[Paper Review] $R$-closed homeomorphisms on surfaces

Tomoo Yokoyama|arXiv (Cornell University)|May 16, 2012

Mathematical Dynamics and Fractals4 references3 citations

TL;DR

This paper classifies minimal sets of $R$-closed homeomorphisms on connected orientable closed surfaces, showing that on surfaces of genus >1, minimal sets are either periodic orbits or extensions of Cantor sets. On the torus, non-minimal, non-periodic maps yield finite unions of essential circloids or Cantor set extensions. On the sphere, non-periodic orientation-preserving maps have exactly two fixed points and circloids as minimal sets, with the orbit class space homeomorphic to $[0,1]$. The results extend to codimension-two foliations via suspension constructions.

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

Motivation & Objective

• To classify the structure of minimal sets for $R$-closed homeomorphisms on compact orientable surfaces.
• To determine the topological type of minimal sets—periodic orbits, circloids, or extensions of Cantor sets—depending on the surface genus.
• To establish the orbit class space structure $M/\widetilde{f}$ for non-minimal, non-periodic maps on $\mathbb{T}^2$ and $\mathbb{S}^2$.
• To apply the classification to codimension-two foliations via suspension of $R$-closed homeomorphisms.

Proposed method

• Uses the equivalence between $R$-closedness and pointwise almost periodicity in compact metrizable spaces to analyze orbit closures.
• Applies topological tools such as sequential compactness and connectedness of orbit closures under $R$-closedness.
• Employs the concept of circloids and annular continua to characterize minimal sets in terms of topological embedding and invariance.
• Leverages the orbit class space $M/\widilde{f}$ as a quotient space to deduce global topological structure.
• Applies results from dynamical systems theory, including theorems on minimal sets and holonomy in codimension-two foliations.
• Uses suspension constructions to lift $R$-closed homeomorphisms to codimension-two foliations, preserving $R$-closedness.

Experimental results

Research questions

• RQ1 What is the topological structure of minimal sets for $R$-closed homeomorphisms on surfaces of genus greater than one?
• RQ2 How do the minimal sets of $R$-closed homeomorphisms on the torus $\mathbb{T}^2$ differ when the map is neither minimal nor periodic?
• RQ3 What is the structure of minimal sets for non-periodic, orientation-preserving $R$-closed homeomorphisms on $\mathbb{S}^2$?
• RQ4 How does the orbit class space $M/\widetilde{f}$ reflect the dynamics of $R$-closed surface homeomorphisms?
• RQ5 Can $R$-closed homeomorphisms on $\mathbb{T}^2$ or $\mathbb{S}^2$ induce non-minimal, non-compact codimension-two foliations?

Key findings

• On surfaces of genus >1, every minimal set of an $R$-closed homeomorphism is either a periodic orbit or an extension of a Cantor set.
• On the torus $\mathbb{T}^2$, if $f$ is neither minimal nor periodic, then either all minimal sets are finite disjoint unions of essential circloids or there exists a minimal set that is an extension of a Cantor set.
• On the sphere $\mathbb{S}^2$, if $f$ is non-periodic and orientation-preserving, then the minimal sets consist of exactly two fixed points and other circloids, with $\mathbb{S}^2/\widetilde{f} \cong [0,1]$.
• For orientation-reversing $f$ on $\mathbb{S}^2$, the minimal sets of $f^2$ are two fixed points and circloids, and $\mathbb{S}^2/\widetilde{f}$ is a closed interval.
• Suspension of non-minimal, non-periodic $R$-closed homeomorphisms on $\mathbb{T}^2$ or $\mathbb{S}^2$ yields codimension-two $R$-closed foliations that are neither minimal nor compact.
• The orbit class space $M/\widetilde{f}$ is Hausdorff and homeomorphic to a closed interval or a circle, depending on whether $V$ (the set of connected orbit closures) has boundary components.