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

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 $\overline{\{a\}}\cap\overline{\{b\}}=\varnothing$, $\bullet$ $\bigcap\{\overline{\{a\}}:a\in X\}=\varnothing$.