Compact dynamical foliations

Abstract

According to the work of Dennis Sullivan [A counterexample to the periodic orbit conjecture.Publ. Math. Inst. Hautes Études Sci. 46(1976), 5–14], there exists a smooth flow on the five-sphere all of whose orbits are periodic, although there is no uniform bound on their periods. The question addressed in this article is whether these type of examples can occur in the partially hyperbolic context. That is, does there exist a partially hyperbolic diffeomorphism of a compact manifold such that all the leaves of its center foliation are compact but there is no uniform bound for their volumes? We develop tools to attack the previous question and show that it has a negative answer provided that all periodic leaves have finite holonomy.