Codimension one partially hyperbolic diffeomorphisms with a uniformly compact center foliation

TL;DR

This paper studies partially hyperbolic diffeomorphisms with compact center foliations, showing under certain conditions that the quotient space is a torus with hyperbolic automorphism, and establishing spectral decomposition and invariant measures.

Contribution

It proves the holonomy vanishes, the quotient is a torus with hyperbolic automorphism, and extends results without restrictions on dimensions, including spectral decomposition and invariant measures.

Findings

Holonomy of the center foliation vanishes.

The quotient space is a torus with a hyperbolic automorphism.

Existence of a holonomy invariant family of measures on unstable leaves.

Abstract

We consider a partially hyperbolic C1-diffeomorphism f on a smooth compact manifold M with a uniformly compact f-invariant center foliation. We show that if the unstable bundle is one-dimensional and oriented, then the holonomy of the center foliation vanishes everywhere, the quotient space of the center foliation is a torus and f induces a hyperbolic automorphism on it, in particular, f is centrally transitive. We actually obtain further interesting results without restrictions on the unstable, stable and center dimension: we prove a kind of spectral decomposition for the chain recurrent set of the quotient dynamics, and we establish the existence of a holonomy invariant family of measures on the unstable leaves (Margulis measure).