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