Compact non-uniformizable Li-Yorke chaotic dynamical systems via an example

TL;DR

This paper extends the concept of Li-Yorke chaos to non-uniform compact dynamical systems, specifically analyzing shift maps on finite topological spaces and establishing equivalent conditions for chaos.

Contribution

It introduces a framework for understanding Li-Yorke chaos in non-uniform compact spaces and proves equivalences for shift maps on finite topological spaces.

Findings

Li-Yorke chaos is equivalent to the existence of scrambled pairs in finite spaces.

Shift maps have at least one non-asymptotic pair if and only if they are Li-Yorke chaotic.

Certain topological conditions on the space characterize chaos in these systems.

Abstract

The main aim of this paper is extending the concept of scambled pair and Li--Yorke chaos to non--uniform compact dynamical systems. We show for finite (compact Alexandroff) topological space $X$ with at least two elements the following statements are equivalent: $\bullet$ one--sided shift $\sigma:X^{\mathbb{N}}\to X^\mathbb{N}$ is Li--Yorke chaotic, $\bullet$ one--sided shift $\sigma:X^{\mathbb{N}}\to X^\mathbb{N}$ has at least one scrambled pair, $\bullet$ one--sided shift $\sigma:X^{\mathbb{N}}\to X^\mathbb{N}$ has at least one non--asymptotic pair, $\bullet$ there exists $a,b\in X$ such that $\overline{\{a\}}\cap\overline{\{b\}}=\varnothing$, $\bullet$ $\bigcap\{\overline{\{a\}}:a\in X\}=\varnothing$.