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Über dieses Buch

This volume collects thirteen expository or survey articles on topics including Fractal Geometry, Analysis of Fractals, Multifractal Analysis, Ergodic Theory and Dynamical Systems, Probability and Stochastic Analysis, written by the leading experts in their respective fields. The articles are based on papers presented at the International Conference on Advances on Fractals and Related Topics, held on December 10-14, 2012 at the Chinese University of Hong Kong. The volume offers insights into a number of exciting, cutting-edge developments in the area of fractals, which has close ties to and applications in other areas such as analysis, geometry, number theory, probability and mathematical physics.

Inhaltsverzeichnis

Mandelbrot Cascades and Related Topics

This article is an extended version of the talk given by the author at the conference “Advances in fractals and related topics”, in December 2012 at the Chinese Hong-Kong University. It gathers recent advances in Mandelbrot cascades theory and related topics, namely branching random walks, directed polymers on disordered trees, multifractal analysis, and dynamical systems.

Julien Barral

Law of Pure Types and Some Exotic Spectra of Fractal Spectral Measures

Let

$$\mu$$

μ

be a Borel probability measure with compact support in

$${\mathbb R}^d$$

R

d

and let

$$E(\Lambda )=\{e^{-2\pi \lambda \cdot x}: \lambda \in \Lambda \}$$

E

(

Λ

)

=

{

e

-

2

π

λ

·

x

:

λ

Λ

}

. We make a review on some recent progress about spectral measures. We first show that the law of pure types holds for spectral measures, i.e. if

$$E(\Lambda )$$

E

(

Λ

)

is a frame for

$$L^2(\mu )$$

L

2

(

μ

)

, then

$$\mu$$

μ

is discrete or absolutely continuous or singular continuous with respect to Lebesgue measure (see [

HLL13

]). And we discuss the spectral properties of Cantor measures (see [

DaHL13

]), where we focus on some exotic properties of the spectra of some Cantor measures.

Xin-Rong Dai, Xing-Gang He, Chun-Kit Lai

The Role of Transfer Operators and Shifts in the Study of Fractals: Encoding-Models, Analysis and Geometry, Commutative and Non-commutative

We study a class of dynamical systems in

$$L^2$$

L

2

spaces of infinite products

$$X$$

X

. Fix a compact Hausdorff space

$$B$$

B

. Our setting encompasses such cases when the dynamics on

$$X = B^\mathbb {N}$$

X

=

B

N

is determined by the one-sided shift in

$$X$$

X

, and by a given transition-operator

$$R$$

R

. Our results apply to any positive operator

$$R$$

R

in

$$C(B)$$

C

(

B

)

such that

$$R1 = 1$$

R

1

=

1

. From this we obtain induced measures

$$\Sigma$$

Σ

on

$$X$$

X

, and we study spectral theory in the associated

$$L^2(X,\Sigma )$$

L

2

(

X

,

Σ

)

. For the second class of dynamics, we introduce a fixed endomorphism

$$r$$

r

in the base space

$$B$$

B

, and specialize to the induced solenoid

$$\mathrm{Sol }(r)$$

Sol

(

r

)

. The solenoid

$$\mathrm{Sol }(r)$$

Sol

(

r

)

is then naturally embedded in

$$X = B^\mathbb {N}$$

X

=

B

N

, and

$$r$$

r

induces an automorphism in

$$\mathrm{Sol }(r)$$

Sol

(

r

)

. The induced systems will then live in

$$L^2(\mathrm{Sol }(r), \Sigma )$$

L

2

(

Sol

(

r

)

,

Σ

)

. The applications include wavelet analysis, both in the classical setting of

$$\mathbb {R}^n$$

R

n

, and Cantor-wavelets in the setting of fractals induced by affine iterated function systems (IFS). But our solenoid analysis includes such hyperbolic systems as the Smale-Williams attractor, with the endomorphism

$$r$$

r

there prescribed to preserve a foliation by meridional disks. And our setting includes the study of Julia set-attractors in complex dynamics.

