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

TL;DR

This paper proves that for graph maps, the existence of a single scrambled pair guarantees Li-Yorke chaos, extending known results from interval and circle maps to more general graph structures.

Contribution

It establishes that a scrambled pair implies Li-Yorke chaos for graph maps and shows that on countable spaces, a scrambled pair leads to an infinite scrambled set.

Findings

A scrambled pair implies Li-Yorke chaos for graph maps.

On countable spaces, a scrambled pair guarantees an infinite scrambled set.

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., $\liminf_{n\to+\infty}d(f^n(x),f^n(y))=0$ 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.