Details

Subject(s)

• Self-similar processes [Browse]

Author

• Simon, Károly [Browse]
• Solomyak, Boris, 1959- [Browse]

Series

• Mathematical surveys and monographs ; Volume 276. [More in this series]
• Mathematical Surveys and Monographs ; Volume 276

Summary note

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
• Chapter 9. Elements of Ergodic Theory
• 9.1. Ergodic theorems
• 9.2. Multiplicative Ergodic Theorem (Oseledeč Theorem)
• 9.3. Conditional expectation and martingales
• 9.4. Measurable partitions, Lebesgue spaces, Rokhlin's theorem
• 9.5. Entropy
• Chapter 10. Self-affine sets and measures
• 10.1. Introduction
• 10.2. Singular value function.
• 10.3. Self-affine Generalized Projection Scheme
• 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.

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ISBN

1-4704-7550-2

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