Further developments in fractals and related fields mathematical foundations and connections Julien Barral, Stéphane Seuret, editors

Format:

Book

Contributor:

Barral, Julien

Seuret, Stéphane

Series:

Trends in mathematics

Trends in Mathematics

Language:

English

Subjects (All):

Fractals.

Mathematics.

Geometry.

Abstract Harmonic Analysis.

Functional Analysis.

Partial Differential Equations.

Dynamical Systems and Ergodic Theory.

Probability Theory and Stochastic Processes.

fractals.

Local Subjects:

Mathematics.

Geometry.

Abstract Harmonic Analysis.

Functional Analysis.

Partial Differential Equations.

Dynamical Systems and Ergodic Theory.

Probability Theory and Stochastic Processes.

Genre:

Conference papers and proceedings

Physical Description:

1 online resource

Place of Publication:

New York Birkhäuser ©2013

Language Note:

English

System Details:

PDF

text file

Summary:

This volume, following in the tradition of a similar 2010 publication by the same editors, is an outgrowth of an international conference, "Fractals and Related Fields II," held in June 2011. The book provides readers with an overview of developments in the mathematical fields related to fractals, including original research contributions as well as surveys from many of the leading experts on modern fractal theory and applications

Contents:

The Rauzy Gasket Pierre Arnoux, Štěpán Starosta On the Hausdorff Dimension of Graphs of Prevalent Continuous Functions on Compact Sets Frédéric Bayart, Yanick Heurteaux Hausdorff Dimension and Diophantine Approximation Yann Bugeaud Singular Integrals on Self-similar Subsets of Metric Groups Vasilis Chousionis, Pertti Mattila Multivariate Davenport Series Arnaud Durand, Stéphane Jaffard Dimensions of Self-affine Sets: A Survey Kenneth Falconer The Multifractal Spectra of V-Statistics Ai-hua Fan, Jörg Schmeling, Meng Wu Projections of Measures Invariant Under the Geodesic Flow Maarit Järvenpää Multifractal Tubes Lars Olsen The Multiplicative Golden Mean Shift Has Infinite Hausdorff Measure Yuval Peres, Boris Solomyak The Law of Iterated Logarithm and Equilibrium Measures Versus Hausdorff Measures for Dynamically Semi-regular Meromorphic Functions Bartłomiej Skorulski, Mariusz Urbański Cookie-Cutter-Like Sets with Graph-Directed Construction Shen Fan, Qing-Hui Liu, Zhi-Ying Wen Recent Developments on Fractal Properties of Gaussian Random Fields Yimin Xiao

Machine generated contents note: Rauzy Gasket Stepan Starosta

1. Introduction

2. Preliminaries

2.1. Background: Complexity and Sturmian Words

2.2. Arnoux-Rauzy Words and Episturmian Words: Definition

2.3. Ternary AR Words: Renormalization

3. Rauzy Gasket

3.1. Rauzy Gasket as an Iterated Function System

3.2. Symbolic Dynamics for the Rauzy Gasket

4. Relation with the Sierpinski Gasket and a Generalization of the Question Mark Function

4.1. Minkowski Question Mark Function

4.2. Sierpinski Gasket

4.3. Generalization of the Minkowski Question Mark Function

5. Apollonian Gasket

6. Relation with the Fully Subtractive Algorithm

6.1. Fully Subtractive Algorithm

6.2. Fully Subtractive Algorithm as an Extension of the Rauzy Gasket

6.3. Two Properties of the Rauzy Gasket

7. Final Remarks References On the Hausdorff Dimension of Graphs of Prevalent Continuous Functions on Compact Sets Yanick Heurteaux

2. Prevalence

3. On the Graph of a Perturbed Fractional Brownian Motion

4. Proof of Theorem 3

5. Case of α-Holderian Functions References Hausdorff Dimension and Diophantine Approximation Yann Bugeaud

2. Three Families of Exponents of Approximation

3. Approximation to Points in the Middle Third Cantor Set References Singular Integrals on Self-similar Subsets of Metric Groups Pertti Mattila

