized projection scheme
6.7. Applications of the Generalized Projection Scheme
6.8. A survey of more recent results
Chapter 7. Further properties of self-similar IFS with overlaps
7.1. Hausdorff and packing measure for self-similar IFS: Introduction
7.2. Positive packing measure and intersection numbers
7.3. Zero Hausdorff measure and dimension drop
7.4. Singularity of self-similar measures without exact overlaps
7.5. Planar self-similar sets of positive Lebesgue measure and empty interior
7.6. Positive Hausdorff measure and the weak separation property
7.7. Examples
Chapter 8. Fourier-analytic and number-theoretic methods
8.1. Pisot-Vijayaraghavan and Garsia numbers
8.2. Fourier dimension and its variant
8.3. Fourier decay
8.4. Applications of Fourier decay
8.5. Growth rate of -expansions and local dimensions of Bernoulli convolutions
8.6. Garsia entropy and dimension of Bernoulli convolutions for algebraic parameters
10.4. Self-affine transversality and the proof of Theorem 10.1.2
10.5. Thermodynamic formalism
10.6. Dominated matrices
10.7. A survey of some related results
Chapter 11. Diagonally self-affine IFS
11.1. The Feng-Hu Theorem in full generality
11.2. Feng-Hu Theorem in ℝ² assuming separation
11.3. Dimension of homogeneous self-similar measures
11.4. Box dimension of self-affine sets with aligned cylinders
11.5. Applications: Self-affine carpets
Chapter 12. Exact dimensionality and dimension conservation
12.1. Random walk on the projective line and the Furstenberg measure
12.2. Slices and dimension conservation
12.3. The Ledrappier-Young formula
12.4. Dimension conservation for the natural measure
Chapter 13. Local entropy averages and projections of self-affine sets and measures
13.1. Local entropy averages: Symbolic setting
13.2. Local entropy averages: Concrete realization
13.3. Orthogonal projections of self-affine measures
Chapter 14. Nonlinear conformal iterated functions systems
14.1. Existence and uniqueness of invariant measure in a general case
14.2. Hyperbolic IFS on the line
14.3. Equivalent conditions for OSC and WSP for nonlinear Iterated Function Systems
14.4. Families of nonlinear overlapping IFS's
14.5. Translation families and topologically generic IFS with overlaps
Appendix A. Some elements of Linear algebra
A.1. Singular values and SVD
A.2. The -fold exterior product of ℝ^{ }.
Appendix B. Some elements of measure theory
B.1. Measure spaces
B.2. Measures on metric spaces
B.3. Covering Theorems
B.4. Density theorems
B.5. Integral in terms of the distribution function
Appendix C. Some elements of Harmonic Analysis
Appendix D. Some facts about algebraic numbers
Bibliography.
Index
Basic notation.
ISBN
1-4704-7550-2
Statement on language in description
Princeton University Library aims to describe library materials in a manner that is respectful to the individuals and communities who create, use, and are represented in the collections we manage.
Read more...