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

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.