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arXiv · 1205.3634

$R$-closed homeomorphisms on surfaces

Tomoo Yokoyama
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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 $\mathbb{S}^2/\widetilde{f} \cong [0, 1]$.

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Tomoo Yokoyama. 2012-05-16. $R$-closed homeomorphisms on surfaces. https://arxiv.org/abs/1205.3634

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