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 / w is either [ 0 , 1 ] , or S 1 . (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 F on a compact connected manifold: (1) F is pointwise almost periodic. (2) F is minimal or compact. (3) F 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.