Relatively pointwise recurrence on local dendrites

Abstract

Let X be a local dendrite and let f : X → X be a continuous map. Denote by P ( f ) and R ( f ) the sets of periodic and recurrent points of f, respectively. We say that f is relatively pointwise recurrent if R ( f ) ¯ = X . We show that, if X is almost meshed (i.e the union of the interior of all free arcs is dense in X), then f is relatively pointwise recurrent if and only if either f is topologically conjugate to an irrational rotation on the circle or it has a dense set of periodic points.