Recurrent and Non-wandering properties for foliations

TL;DR

This paper investigates the properties of recurrence and non-wandering in codimension one foliations on closed 3-manifolds, establishing inclusion relations and characterizing minimal and compact foliations based on leaf properties.

Contribution

It introduces the concepts of recurrence and non-wandering for decompositions and characterizes foliations with specific properties in terms of leaf dynamics and fundamental group growth.

Findings

Inclusion relations among minimal, compact, almost periodic, recurrent, non-wandering, and Reebless foliations.

Characterization of minimal and compact foliations based on leaf end count and holonomy.

All leaves have the same polynomial growth if and only if the foliation has no holonomy and contains a leaf with polynomial growth.

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.