Abstract
Let X be a compact connected metric space and f : X -> 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 -> X for which omega(f) is continuous, f vertical bar X-infinity is equicontinuous but f is not equicontinuous on the whole space X. Finally, we answer to an open question raised by Acosta and Fernandez, (Equicontinuous mappings on finite trees, Fund. Math) by providing a class of dendrites on which the equicontinuity of f vertical bar X-infinity imply the equicontinuity of f.