Completely Li--Yorke chaotic homeomorphisms with positive entropy

TL;DR

This paper constructs examples of homeomorphisms on compact metric spaces that are completely Li--Yorke chaotic and have positive entropy, answering an open question about their coexistence.

Contribution

It demonstrates that completely Li--Yorke chaotic homeomorphisms with positive entropy can exist, expanding understanding of chaos and entropy in dynamical systems.

Findings

Constructed Li--Yorke chaotic homeomorphisms with positive entropy

Showed such chaos can coexist with positive entropy in compact spaces

Provided a method to associate chaotic homeomorphisms with given positive entropy maps

Abstract

It is an open problem whether a homeomorphism on a compact metric space satisfying that each proper pair is either positively or negatively Li--Yorke, called completely Li--Yorke chaotic, can have positive entropy. In the present paper, an affirmative answer to this question is given. In fact, for each homeomorphism $T$ with positive entropy such that each proper pair is not two-sided asymptotic, a completely Li--Yorke chaotic homeomorphism with positive entropy associated with the given homeomorphism can be constructed.