Dorin Ervin Dutkay, Palle E. T. Jorgensen

Generalized Energy Inequalities and Higher Multifractal Moments

We present a class of generalized energy inequalities which may be used to investigate higher multifractal moments, in particular

$$L^q$$

-dimensions of images of measures under Brownian-type processes,

$$L^q$$

-dimensions of almost self-affine measures, and moments of random cascade measures.

Kenneth Falconer

Some Aspects of Multifractal Analysis

The aim of this survey is to present some aspects of multifractal analysis around the recently developed subject of multiple ergodic averages. Related topics include dimensions of measures, oriented walks, Riesz products etc. The exposition on the multifractal analysis of multiple ergodic averages is mainly based on [

FLM12

,

KPS12

,

FSW00

].

Ai-Hua Fan

Heat Kernels on Metric Measure Spaces

In this section we shall discuss the notion of the heat kernel on a metric measure space

$$( M,d,\mu )$$

(

M

,

d

,

μ

)

.

Alexander Grigor’yan, Jiaxin Hu, Ka-Sing Lau

Stochastic Completeness of Jump Processes on Metric Measure Spaces

We give criteria for stochastic completeness of jump processes on metric measure spaces and on graphs in terms of volume growth.

Alexander Grigor’yan, Xueping Huang

Self Similar Sets, Entropy and Additive Combinatorics

Hoc12

], that self-similar sets whose dimension is smaller than the trivial upper bound have “almost overlaps” between cylinders. We give a heuristic derivation of the theorem using elementary arguments about covering numbers. We also give a short introduction to additive combinatorics, focusing on inverse theorems, which play a pivotal role in the proof. Our elementary approach avoids many of the technicalities in [

Hoc12

], but also falls short of a complete proof; in the last section we discuss how the heuristic argument is turned into a rigorous one.

Michael Hochman

Quasisymmetric Modification of Metrics on Self-Similar Sets

Using the notions of scales and their gauge functions associated with self-similar sets, we give a necessary and sufficient condition for two metrics on a self-similar set being quasisymmetric to each other. As an application, we construct metrics on the Sierpinski carpet which is quasisymmetric with respect to the Euclidean metrics and obtain an upper estimate of the conformal dimension of the Sierpinski carpet.

Jun Kigami

Recent Progress on Dimensions of Projections

This is a survey on recent progress on the question: how do projections effect dimensions generically? I shall also discuss briefly dimensions of plane sections.

Pertti Mattila

The Geometry of Fractal Percolation

A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions:

$$\bullet$$

the statements work for all directions, not almost all,

$$\bullet$$

the statements are true for more general projections, for example radial projections onto a circle,

$$\bullet$$

in the case

$$\dim _H >1$$

dim

H

>

1

, each projection has not only positive Lebesgue measure but also has nonempty interior.

Michał Rams, Károly Simon

Self-affine Sets and the Continuity of Subadditive Pressure

The affinity dimension is a number associated to an iterated function system of affine maps, which is fundamental in the study of the fractal dimensions of self-affine sets. De-Jun Feng and the author recently solved a folklore open problem, by proving that the affinity dimension is a continuous function of the defining maps. The proof also yields the continuity of a topological pressure arising in the study of random matrix products. I survey the definition, motivation and main properties of the affinity dimension and the associated SVF topological pressure, and give a proof of their continuity in the special case of ambient dimension two.

Pablo Shmerkin

Stability Properties of Fractal Curvatures

Lipschitz-Killing curvatures of singular sets are known from geometric measure theory. These are extensions of classical notions from convex and differential geometry. In recent years their fractal versions have been introduced via approximation by parallel sets of small distances. In the present paper stability properties of the corresponding limits under small perturbations of these neighborhoods are studied. The well-known Minkowski content may be considered as marginal case.

Martina Zähle

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