2. One-Dimensional Case

3. Higher-Dimensional Case

4. Self-similar Sets and Singular Integrals

5. Self-similar Sets in Heisenberg Groups

6. Riesz-Type Kernels in Heisenberg Groups

7. δh-Removability and Singular Integrals

8. δh-Removable Self-similar Cantor Sets in Hn

9. Concluding Comments References Multivariate Davenport Series Stephane Jaffard

2. Relationships Between Davenport and Fourier Series

3. Discontinuities of Davenport Series

4. Jump Operator

5. Pointwise Holder Regularity

6. Sparse Davenport Series

6.1. Sparse Sets and Link with Lacunary and Hadamard Sequences

6.2. Decay of Sequences with Sparse Support and Behavior of the Jump Operator

6.3. Pointwise Regularity of Sparse Davenport Series

7. Implications for Multifractal Analysis

8. Convergence and Global Regularity of Davenport Series

8.1. Preliminaries on Multivariate Arithmetic Functions

8.2. Davenport Expansions Versus Fourier Expansions

8.3. Regularity of the Sum of a Davenport Series

9. Concluding Remarks and Open Problems

9.1. Optimality of Lemma 2

9.2. Hecke's Functions

9.3. Spectrum of Singularities of Compensated Pure Jumps Functions

9.4. p-Exponent

9.5. Directional Regularity

10. Proof of Theorem 1

11. Proof of Theorems 2 and 3

11.1. Locations of the Singularities

11.2. Size and Large Intersection Properties of the Sets La(α), Connection with the Duffin-Schaeffer and Catlin Conjectures

11.3. End of the Proof References Dimensions of Self-affine Sets: A Survey Kenneth Falconer

1.1. Basic Definitions

2. Affinity Dimension

2.1. Cutting up Ellipses

2.2. Affinity Dimension

2.3. Generic Results

2.4. Sets with Dimension Attaining the Affinity Dimension

3. Self-affine Carpets

3.1. Bedford-McMullen Carpets

3.2. Other Carpets

3.3. Box-Like Sets

4. Self-affine Functions

5. Related Topics

5.1. Multifractal Analysis of Measures on Self-affine Sets

5.2. Nonlinear Analogues References Multifractal Spectra of V-Statistics Meng Wu

2. V-Statistics

3. Topological Entropy

4. Proof of Theorem 1

5. Example: Shift Dynamics References Projections of Measures Invariant Under the Geodesic Flow Maarit Jarvenpaa

2. Projections of Measures Invariant Under the Geodesic Flow

3. Quantum Unique Ergodicity References Multifractal Tubes Lars Olsen

1. Fractal Tubes

2. Multifractals

2.1. Multifractal Spectra

2.2. Renyi Dimensions

2.3. Multifractal Formalism

3. Multifractal Tubes

3.1. Multifractal Tubes

3.2. Multifractal Tubes of Self-similar Measures

3.3. How Does One Prove Theorem 2 on the Asymptotic Behaviour of Multifractal Tubes of Self-similar Measures

4. Multifractal Tube Measures

4.1. Multifractal Tube Measures

4.2. Multifractal Tube Measures of Self-similar Measures References Multiplicative Golden Mean Shift Has Infinite Hausdorff Measure Boris Solomyak

2. Preliminaries and the Scheme of the Proof

3. Lower Estimates of Hausdorff Measure

4. Upper Bound for Hausdorff Measure References Law of Iterated Logarithm and Equilibrium Measures Versus Hausdorff Measures for Dynamically Semi-regular Meromorphic Functions Mariusz Urbariski

3. Law of Iterated Logarithm: Abstract Setting

4. Law of Iterated Logarithm: Meromorphic Functions

5. Equilibrium States Versus Hausdorff Measures References Cookie-Cutter-Like Sets with Graph-Directed Construction Zhi-Ying Wen

1.1. Cookie-Cutter and Cookie-Cutter-Like Constructions

1.2. Graph-Directed Construction

1.3. Cookie-Cutter-Like Sets with Graph-Directed Construction (GCCL)

2. Main Results

2.1. Basic Assumption

2.2. Main Theorems

3. Four Properties of GCCL

3.1. More on Coding Space and Other Notations

3.2. Proofs of Four Properties

4. Proofs of the Theorems

4.1. Proof of Theorem 1

4.2. Proof of Theorem 2

4.3. Proof of Theorem 3 References Recent Developments on Fractal Properties of Gaussian Random Fields Yimin Xiao

2. Gaussian Random Fields

2.1. Space-Anisotropic Gaussian Random Fields

2.2. Time-Anisotropic Gaussian Random Fields

2.3. Assumptions

3. Analytic Results

3.1. Exact Modulus of Continuity and LIL

3.2. Chung's LIL and Modulus of Nondifferentiability

3.3. Regularity of Local Times

4. Fractal Properties

4.1. Hausdorff Dimension Results

4.2. Fourier Dimension and Salem Sets

4.3. Packing Dimension Results

4.4. Uniform Dimension Results

4.5. Exact Hausdorff Measure Functions

4.6. Exact Packing Measure Functions

4.7. Hitting Probabilities and Intersections of Gaussian Random Fields

Notes:

Includes bibliographical references

Print version record

Other Format:

Print version Further developments in fractals and related fields

ISBN:

9780817684006

081768400X

1299336108

9781299336100

0817683992

9780817683993

OCLC:

828794669

Publisher Number:

10.1007/978-0-8176-8

Access Restriction:

Restricted for use by site license