On the topological entropy of continuous and almost continuous functions
type: Item
instance of: scholarly article
title: On the topological entropy of continuous and almost continuous functions
zbMATH Open document ID: 1227.54022
author: Anna Loranty
author: Anna Bąkowska
author: Ryszard Pawlak
published in: Topology and its Applications
publication date: 2011-10-11T00:00:00Z
review text: The paper under review can be divided into two parts. In the first part the authors introduce new notions of an entropy point and a strong entropy point for a function \(f:X\to X\), where \(X\) is a compact metric space. (Note that those definitions are different from other notions of entropy point which occur in the literature, because they are connected with the entropy of \(f\)-bundles. In particular, each strong entropy point is a fixed point of \(f\).) In the main theorem of this part the authors show that if \(f,g:X\to X\) are conjugate then \(f\) has an entropy point (a strong entropy point) iff \(g\) has the same property. In the second part the authors consider approximations of a function \(f:X\to X\) (here \(X\) is a compact manifold) by functions having strong entropy points. Two kinds of approximation are considered: the \(\Gamma\)-approximation and the uniform approximation. Recall that a function \(g:X\to X\) is \(\Gamma\)-approximated by functions belonging to the class \(\mathcal{F}\) if for each open set \(U\subset X\times X\) containing the graph of \(g\) there exists \(f\in\mathcal{F}\) with the graph contained in \(U\). (In particular, \(g:X\to X\) is \(\Gamma\)-approximated by continuous functions iff it is almost continuous in the sense of Stallings.) The main results of this part: every almost continuous function can be \(\Gamma\)-approximated by (almost) continuous functions having strong entropy points; and every continuous function can be uniformly approximated by continuous functions having strong entropy points.
reviewed by: Tomasz Natkaniec
Mathematics Subject Classification ID: 54C70
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 54C65
zbMATH DE Number: 5956306
zbMATH Keywords: almost continuity in the sense of Stallings
zbMATH Keywords: \(f\)-bundle
zbMATH Keywords: multivalued function
zbMATH Keywords: entropy
zbMATH Keywords: entropy point
zbMATH Keywords: strong entropy point
zbMATH Keywords: \(\Gamma\)-approximation
zbMATH Keywords: uniform approximation
zbMATH Keywords: fixed point.
MaRDI profile type: Publication
full work available at URL: j.topol.2011.06.049
OpenAlex ID: W2004826575
cites work: Q4858735
cites work: Some generic properties in fixed point theory
cites work: Q4098245
cites work: Q3805354
cites work: Continuous Selections for Weakly Hausdorff Lower Semicontinuous Multifunctions
cites work: Erratum to "Entropy for Group Endomorphisms and Homogeneous Spaces"
cites work: Set valued mappings, continuous selections, and metric projections
cites work: Dynamical systems generated by functions with connected \(G_\delta\) graphs
cites work: Q5592909
cites work: Q4423214
cites work: Q3836181
cites work: A nonlinear Banach–Steinhaus theorem and some meager sets in Banach spaces
cites work: On a question of Borsuk concerning non-continuous retracts II
cites work: Almost continuous functions on $I^n$
cites work: Ultra-methods in metric fixed point theory
cites work: Handbook of metric fixed point theory
cites work: Continuous selection theorems for nonlower semicontinuous multifunctions.
cites work: Introduction to Topological Manifolds
cites work: Period Three Implies Chaos
cites work: On the Porosity of the Set of ω-Nonexpansive Mappings without Fixed Points
cites work: Topological entropy and the preimage structure of maps
cites work: On the entropy of Darboux functions
cites work: Generic aspects of metric fixed point theory
cites work: Q4793526
cites work: Fixed point theorems for connectivity maps
DOI: 10.1016/J.TOPOL.2011.06.049
Recommended article: Topological entropy of continuous functions on topological spaces
Recommended article: Some fundamental properties of continuous functions and topological entropy
Recommended article: Estimates of topological entropy of continuous maps with applications
Recommended article: Q4715992
Recommended article: On the topological entropy of a semigroup of continuous maps
Recommended article: On a class of functions of the interval with zero topological entropy
Recommended article: Topological \(r\)-entropy and measure-theoretic \(r\)-entropy of a continuous map
Recommended article: Topological entropy on set-valued functions
Recommended article: On the set of continuity of the topological entropy of parameter-dependent mappings of the interval
Recommended article: Topological entropy for nonuniformly continuous maps
description: scientific article; zbMATH DE number 5956306