Recurrence, pointwise almost periodicity and orbit closure relation for flows and foliations

TL;DR

This paper characterizes recurrence and almost periodicity for flows and foliations on surfaces and manifolds, linking these properties to minimality, orbit closure, and holonomy conditions.

Contribution

It provides new characterizations of recurrence, almost periodicity, and orbit closure relations for flows and codimension one foliations on compact manifolds.

Findings

Recurrence is equivalent to almost periodicity and minimality for flows.

Orbit closure relation is closed if and only if the flow is recurrent or minimal.

Pointwise almost periodic foliations are either minimal or compact, with $R$-closure equivalence.

Abstract

In this paper, we obtain a characterizations of the recurrence of a continuous vector field $w$ of a closed connected surface $M$ as follows. The following are equivalent: 1) $w$ is pointwise recurrent. 2)$w$ is pointwise almost periodic. 3) $w$ is minimal or pointwise periodic. Moreover, if $w$ is regular, then the following are equivalent: 1) $w$ is pointwise recurrent. 2)$w$ is minimal or the orbit space $M/w$ is either $[0,1]$, or $S^1$. 3) $R$ is closed (where $R := \{(x,y) \in M \times M \mid y \in \bar{O(x)} \}$ is the orbit closure relation). On the other hand, we show that the following are equivalent for a codimension one foliation $\mathcal{F}$ on a compact manifold: 1) $\mathcal{F}$ is pointwise almost periodic. 2) $\mathcal{F}$ is minimal or compact. 3) $\mathcal{F}$ is $R$-closed. Also we show that if a foliated space on a compact metrizable space is either minimal or is both compact and without infinite holonomy, then it is $R$-closed.