Recurrence, pointwise almost periodicity and orbit closure relation for flows and foliations (Q386143)

The author characterizes the pointwise recurrence of a continuous vector field \(w\) of a closed connected surface \(M\) in terms of pointwise almost periodicity and in terms of minimality or pointwise periodicity. In the case where \(w\) is regular, the characterization of pointwise recurrence involves the orbit space \(M/w\) and the orbit closure relation. If \({\mathcal F}\) is a codimension one foliation on a compact connected manifold, the author proves that the pointwise almost periodicity of the foliation is equivalent to the minimality or the compactness of \({\mathcal F}\) and also to the \(R\)-closedness of the foliation (\(R\) is the orbit closure relation). He also shows 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.