Recurrent and Non-wandering properties for foliations

Abstract

In this paper, we define the recurrence and "non-wandering" for decompositions. The following inclusion relations hold for codimension one foliations on closed $3$-manifolds: $\{$minimal$\} \sqcup \{$compact$\}$ $\subsetneq$ $\{$pointwise almost periodic$\}$ $\subsetneq$ $\{$recurrent$\}$ $\subsetneq$ $\{$non-wandering$\}$ $\subsetneq$ $\{$Reebless$\}$. A non-wandering codimension one $C^2$ foliation on a closed connected $3$-manifold which has no leaf with uncountably many ends is minimal (resp. compact) if and only if it has no compact (resp. locally dense) leaves. In addition, the fundamental groups of all leaves of a codimension one transversely orientable $C^2$ foliation $\mathcal{F}$ on a closed $3$-manifold have the same polynomial growth if and only if $\mathcal{F}$ is without holonomy and has a leaf whose fundamental group has polynomial growth.