Equicontinuity and sensitivity in mean forms
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.
scientific article; zbMATH DE number 7491605
scientific article