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