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

TL;DR

This paper demonstrates that positive entropy dynamical systems contain large sets exhibiting mean Li-Yorke chaos, with dense intersections among various chaotic and entropy-related tuples, revealing complex chaotic structures.

Contribution

It introduces a multivariant version of mean Li-Yorke chaos in positive entropy systems and proves density results for intersections of chaotic and entropy tuples.

Findings

Existence of large sets with mean Li-Yorke chaos in positive entropy systems

Density of intersections of asymptotic, mean Li-Yorke, and entropy tuples

Chaotic structures are densely intertwined with entropy in these 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.