Pointwise-recurrent maps on uniquely arcwise connected locally arcwise connected spaces

TL;DR

This paper characterizes pointwise-recurrent self-maps on uniquely arcwise connected locally arcwise connected spaces, linking recurrence to periodicity of cutpoints and endpoints, and introduces the concept of ray completeness.

Contribution

It establishes a precise criterion for recurrence based on the periodicity of special points and introduces the notion of ray complete spaces where weak and classical adding machines coincide.

Findings

Pointwise-recurrence is characterized by periodic cutpoints.

Endpoints are either periodic or belong to topological weak adding machines.

In ray complete spaces, weak adding machines are equivalent to classical adding machines.

Abstract

We prove that self-mappings of uniquely arcwise connected locally arcwise connected spaces are pointwise-recurrent if and only if all their cutpoints are periodic while all endpoints are either periodic or belong to what we call "topological weak adding machines". We also introduce the notion of a \emph{ray complete} uniquely arcwise connected locally arcwise connected space and show that for them the above "topological weak adding machines" coincide with classical adding machines (e.g., this holds if the entire space is compact).