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2017 | Buch

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Recent Developments in Fractals and Related Fields

Conference on Fractals and Related Fields III, île de Porquerolles, France, 2015

herausgegeben von: Julien Barral, Stéphane Seuret

Verlag: Springer International Publishing

Buchreihe : Trends in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This contributed volume provides readers with an overview of the most recent developments in the mathematical fields related to fractals, including both original research contributions, as well as surveys from many of the leading experts on modern fractal theory and applications. It is an outgrowth of the Conference of Fractals and Related Fields III, that was held on September 19-25, 2015 in île de Porquerolles, France. Chapters cover fields related to fractals such as harmonic analysis, multifractal analysis, geometric measure theory, ergodic theory and dynamical systems, probability theory, number theory, wavelets, potential theory, partial differential equations, fractal tilings, combinatorics, and signal and image processing. The book is aimed at pure and applied mathematicians in these areas, as well as other researchers interested in discovering the fractal domain.

Inhaltsverzeichnis

Frontmatter

New Exponents for Pointwise Singularity Classification

Abstract

We introduce new tools for pointwise singularity classification: We investigate the properties of the two-variable function which is defined at every point as the p-exponent of a fractional integral of order t; new exponents are derived which are not of regularity type but give a more precise description of the behavior of the function near a singularity. We revisit several classical examples (deterministic and random) of multifractal functions for which the additional information supplied by this classification is derived. Finally, a new example of multifractal function is studied, where these exponents prove pertinent.

Patrice Abry, Stéphane Jaffard, Roberto Leonarduzzi, Clothilde Melot, Herwig Wendt

The Two-Dimensional Density of Bernoulli Convolutions

Abstract

Bernoulli convolutions form a one-parameter family of self-similar measures on the unit interval. We suggest to study their two-dimensional density which has an intricate combinatorial structure. Visualizing this structure we discuss results of Erdös, Jóo, Komornik, Sidorov, de Vries, Jordan, Shmerkin and Solomyak, Feng and Wang. We emphasize the role of finite orbits of associated multivalued maps and mention a few new properties and examples.

Christoph Bandt

Iterated Functions Systems, Blenders, and Parablenders

Abstract

We recast the notion of parablender introduced in Berger (Invent. Math. 205, 121–172 (2016)) as a parametric IFS. This is done using the concept of open covering property and looking to parametric IFS as systems acting on jets.

Pierre Berger, Sylvain Crovisier, Enrique Pujals

Multifractal Properties of Convex Hulls of Typical Continuous Functions

Abstract

We study the singularity (multifractal) spectrum of the convex hull of the typical/generic continuous functions defined on [0, 1]^d. We denote by E _ϕ31 ^h the set of points at which \(\phi: [0,1]^{d} \rightarrow \mathbb{R}\) has a pointwise Hölder exponent equal to h. Let H _f be the convex hull of the graph of f, the concave function on the top of H _f is denoted by ϕ _1,f(x) = max{y: (x, y) ∈ H _f} and ϕ _2,f(x) = min{y: (x, y) ∈ H _f} denotes the convex function on the bottom of H _f. We show that there is a dense G _δ subset \(\mathcal{G} \subset C[0,1]^{d}\) such that for \(f \in \mathcal{ G}\) the following properties are satisfied. For i = 1, 2 the functions ϕ _i, f and f coincide only on a set of zero Hausdorff dimension, the functions ϕ _i, f are continuously differentiable on (0, 1)^d, \(\mathbf{E}_{\phi _{ i,f}}^{0}\) equals the boundary of [0, 1]^d, \(\dim _{H}\mathbf{E}_{\phi _{ i,f}}^{1} = d - 1\), \(\dim _{H}\mathbf{E}_{\phi _{ i,f}}^{+\infty } = d\) and \(\mathbf{E}_{\phi _{ i,f}}^{h} =\emptyset\) if h ∈ (0, +∞)∖{1}.

Zoltán Buczolich

Fourier Bases and Fourier Frames on Self-Affine Measures

Abstract

This paper gives a review of the recent progress in the study of Fourier bases and Fourier frames on self-affine measures. In particular, we emphasize the new matrix analysis approach for checking the completeness of a mutually orthogonal set. This method helps us settle down a long-standing conjecture that Hadamard triples generate self-affine spectral measures. It also gives us non-trivial examples of fractal measures with Fourier frames. Furthermore, a new avenue is open to investigate whether the Middle-Third-Cantor measure admits Fourier frames.

Dorin Ervin Dutkay, Chun-Kit Lai, Yang Wang

Self-Similar Sets: Projections, Sections and Percolation

Abstract

We survey some recent results on the dimension of orthogonal projections of self-similar sets and of random subsets obtained by percolation on self-similar sets. In particular we highlight conditions when the dimension of the projections takes the generic value for all, or very nearly all, projections. We then describe a method for deriving dimensional properties of sections of deterministic self-similar sets by utilising projection properties of random percolation subsets.

