Self-conformal multifractal measures
type: Item
instance of: scholarly article
title: Self-conformal multifractal measures
zbMATH Open document ID: 0912.28007
author: Norbert Patzschke
published in: Advances in Applied Mathematics
publication date: 1999-05-02T00:00:00Z
review text: Let \((S,p)\) be an iterated function system generated by selfmappings \(S_1,\dots, S_n\) of an open and connected subset of a smooth Riemannian manifold and a probability vector \((p_1,\dots, p_n)\). The mappings \(S_1,\dots, S_n\) are assumed to be conformal \(C^{1+\gamma}\) diffeomorphisms with \(\gamma\in (0,1)\) and to satisfy \(0<| S_i'|< 1\) for \(i= 1,\dots, n\) and the following Morans open set condition: there is an open (and bounded) set \(V\) with pairwise disjoint images \(S_1(V),\dots, S_n(V)\subset V\). It is well known that there is a unique compact \(\Xi\) with \(\Xi= \bigcup^n_{i= 1} S_i(\Xi)\) (attractor of \((S,p)\)) and a unique probability measure \(\varphi\) on \(\text{cl }V\) satisfying \(\varphi= \sum^n_{i=1} p_i\varphi\circ S^{-1}_i\) (fractal measure of \((S,p)\) with the support \(\Xi\)). The main result of the paper (Theorem 4.7) describes the local structure of the attractor \(\Xi\) by computing the Hausdorff and packing dimensions of the decomposition parts \(\Xi_a\) of \(\Xi\) defined for \(a\in\mathbb{R}\) by \(\Xi_a= \{x\in \Xi: \lim_{r\to 0}\log\varphi(B(x, r))/\log r=a\}\). The formulae obtained for these dimensions have been proved under the assumption \(V\cap\Xi\neq \emptyset\) and make use of some functions defined via the so-called equilibrium measures.
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 37D45
zbMATH DE Number: 1139872
zbMATH Keywords: IFS
zbMATH Keywords: conformal selfmapping of a Riemannian manifold
zbMATH Keywords: attractor
zbMATH Keywords: iterated function system
zbMATH Keywords: Morans open set condition
zbMATH Keywords: Hausdorff and packing dimensions
reviewed by: Witold Jarczyk
MaRDI profile type: Publication
OpenAlex ID: W1987395549
cites work: Random Self-Similar Multifractals
cites work: Q3972277
cites work: Equilibrium states and the ergodic theory of Anosov diffeomorphisms
cites work: Multifractal decompositions of Moran fractals
cites work: The multifractal spectrum of statistically self-similar measures
cites work: An exact formula for the measure dimensions associated with a class of piecewise linear maps
cites work: On Bandt's tangential distribution for self-similar measures
cites work: Q3731958
cites work: Singularity spectrum for recurrent IFS attractors
cites work: Q4301606
cites work: Random self-affine multifractal Sierpinski sponges in \(\mathbb R^d\)
cites work: The singularity spectrumf(α) for cookie-cutters
cites work: An improved multifractal formalism and self-similar measures
cites work: Q4185090
cites work: Q4287683
DOI: 10.1006/AAMA.1997.0557
description: scientific article; zbMATH DE number 1139872