$R$-closed homeomorphisms on surfaces

Abstract

Let $f$ be an $R$-closed homeomorphism on a connected orientable closed surface $M$. In this paper, we show that If $M$ has genus more than one, then each minimal set is either a periodic orbit or an extension of a Cantor set. If $M={\mathbb{T}}^2$ and $f$ is neither minimal nor periodic, then either each minimal set is finite disjoint union of essential circloids or there is a minimal set which is an extension of a Cantor set. If $M={\mathbb{S}}^2$ and $f$ is not periodic but orientation-preserving (resp. reversing), then the minimal sets of $f$ (resp. $f^2$) are exactly two fixed points and other circloids and ${\mathbb{S}}^2/\tilde{f}\cong [0,1]$.

References in corpus (2)

• Aperiodic invariant continua for surface homeomorphisms
• A classification of minimal sets for surface homeomorphisms

Cited by in corpus (2)

• R-closedness and Upper semicontinuity
• Toral or non locally connected minimal sets for suspensions of $R$-closed surface homeomorphisms