Normally Hyperbolic Circle Foliations

TL;DR

This paper proves that normally hyperbolic circle foliations have a finite upper bound on leaf length, contrasting with Sullivan's example, and explores dynamical consequences of bounded volume leaves in partially hyperbolic systems.

Contribution

It establishes that normally hyperbolic circle foliations cannot have unbounded leaf length and links bounded volume leaves to dynamical coherence and plaque expansiveness.

Findings

No unbounded leaf length in normally hyperbolic circle foliations.

Bounded volume leaves imply dynamical coherence.

Provides short proofs of dynamical consequences.

Abstract

In 1976 D. Sullivan gave an example of a flow on a compact manifold such that each one of its orbits is a circle and with the surprising property that there is no finite upper bound for their length. The aim of this article is to show that these type of examples do not appear as normally hyperbolic foliations. Namely, we prove that if a circle foliation is the center foliation of a (dynamically coherent) partially hyperbolic diffeormophism, then there is a finite upper bound for the length of the leaves. We also give short proofs of some dynamical consequences in the converse case: if the center foliation of a partially hyperbolic diffeomorphism $f$ is by compact leaves with uniformly bounded volume, then $f$ is dynamically coherent and plaque expansive.