Toral or non locally connected minimal sets for suspensions of $R$-closed surface homeomorphisms

TL;DR

This paper classifies minimal sets for suspensions of $R$-closed surface homeomorphisms, showing they are either toral and minimal or non-locally connected, and proves positive iterations preserve $R$-closedness.

Contribution

It provides a classification of minimal sets for suspensions of $R$-closed surface homeomorphisms and demonstrates that positive iterations maintain $R$-closedness.

Findings

Minimal sets are either toral and minimal or non-locally connected.

Positive iterations of $R$-closed homeomorphisms are also $R$-closed.

The classification applies to suspensions of $R$-closed surface homeomorphisms.

Abstract

Let $M$ be an orientable connected closed surface and $f$ be an $R$-closed homeomorphism on $M$ which is isotopic to identity. Then the suspension of $f$ satisfies one of the following condition: 1) the closure of each element of it is minimal and toral. 2) there is a minimal set which is not locally connected. Moreover, we show that any positive iteration of an $R$-closed homeomorphism on a compact metrizable space is $R$-closed.