When is a dynamical system mean sensitive?

Abstract

This article is devoted to study which conditions imply that a topological dynamical system is mean sensitive and which do not. Among other things we show that every uniquely ergodic, mixing system with positive entropy is mean sensitive. On the other hand we provide an example of a transitive system which is cofinitely sensitive or Devaney chaotic with positive entropy but fails to be mean sensitive.As applications of our theory and examples, we negatively answer an open question regarding equicontinuity/sensitivity dichotomies raised by Tu, we introduce and present results of locally mean equicontinuous systems and we show that mean sensitivity of the induced hyperspace does not imply that of the phase space.2010 Mathematics Subject Classification. 54H20, 37B10, 54B20, 37B05.

A Note on Mean Equicontinuity

“…Answering an open question in [15], it was proved by Li, Tu and Ye in [12] that a minimal mean equicontinuous system has discrete spectrum. We refer to [6,7,8,9,10,13] for further study on mean equicontinuity and related subjects.…”

Mean proximality and mean Li-Yorke chaos

“…Our aim is to add something to the version of Devaney chaos such that the chaos in the new sense implies mean Li-Yorke chaos. It is worth mentioning that there are Devaney chaotic examples (even with positive entropy) that are not mean sensitive [13]. So the first extra hypothesis we add is mean sensitivity.…”

Mean equicontinuity, complexity and applications

“…More complicated chaotic behaviors for some almost mean equicontinuous t.d.s. were discussed in [17,48,16]. To seek well-behaved local notions, the authors in [46] introduced the notion of Weyl-mean equicontinuity and its local version almost Weyl-mean equicontinuity.…”

“…This result was generalized to countable abelian group action by Zhu, Huang and Lian [65] recently. See [17] and [45] for more special kinds of local version of mean equicontinuity with zero entropy.…”

“…Similarly, one may define the notion of mean sensitivity, as was done in [46]. We will discuss the various notions of sensitivity related to various notions of mean equicontinuity briefly in Subsection 3.2, and we refer [46,31,1,15,17,18,44,49] for further readings.…”