Li-Yorke chaos for dendrite maps with zero topological entropy and $\omega$-limit sets

Abstract

Let $X$ be a dendrite with set of endpoints $E(X)$ closed and let $f:X\to X$ be a continuous map with zero topological entropy. Let $P(f)$ be the set of periodic points of $f$. We prove that if $L$ is an infinite $\omega$-limit set of $f$ then $L\cap P(f)\subset E(X{)}^{\prime }$, where $E(X{)}^{\prime }$ is the set of all accumulations points of $E(X)$. Furthermore, if $E(X)$ is countable and $L$ is uncountable then $L\cap P(f)=\varnothing$. We also show that if $E(X{)}^{\prime }$ is finite then any uncountable $\omega$-limit set of $f$ has a decomposition and as a consequence if $f$ has a Li-Yorke pair $(x,y)$ with $\omega \_f(x)$ or $\omega \_f(y)$ is uncountable then $f$ is Li-Yorke chaotic.