Pre-image entropy for maps on noncompact topological spaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 1313.37010
publication date: 2013-09-26T00:00:00Z
Mathematics Subject Classification ID: 37B40
zbMATH DE Number: 6212348
zbMATH Keywords: pre-image entropy
zbMATH Keywords: locally compact space
zbMATH Keywords: Alexandroff compactification
zbMATH Keywords: hyperspace dynamical system
MaRDI profile type: Publication
author: Lei Liu
title: Pre-image entropy for maps on noncompact topological spaces
published in: Acta Mathematica Universitatis Comenianae. New Series
review text: Topological pre-image entropy for a continuous self-map \(f\) of a metrizable, in general noncompact, topological space \(X\) is introduced. If there is no nonempty compact \(f\)-invariant subset \(F\) of \(X\), the preimage entropy of \(f\) is defined to be zero. Otherwise, for every such \(F\), one computes the pre-image entropy in the sense of \textit{W-Ch. Cheng} and \textit{Sh. E. Newhouse} [Ergodic Theory Dyn. Syst. 25, 1091--1113 (2005; Zbl 1098.37012)] for the restriction of the map \(f\) to the set \(F\) and then the preimage entropy of \(f\) is defined to be the supremum of the obtained values over all such sets \(F\). If \(X\) is compact then the pre-image entropy of \(f\) defined in such a way of course coincides with that in the sense of Cheng and Newhouse [loc. cit.]. Further, the pre-image entropy does not depend on the choice of a metric on the space \(X\) and it is an invariant of topological conjugacy. The author also proves some other properties of it.
description: scientific article; zbMATH DE number 6212348