Resumen de On The Omega-Limit Map on 1-Dimensional Continua

Ghassen Askri

• 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 ω 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 ω f (a) is infinite, then ω f is continuous at a if, and only if, f is equicontinuous at a. We show that the later result fails whenever ω 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 ω f is continuous, f|X∞ 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∞ imply the equicontinuity of f .