Recent Developments in Fractals and Related Fields

The Applied and Numerical Harmonic Analysis (ANHA) book series aims to provide the engineering, mathematical, and scienti?c communities with s- ni?cant developments in harmonic analysis, ranging from abstract harmonic analysis to basic applications. The title of the series re?ects the importance of applications and numerical implementation, but richness and relevance of applications and implementation depend fundamentally on the structure and depth of theoretical underpinnings. Thus, from our point of view, the int- leaving of theory and applications and their creative symbiotic evolution is axiomatic. Harmonic analysis is a wellspring of ideas and applicability that has ?o- ished, developed, and deepened over time within many disciplines and by means of creative cross-fertilizationwith diverse areas. The intricate and f- damental relationship between harmonic analysis and ?elds such as signal processing, partial di?erential equations (PDEs), and image processing is - ?ected in our state-of-the-art ANHA series. Our vision of modern harmonic analysis includes mathematical areas such as wavelet theory, Banach algebras, classical Fourier analysis, time-frequency analysis, and fractal geometry, as well as the diverse topics that impinge on them.

Provides an overview of recent developments in the mathematical fields related to fractals Includes original research contributions as well as surveys written by experts in their respective fields Readers will find interesting and motivating results as well as new avenues for further research Includes supplementary material: sn.pub/extras

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This bookan outgrowth of an international conference held in honor of Jacques Peyrièreprovides readers with an overview of recent developments in the mathematical fields related to fractals. Included are original research contributions as well as surveys written by experts in their respective fields.

The chapters are thematically organized into five major sections:

• Geometric Measure Theory and Multifractals;

• Harmonic and Functional Analysis and Signal Processing;

• Dynamical Systems and Analysis on Fractals;

• Stochastic Processes and Random Fractals;

• Combinatorics on Words.

Recent Developments in Fractals and Related Fields is aimed at pure and applied mathematicians working in the above-mentioned areas as well as other researchers interested in discovering the fractal domain. Throughout the volume, readers will find interesting and motivating results as well as new avenues for further research.

Contributors:

J.-P. Allouche, A.M. Atto, J.-M. Aubry, F. Axel, A. Ayache, C. Bandt, F. Bastin, P.R. Bertrand, A. Bonami, G. Bourdaud, Z. Buczolich, M. Clausel, Y. Demichel, X.-H. Dong, A. Durand, A.H. Fan, U.R. Freiberg, S. Jaffard, E. Järvenpää, A. Käenmäki, A. Karoui, H. Kempka, T. Langlet, K.-S. Lau, B. Li, M.M. France, S. Nicolay, E. Olivier, D. Pastor, H. Rao, S.Gy. Révész, J. Schmeling, A. Sebbar, P. Shmerkin, E.C. Waymire, Z.-X. Wen, Z.-Y. Wen, S.C. Williams, S. Winter

Inhalt
Geometric Measure Theory and Multifractals.- Occupation Measure and Level Sets of the WeierstrassCellerier Function.- Space-Filling Functions and Davenport Series.- Dimensions and Porosities.- On Upper Conical Density Results.- On the Dimension of Iterated Sumsets.- Geometric Measures for Fractals.- Harmonic and Functional Analysis and Signal Processing..- A Walk from Multifractal Analysis to Functional Analysis with Spaces, and Back.- Concentration of the Integral Norm of Idempotents.- Le calcul symbolique dans certaines algèbres de type Sobolev.- Lp-Norms and Fractal Dimensions of Continuous Function Graphs.- Uncertainty Principles, Prolate Spheroidal Wave Functions, and Applications.- 2-Microlocal Besov Spaces.- Refraction on Multilayers.- Wavelet Shrinkage: From Sparsity and Robust Testing to Smooth Adaptation.- Dynamical Systems and Analysis on Fractals..- Simple Infinitely Ramified Self-Similar Sets.- Quantitative Uniform Hitting in Exponentially Mixing Systems.- Some Remarks on the Hausdorff and Spectral Dimension of V-Variable Nested Fractals.- Cantor Boundary Behavior of Analytic Functions.- Measures of Full Dimension on Self-Affine Graphs.- Stochastic Processes and Random Fractals.- A Process Very Similar to Multifractional Brownian Motion.- Gaussian Fields Satisfying Simultaneous Operator Scaling Relations.- On Randomly Placed Arcs on the Circle.- T-Martingales, Size Biasing, and Tree Polymer Cascades.- Combinatorics on Words.- Univoque Numbers and Automatic Sequences.- A Crash Look into Applications of Aperiodic Substitutive Sequences.- Invertible Substitutions with a Common Periodic Point.- Some Studies on Markov-Type Equations.