Dynamics of homeomorphisms of regular curves

TL;DR

This paper investigates the structure and properties of homeomorphisms on regular curves, establishing the closedness of minimal sets, characterizing non-wandering sets, and analyzing the continuity of omega-limit maps, with implications for minimal set uniqueness.

Contribution

It proves the closedness of minimal sets, characterizes non-wandering sets, and links omega-limit map continuity to limit set equality, advancing understanding of dynamical behavior on regular curves.

Findings

Minimal sets form a closed subset in the hyperspace of closed sets.

The non-wandering set equals the set of recurrent points.

Uniqueness of infinite minimal set when no periodic points exist.

Abstract

In this paper, we prove first that the space of minimal sets of any homeomorphisms $f:X\to X$ of a regular curve $X$ is closed in the hyperspace $2^X$ of closed subsets of $X$ endowed with the Hausdorff metric, and the non-wandering set $\Omega (f)$ is equal to the set of recurrent points of $f$. Second, we study the continuity of the map $\omega_f:X\to 2^X;x\mapsto \omega_f (x) $, we show for instance the equivalence between the continuity of $\omega_f $ and the equality between the $\omega$-limit set and the $\alpha $-limit set of every point in $X $. Finally, we prove that there is only one (infinite) minimal set when there is no periodic point.