$\overline{R} = \overline{P}$ FOR MAPS OF DENDRITES X WITH Card(End(X)) < c

Abstract

Let X be a dendrite and f : X → X be a continuous map. Denote by R(f) and P(f) the sets of recurrent points and periodic points of f respectively. In this paper we show that, if the cardinal number Card ( End (X)) of the set of endpoints of X is less than the cardinal number c of the continuum, then [Formula: see text]. From this we derive that, if Card ( End (X)) < c, then f is chaotic in the sense of Devaney if and only if f is transitive.