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 EX represent the subset of endpoints of X. Let APf denote the subset of almost periodic points of f, Rf be the subset of recurrent points of f, and Pf be the subset of periodic points of f. In this work, it is shown that Rf¯=APf¯ if and only if EX is countable. Also, we show that if EX is countable, then Rf=X (respectively, Rf¯=X) if and only if either X=S1, and f is a homeomorphism topologically conjugate to an irrational rotation, or Pf=X (respectively, Pf¯=X). In this setting, we derive that if EX is countable, then, on local dendrites ≠S1, transitivity = chaos.