Compact Dynamical Foliations

TL;DR

This paper investigates whether partially hyperbolic diffeomorphisms can have all compact center leaves with unbounded volume, concluding negatively under the condition of finite holonomy.

Contribution

It introduces new tools to analyze the structure of center foliations and proves that such unbounded volume leaves cannot occur if all periodic leaves have finite holonomy.

Findings

No examples of partially hyperbolic diffeomorphisms with unbounded volume compact leaves under finite holonomy.

Provides a negative answer to the existence of such foliations in the partially hyperbolic setting.

Develops analytical tools for studying the volume and holonomy of center leaves.

Abstract

According to the work of Dennis Sullivan, there exists a smooth flow on the 5-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, if 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 negative answer provided that all periodic leaves have finite holonomy.