Small sets of divergence points are dimensionless
type: Item
instance of: scholarly article
title: Small sets of divergence points are dimensionless
zbMATH Open document ID: 1060.37021
author: L. Olsen
published in: Monatshefte für Mathematik
publication date: 2004-03-07T00:00:00Z
review text: The framework of this paper is as follows: one denotes by \(\Sigma\) a subshift modeled by a strongly connected graph, and by \(S:\Sigma\to\Sigma\) the associated shift. \({\mathcal P}(\Sigma)\) is the family of probability measures on \(\Sigma\). For \(n\in \mathbb N\), \(L_{n}:\Sigma\to{\mathcal P}(\Sigma)\) is the \(n\)th empirical measure, defined by \[ L_{n}\omega={{1}\over{n}}\sum_{k=0}^{n-1}\delta_{S^{k}\omega}, \] where \(\delta_{x}\) is the Dirac measure at \(x\). A continuous deformation of \(L_{n}\) is a pair \((X,\Xi)\), where \(X\) is a metric space, and \( \Xi:{\mathcal P}(\Sigma)\to X\) is a continuous map with respect to the weak topology. For two subsets \(T, U\) in \(X\) one defines \[ \Delta(T,U)=\{\omega\in\Sigma| T\subseteq A(\Xi L_{n} \omega)\subseteq U\}, \] where the symbol \(A\) denotes the set of accumulation points of the subsequent sequence. The author proves that the set \(\Delta(T,U)\) is dimensionless, i.e., it does not have an exact dimension function. More precisely, if \(t\) is its Hausdorff dimension and \(h\) is a dimension function of the form \(h(r)=r^{t}L(t)\), with \(L\) a slowly varying function, then if \(t<\text{dim} \Sigma\), we have \[ {\mathcal H}^{h}( \Delta(T,U))=0, \,\,\,\text{or}\,\,\, {\mathcal H}^{h}( \Delta(T,U))=\infty, \] in particular, \({\mathcal H}^{t}( \Delta(T,U))=0\) or \({\mathcal H}^{t}( \Delta(T,U))=\infty\). The main result is used to show that the multifractal decomposition sets of a general type of mixed multifractal spectrum of ergodic averages are dimensionless.
reviewed by: Emilia Petrisor
Mathematics Subject Classification ID: 37D35
Mathematics Subject Classification ID: 37D20
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 2051555
zbMATH Keywords: multifractals
zbMATH Keywords: multifractal spectrum
zbMATH Keywords: local dimension
MaRDI profile type: Publication
full work available at URL: s00605-003-0110-x
OpenAlex ID: W2159031940
DOI: 10.1007/S00605-003-0110-X
description: scientific article; zbMATH DE number 2051555