R-closedness and Upper semicontinuity

TL;DR

This paper investigates the properties of R-closedness and upper semicontinuity in flows on compact spaces, establishing conditions under which these properties are equivalent and analyzing the structure of orbit spaces and minimal sets.

Contribution

It provides new characterizations of R-closedness via upper semicontinuity, explores the topology of orbit class spaces, and classifies possible structures of vector fields on 3-manifolds.

Findings

R-closedness is equivalent to upper semicontinuity of the orbit class decomposition.

Orbit class spaces of certain flows are compact manifolds with conners.

Classifies possible orbit space structures for nontrivial R-closed vector fields on 3-manifolds.

Abstract

Let $\mathcal{F} $ be a pointwise almost periodic decomposition of a compact metrizable space $X$. Then $\mathcal{F} $ is $R$-closed if and only if $\hat{\mathcal{F}} $ is usc. Moreover, if there is a finite index normal subgroup $H$ of an $R$-closed flow $G$ on a compact manifold such that the orbit closures of $H$ consist of codimension $k$ compact connected elements and "few singularities" for $k = 1$ or 2, then the orbit class space of $G$ is a compact $k$-dimensional manifold with conners. In addition, let $v$ be a nontrivial $R$-closed vector field on a connected compact 3-manifold $M$. Then one of the following holds: 1) The orbit class space $M/ \hat{v}$ is $[0,1]$ or $S^1$ and each interior point of $M/ \hat{v}$ is two dimensional. 2) $\mathrm{Per}(v)$ is open dense and $M = \mathrm{Sing}(v) \sqcup \mathrm{Per}(v)$. 3) There is a nontrivial non-toral minimal set. On the other hand, let $G$ be a flow on a compact metrizable space and $H$ a finite index normal subgroup. Then we show that $G$ is $R$-closed if and only if so is $H$.