On The Omega-Limit Map on 1-Dimensional Continua

Abstract

Let X be a compact connected metric space and $f :~ X \rightarrow X$ be a continuous map. In this paper, we prove that if f has a periodic point and $\omega _f$ is continuous then the almost periodic set is a finite union of cyclically permuted subcontinua of X. In particular, AP(f) is connected whenever f has a fixed point. Also we show that for dendrites with closed endpoint set, if $\omega _f (a)$ is infinite, then $\omega _f$ is continuous at a if, and only if, f is equicontinuous at a. We show that the later result fails whenever $\omega _f (a)$ is finite or the endpoints set is not closed. We give an example of a local dendrite map $f :~ X \rightarrow X$ for which $\omega _f$ is continuous, $f_{|X_{\infty }}$ is equicontinuous but f is not equicontinuous on the whole space X. Finally, we answer to an open question raised by Acosta and Fernández, (Equicontinuous mappings on finite trees, Fund. Math) by providing a class of dendrites on which the equicontinuity of $f_{|X_{\infty }}$ imply the equicontinuity of f.