Continuity of the entropy for monotonic mod one transformations
type: Item
instance of: scholarly article
title: Continuity of the entropy for monotonic mod one transformations
zbMATH Open document ID: 0906.54016
DOI: 10.1023/A:1006542931067
published in: Acta Mathematica Hungarica
publication date: 1999-01-24T00:00:00Z
review text: The author considers the problem of continuity of the topological entropy \(h_{\text{top}}\) on the space of piecewise monotone transformations of \(I=[0,1]\). Two piecewise monotone transformations of \(I\) are near if the closures of their graphs in \(I^2\) are near with respect to the Hausdorff distance. It is proven that for a continuous strictly increasing function \(f:I\to\mathbb R\) the topological entropy \(h_{\text{top}}\) is continuous at the piecewise monotone transformation \(T:=f\;(\text{mod} 1)\) (such transformations \(T\) are called monotone mod one transformations), provided \(h_{\text{top}}(T)>0\). The author presents two examples of monotone mod one transformations of \(I\) at which the entropy is discontinuous. Also, he gives an example of a monotonic mod one transformation \(T\) with positive (and thus continuous at \(T\)) entropy such that the maximal measure and the pressure \(p(T,g)\) are discontinuous at \(T\) for some continuous function \(g:I\to\mathbb R\).
Mathematics Subject Classification ID: 54C70
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 1244251
zbMATH Keywords: topological pressure
author: Peter Raith
reviewed by: Taras Banakh
MaRDI profile type: Publication
full work available at URL: a:1006542931067
OpenAlex ID: W162949375
description: scientific article; zbMATH DE number 1244251