For graph maps, one scrambled pair implies Li-Yorke chaos

Abstract:

For a dynamical system $(X,f)$, $X$ being a compact metric space with metric $d$ and $f$ being a continuous map $X\to X$, a set $S\subseteq X$ is scrambled if every pair $(x,y)$ of distinct points in $S$ is scrambled, i.e., \[ \liminf _{n\to +\infty }d(f^n(x),f^n(y))=0 \hbox { and } \limsup _{n\to +\infty }d(f^n(x),f^n(y))>0.\] The system $(X,f)$ is Li-Yorke chaotic if it has an uncountable scrambled set. It is known that for interval and circle maps, the existence of a scrambled pair implies Li-Yorke chaos, in fact, the existence of a Cantor scrambled set. We prove that the same result holds for graph maps. We further show that on compact countable metric spaces one scrambled pair implies the existence of an infinite scrambled set.