On the relation between entropy convergence rates and Baire category
type: Item
instance of: scholarly article
title: On the relation between entropy convergence rates and Baire category
zbMATH Open document ID: 1283.37008
author: Frank Blume
published in: Mathematische Zeitschrift
publication date: 2012-08-13T00:00:00Z
review text: Let \(G\) denote the group of all invertible Lebesgue measure-preserving transformations on the unit interval \([0,1]\). For \(T\in G\) and \(\alpha\) a finite measurable partition of \([0,1]\), let \(\alpha_T^n\) denote the \(n\)th refinement of \(\alpha\) under \(T\). In this paper the author shows that if \(\{a_n\}\) is a sequence of positive integers whose limit superior is \(\infty\) then \[ \{T\in G: \liminf_{n\longrightarrow \infty}\frac{H(\alpha_T^n)}{a_n}>0 \text{ for all finite nontrivial partitions }\alpha\} \] is of first category in \(G\), and if \(\{a_n\}\) is a sequence of positive integers such that \(\displaystyle\lim_{n\longrightarrow \infty}\frac{a_n}{n}=0\) then \[ \{T\in G: \limsup_{n\longrightarrow \infty}\frac{H(\alpha_T^n)}{a_n}>0 \text{ for all finite nontrivial partitions }\alpha\} \] is residual in \(G\).
Mathematics Subject Classification ID: 37A35
Mathematics Subject Classification ID: 37B05
zbMATH DE Number: 6065378
zbMATH Keywords: entropy convergence rate
zbMATH Keywords: Baire category
reviewed by: Yujun Zhu
MaRDI profile type: Publication
full work available at URL: s00209-011-0887-6
OpenAlex ID: W2016797835
cites work: Minimal rates of entropy convergence for completely ergodic systems
cites work: Possible rates of entropy convergence
cites work: In general a measure preserving transformation is mixing
cites work: Q3234755
cites work: Q5565255
cites work: Q4744504
cites work: Q3999363
DOI: 10.1007/S00209-011-0887-6
description: scientific article; zbMATH DE number 6065378