Typical properties of correlation dimension.
type: Item
instance of: scholarly article
title: Typical properties of correlation dimension.
zbMATH Open document ID: 1048.37020
published in: Real Analysis Exchange
publication date: 2004-08-06T00:00:00Z
review text: Consider a polish space~\(X\), with metric~\(\rho\). The family of all Borel probability measures on~\(X\) is denoted by \(\mathcal M\). The authors consider two metrics on~\(\mathcal M\): the supremum metric~\(d_1\) and the Fortet-Mourier distance~\(d_2\), defined as the supremum of all numbers \(| \int_X f\,d\mu_1-\int_Xf\,d\mu_2| \) where \(f\)~ranges in the family of Lipschitz-continuous functions of uniform norm at most~\(1\) and Lipschitz constant \(1\) (with respect to~\(\rho\)). The upper and lower correlation dimensions of a measure \(\mu\) are the \(\lim\sup\) and \(\liminf\) of \({1\over\log r}\log\int_X\mu \bigl( B(x,r)\bigr)\,d\mu(x)\) as \(r\to0\), respectively. The principal result of the paper states that 1)~the upper correlation dimension is zero for all~\(\mu\) in a residual in \((\mathcal M,d_1)\); 2)~the lower correlation dimension is zero for all~\(\mu\) in a residual in \((\mathcal M,d_2)\); and 3)~the upper correlation dimension lies between two values dependent on the so-called entropy dimension of balls for all~\(\mu\) in a residual in \((\mathcal M,d_2)\). Carefully chosen Cantor sets in the real line witness that the estimates obtained in~3) are sharp.
Mathematics Subject Classification ID: 37C45
Mathematics Subject Classification ID: 54E52
Mathematics Subject Classification ID: 28A33
zbMATH DE Number: 2084037
zbMATH Keywords: measure
zbMATH Keywords: correlation dimension
zbMATH Keywords: entropy dimension
zbMATH Keywords: residual set
author: Józef Myjak
author: Ryszard Rudnicki
reviewed by: K. P. Hart
MaRDI profile type: Publication
full work available at URL: realanalexch.28.2.0269
OpenAlex ID: W1508043317
DOI: 10.14321/REALANALEXCH.28.2.0269
description: scientific article; zbMATH DE number 2084037