Equicontinuity and sensitivity in mean forms
type: Item
instance of: scholarly article
title: Equicontinuity and sensitivity in mean forms
zbMATH Open document ID: 1496.37008
author: Jie Li
author: Xiangdong Ye
author: Tao Yu
published in: Journal of Dynamics and Differential Equations
publication date: 2022-03-17T00:00:00Z
review text: This paper deals with topological dynamical systems of the form \((X,T)\), where \(X\) is compact metric and \(T\) a continuous and surjective self map of \(X\). Starting from the observation that equicontinuous systems are `easy' the authors generalize further earlier weakenings equicontinuity: mean equicontinuity [\textit{J. Li} et al., Ergodic Theory Dyn. Syst. 35, No. 8, 2587--2612 (2015; Zbl 1356.37016)] and equicontinuity in the mean [\textit{W. Huang} et al., Ergodic Theory Dyn. Syst. 41, No. 2, 494--533 (2021; Zbl 1456.37015)]; both are equivalent for minimal systems. The further weakenings are called almost mean equicontinuity and almost equicontinuity in the mean; both localize their condition to a point. Both consider the behavior of the means \[M_n(x,y)=\frac1n\sum_{i=0}^nd(T^ix,T^iy)\] of the initial segments of sequences of distances \(d(T^ix,T^iy)\); the former wants \(\limsup_nM_n(x,y)\) to be small for all \(y\) close to a singled out~\(x\); the latter (stronger) condition wants \(\sup_nM_n(x,y)\) to be small.\par The authors provide an example that shows that almost equicontinuity in the mean is indeed stronger than almost mean equicontinuity, and also of a system that is almost equicontinuous in the mean and exhibits Devaney chaos and uniform positive entropy.\par Next there is a treatment of variants of sensitivity: mean sensitivity and sensitivity in the mean. The authors give sufficient conditions for these properties to hold and show that they are equivalent for minimal systems. The paper ends with an investigation of these notions in the context of measure-preserving systems.
reviewed by: K. P. Hart
Mathematics Subject Classification ID: 37B05
Mathematics Subject Classification ID: 37A05
Mathematics Subject Classification ID: 37B02
zbMATH DE Number: 7491605
zbMATH Keywords: mean almost equicontinuity
zbMATH Keywords: almost equicontinuity in the mean
zbMATH Keywords: mean sensitivity
zbMATH Keywords: sensitivity in the mean
MaRDI profile type: Publication
full work available at URL: s10884-021-09945-9
OpenAlex ID: W3127690185
Wikidata QID: Q115383295
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DOI: 10.1007/S10884-021-09945-9
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description: scientific article; zbMATH DE number 7491605