Periodic leaves for diffeomorphisms preserving codimension one foliations

Abstract

We consider the group of diffeomorphisms of a compact manifold $M$ which preserve a codimension one foliation $F$ on $M$. For the $C^2$ case if $F$ has compact leaves with nontrivial holonomy then at least one of these leaves is periodic. Our main result is proved in the context of diffeomorphisms which preserve commutative actions of finitely generated groups on $[0,1]$. Applying this result to foliations almost without holonomy we prove the periodicity of all compact leaves with nontrivial holonomy. We also study the codimension one foliation preserving diffeomorphisms that are $C^2$ close to the identity.