Mean equicontinuity and mean sensitivity

Abstract

ABSTRACT. Answering an open question affirmatively it is shown that every ergodic invariant measure of a mean equicontinuous (i.e. mean-L-stable) system has discrete spectrum. Dichotomy results related to mean equicontinuity and mean sensitivity are obtained when a dynamical system is transitive or minimal.Localizing the notion of mean equicontinuity, notions of almost mean equicontinuity and almost Banach mean equicontinuity are introduced. It turns out that a system with the former property may have positive entropy and meanwhile a system with the later property must have zero entropy.

Measure-theoretic mean equicontinuity and bounded complexity

“…In the study of discrete spectrum, Fomin introduced the concept of stability in the mean in the sense of Lyapunov (mean-L-stability for short) in [6]. It was shown in [22] that mean-L-stability is equivalent to mean equicontinuity. In [22], Li, Tu and Ye showed that every ergodic measure in a mean equicontinuous system has discrete spectrum.…”

Constant length substitutions, iterated function systems and amorphic complexity

“…This notion was introduced by Li, Tu and Ye in [LTY15]. It is immediately seen to be equivalent to the concept of mean Lyapunov-stability which was introduced in 1951 by Fomin [Fom51] in the context of systems with discrete spectrum.…”

Dynamical compactness and sensitivity

“…Since then, many authors studied different properties related to sensitivity (cf. [16], [3], [15], [2], [4], [5], [1], [21], [30]). For the recent development of sensitivity in topological dynamics see for example the survey [31] by Li and Ye. According to the works by Guckenheimer [17], Auslander and Yorke [7] a dynamical system (X, T ) is sensitive if there exists δ > 0 such that for every x ∈ X and every neighborhood U x of x, there exist y ∈ U x and n ∈ N with d(T n x, T n y) > δ.…”