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