Pointwise-recurrent dendrite maps

TL;DR

This paper characterizes the conditions under which continuous maps on dendrites are pointwise-recurrent, linking recurrence to periodicity and the structure of endpoints and branch points.

Contribution

It establishes a precise relationship between the structure of dendrites and the recurrence properties of continuous maps on them.

Findings

f is pointwise-recurrent iff it is a pointwise periodic homeomorphism when E(D) is countable.

If B(D) is discrete, then every point outside endpoints is periodic.

Recurrence properties are characterized by the topological structure of dendrites.

Abstract

Let D be a dendrite and f:D-> D a continuous map. Denote by E(D) and B(D) the sets of endpoints and branch points of D respectively. We show that if E(D) is countable (resp. B(D) is discrete) then f is pointwise-recurrent if and only if f is pointwise periodic homeomorphism (resp. every pointin D\E(D) is periodic).