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

TL;DR

This paper investigates Li-Yorke chaos in dendrite maps with zero topological entropy, establishing conditions on $oldsymbol{ ext{omega}}$-limit sets and endpoint accumulation points that imply chaotic behavior.

Contribution

It provides new results linking $oldsymbol{ ext{omega}}$-limit sets, endpoint accumulation points, and Li-Yorke chaos for dendrite maps with zero entropy.

Findings

Uncountable $oldsymbol{ ext{omega}}$-limit sets intersect periodic points only at endpoint accumulation points.

If $E(X)$ is countable and $oldsymbol{ ext{omega}}$-limit set is uncountable, then it contains no periodic points.

Finite accumulation points of $E(X)$ imply uncountable $oldsymbol{ ext{omega}}$-limit sets have a decomposition, leading to Li-Yorke chaos.

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 $L\cap P(f)=\emptyset$. 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.