Stable sets and mean Li–Yorke chaos in positive entropy systems

Abstract

It is shown that in a topological dynamical system with positive entropy, there is a measure-theoretically "rather big" set such that a multivariant version of mean Li-Yorke chaos happens on the closure of the stable or unstable set of any point from the set. It is also proved that the intersections of the sets of asymptotic tuples and mean Li-Yorke tuples with the set of topological entropy tuples are dense in the set of topological entropy tuples respectively.2000 Mathematics Subject Classification. Primary 37B05; Secondary 54H20.

Mean Li–Yorke chaos along some good sequences

“…. , x n ) ∈ X n : lim inf By a similar argument as in [20,Lemma 2.4], MP n (X , T ) is a G δ subset of X n . It is clear that (x 1 , x 2 , .…”

“…For convenience, our notations will be as close to [20] as possible. For the reader who is not familiar with notions in ergodic theory and dynamical systems, we refer to [12,33].…”

“…It is natural to study "sequence versions" of dynamical notions (e.g., sequence entropy). In order to make our statement explicit, we restate the main theorem of [20] in the following, which concerns the terminology of multivariant mean Li-Yorke chaos. By a topological dynamical system, we mean a pair (X , T ), where X is a compact metric space equipped with a metric d and T is a homeomorphism of X onto itself.…”

“…The proof of Theorem 1.5 is based on the techniques used in [20]. A main new ingredient in this paper is the convergence argument for sequences given in Section 3.…”

Mean Li-Yorke chaos in Banach spaces

“…In recent years, it has become popular in the area of dynamical systems to investigate notions related to averages involving orbits or pseudo-orbits, such as mean Li-Yorke chaos [24,29], mean equicontinuity and mean sensitivity [31], and notions of shadowing with average error in tracing [39].…”