Recurrence and Chaos of Local Dendrite Maps

Abstract

Let $X$ be a local dendrite, and let $f$ be a continuous self-mapping of $X$ . Let $E\left(X\right)$ represent the subset of endpoints of $X$ . Let $AP\left(f\right)$ denote the subset of almost periodic points of $f$ , $R\left(f\right)$ be the subset of recurrent points of $f$ , and $P\left(f\right)$ be the subset of periodic points of $f$ . In this work, it is shown that $\overline{R\left(f\right)}=\overline{AP\left(f\right)}$ if and only if $E\left(X\right)$ is countable. Also, we show that if $E\left(X\right)$ is countable, then $R\left(f\right)=X$ (respectively, $\overline{R\left(f\right)}=X$ ) if and only if either $X={\mathbb{S}}^1$ , and $f$ is a homeomorphism topologically conjugate to an irrational rotation, or $P\left(f\right)=X$ (respectively, $\overline{P\left(f\right)}=X$ ). In this setting, we derive that if $E\left(X\right)$ is countable, then, on local dendrites  $\ne {\mathbb{S}}^1$ , transitivity = chaos.