Kenneth Falconer, Xiong Jin

Some Problems on the Boundary of Fractal Geometry and Additive Combinatorics

Abstract

This paper is an exposition, with some new applications, of our results from Hochman (Ann Math (2) 180(2):773–822, 2014; preprint, 2015, http://arxiv.org/abs/1503.09043) on the growth of entropy of convolutions. We explain the main result on \(\mathbb{R}\), and derive, via a linearization argument, an analogous result for the action of the affine group on \(\mathbb{R}\). We also develop versions of the results for entropy dimension and Hausdorff dimension. The method is applied to two problems on the border of fractal geometry and additive combinatorics. First, we consider attractors X of compact families \(\Phi\) of similarities of \(\mathbb{R}\). We conjecture that if \(\Phi\) is uncountable and X is not a singleton (equivalently, \(\Phi\) is not contained in a 1-parameter semigroup) then dimX = 1. We show that this would follow from the classical overlaps conjecture for self-similar sets, and unconditionally we show that if X is not a point and \(\dim \Phi> 0\) then dimX = 1. Second, we study a problem due to Shmerkin and Keleti, who have asked how small a set \(\emptyset \neq Y \subseteq \mathbb{R}\) can be if at every point it contains a scaled copy of the middle-third Cantor set K. Such a set must have dimension at least dimK and we show that its dimension is at least dimK + δ for some constant δ > 0.

Michael Hochman

Random Covering Sets, Hitting Probabilities and Variants of the Covering Problem

Abstract

We discuss various types of problems related to random covering sets. These include dimensional properties of random covering sets in Riemann manifolds as well as hitting probabilities of typical random covering sets in Ahlfors regular metric spaces.

Maarit Järvenpää

Small Union with Large Set of Centers

Abstract

Let \(T \subset \mathbb{R}^{n}\) be a fixed set. By a scaled copy of T around \(x \in \mathbb{R}^{n}\) we mean a set of the form x + rT for some r > 0. In this survey paper we study results about the following type of problems: How small can a set be if it contains a scaled copy of T around every point of a set of given size? We will consider the cases when T is circle or sphere centered at the origin, Cantor set in \(\mathbb{R}\), the boundary of a square centered at the origin, or more generally the k-skeleton (0 ≤ k < n) of an n-dimensional cube centered at the origin or the k-skeleton of a more general polytope of \(\mathbb{R}^{n}\). We also study the case when we allow not only scaled copies but also scaled and rotated copies and also the case when we allow only rotated copies.

Tamás Keleti

Some Recent Developments of Self-Affine Tiles

Abstract

A self-affine set \(T:= T(A,\mathcal{D})\) is the attractor of an affine pair \((A,\mathcal{D})\), where A is an expanding matrix on \(\mathbb{R}^{s}\) with integral entries, and \(\mathcal{D} \subset \mathbb{Z}^{s}\) is a finite set; T is called a self-affine tile if it is also a tile, and call such \(\mathcal{D}\) a tile digit set. In this survey, we review some recent developments on the structure and characterizations of the tile digit sets \(\mathcal{D}\) for a given A. We also discuss the celebrated Fuglede’s spectral set problem on the self-affine tiles.

Chun-Kit Lai, Ka-Sing Lau

A Class of Random Cantor Measures, with Applications

Abstract

We survey some of our recent results on the geometry of spatially independent martingales, in a more concrete setting that allows for shorter, direct proofs, yet is general enough for several applications and contains the well-known fractal percolation measure. We study self-convolutions and Fourier decay of measures in our class, and present applications of these results to the restriction problem for fractal measures, and the connection between arithmetic structure and Fourier decay.

Pablo Shmerkin, Ville Suomala

A Survey on the Dimension Theory in Dynamical Diophantine Approximation

Abstract

Dynamical Diophantine approximation studies the quantitative properties of the distribution of the orbits in a dynamical system. More precisely, it focuses on the size of dynamically defined limsup sets in the sense of measure and dimension. This quantitative study is motivated by the qualitative nature of the density of the orbits and the connections with the classic Diophantine approximation. In this survey, we collect some recent progress on the dimension theory in dynamical Diophantine approximation. This includes the systems of rational maps on its Julia set, linear map on the torus, beta dynamical system, continued fractions as well as conformal iterated function systems.

Baowei Wang, Jun Wu

(S)PDE on Fractals and Gaussian Noise

Abstract

In the first part of this paper we give a survey on results from Hinz and Zähle (Potential Anal 36:483–515, 2012) and Issoglio and Zähle (Stoch PDE Anal Comput 3:372–389, 2015) for nonlinear parabolic (S)PDE on certain metric measure spaces of spectral dimensions less than 4 with applications to fractals. We consider existence, uniqueness, and fractional regularity properties of mild function solutions in the pathwise sense. In the second part we apply this to the special case of fractal Laplace operators as generators and Gaussian random noises.

Furthermore, we show that random space-time fields Y (t, x) like fractional Brownian sheets with Hurst exponents H in time and K in space on general Ahlfors regular compact metric measure spaces X possess a modification whose sample paths are elements of C ^α([0, t ₀], C ^β(X)) for all α < H and β < K. This is used in the above special case of SPDE on fractals.

Martina Zähle

Titel: Recent Developments in Fractals and Related Fields
herausgegeben von: Julien Barral
Stéphane Seuret
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-57805-7
Print ISBN: 978-3-319-57803-3
DOI: https://doi.org/10.1007/978-3-319-57805-7