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

In this paper, we obtain a characterization 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/wM/w is either [0,1][0,1], or S1S1. (3) R   is closed (where R:={(x,y)∈M×M|y∈O(x)¯} is the orbit closure relation). On the other hand, we show that the following are equivalent for a codimension one foliation FF on a compact connected manifold: (1) FF is pointwise almost periodic. (2) FF is minimal or compact. (3) FF is R-closed. Also we show that if a foliated space on a compact metrizable space is either minimal or both compact and without infinite holonomy, then it is R-closed.