The structures of pointwise recurrent quasi-graph maps

TL;DR

This paper characterizes pointwise recurrent continuous maps on quasi-graphs, showing they are either conjugate to irrational rotations on circles or are periodic homeomorphisms, revealing the structure of such dynamical systems.

Contribution

It provides a complete classification of pointwise recurrent maps on quasi-graphs, identifying two distinct types based on their topological and dynamical properties.

Findings

Pointwise recurrent maps on quasi-graphs are either conjugate to irrational rotations or are periodic.

The classification simplifies understanding the dynamics of continuous maps on quasi-graphs.

The results connect topological conjugacy with recurrence properties in dynamical systems.

Abstract

We show that a continuous map $f$ from a quasi-graph $G$ to itself is pointwise recurrent if and only if one of the following two statements holds: (1) $X$ is a simple closed curve and $f$ is topologically conjugate to an irrational rotation on the unit circle $\mathbb S^1$; (2) $f$ is a perodic homeomorphism.