On conservative partially hyperbolic abelian actions with compact center foliation

TL;DR

This paper studies smooth volume-preserving partially hyperbolic actions with compact center foliation, establishing conditions under which such actions are fiber bundle extensions of Anosov actions or have pathological center foliations.

Contribution

It provides a classification theorem for these actions, showing they are either fiber bundle extensions of Anosov actions or have pathological center foliations under certain conditions.

Findings

Actions are fiber bundle extensions of Anosov actions under irreducibility, bunching, and quasiconformality.

Global dichotomy: circle extensions are either products of rotations and linear Anosov actions or have pathological foliations.

Results generalize understanding of partially hyperbolic actions with compact center foliation.

Abstract

We consider smooth partially hyperbolic volume preserving Z^k actions on smooth manifolds, with uniformly compact center foliation. We show that under certain irreducibility condition on the action, bunching and uniform quasiconformality conditions, the action is a smooth fiber bundle extension of an Anosov action, or the center foliation is pathological. We obtain several corollaries of this result. For example, we prove a global dichotomy result that any smooth conservative circle extension over a maximal Cartan action is either essentially a product of an action by rotations and a linear Anosov action on the torus, or has a pathological center foliation.