Partially hyperbolic diffeomorphisms with compact center foliations

TL;DR

This paper investigates the structure of partially hyperbolic diffeomorphisms with compact center leaves, ruling out certain complex foliations and establishing conditions under which the system fibers over an Anosov automorphism.

Contribution

It proves that Sullivan's circle foliation cannot be the center foliation of such diffeomorphisms and identifies conditions for the system to fiber over an Anosov automorphism.

Findings

Sullivan's example cannot be the center foliation.

Finite covers of f fiber over Anosov automorphisms under specified conditions.

Characterization of center foliation structures in partially hyperbolic systems.

Abstract

Let f:M->M be a partially hyperbolic diffeomorphism such that all of its center leaves are compact. We prove that Sullivan's example of a circle foliation that has arbitrary long leaves cannot be the center foliation of f. This is proved by thorough study of the accessible boundaries of the center-stable and the center-unstable leaves. Also we show that a finite cover of f fibers over an Anosov toral automorphism if one of the following conditions is met: 1. the center foliation of f has codimension 2, or 2. the center leaves of f are simply connected leaves and the unstable foliation of f is one-dimensional.