Li–Yorke sensitivity

Abstract

Bau-Sen Du introduced a notion of chaos which is stronger than LiYorke sensitivity. A TDS (X, f ) is called chaotic if there is a positive ε such that for any x and any nonempty open set V of X there is a point y in V such that the pair (x, y) is proximal but not ε-asymptotic. In this article, we show that a TDS (T, f ) is transitive but not mixing if and only if (T, f ) is Li-Yorke sensitive but not chaotic, where T is a tree. Moreover, we compare such chaos with other notions of chaos.

Dynamical compactness and sensitivity

“…The system (X, T ) is proximal if and only if (X, T ) has the unique fixed point, which is the only minimal point of (X, T ) (e.g. see [5]). …”

“…Furstenberg started a systematic study of transitive dynamical systems in his paper on disjointness in topological dynamics and ergodic theory [11], and the theory was further developed in [13] and [12]. The main motivation for this paper comes from [39], [2], [24], [5], [14], [22], [38], [37] and recent papers [29], [34] and [31], which discusses a dynamical property called transitive compactness examined firstly for weakly mixing systems in [5]: transitive compactness is quite related to but different from the property of transitivity, it will be equivalent to weak mixing under some weak conditions, and it presents some kind of sensitivity of the system.…”

“…The following property of dynamical systems, which was first examined in [5] for topologically weakly mixing systems, links the Auslander point dynamics property with topological transitivity. Let N T be the set of all subsets of Z + containing some N T (U, V ), where U, V are opene subsets of X.…”

“…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.…”

“…Nevertheless, as was shown in [26], for minimal topologically weakly mixing systems all these Lyapunov numbers are the same. The motivation of [26] comes from the following proposition according to [5]:…”