Codimension one compact center foliations are uniformly compact

TL;DR

This paper proves that for a specific class of partially hyperbolic dynamical systems with compact center leaves and one-dimensional unstable bundle, the volume of these leaves is uniformly bounded across the manifold.

Contribution

It establishes a uniform bound on the volume of center leaves in partially hyperbolic systems with one-dimensional unstable bundle, under compactness and coherence assumptions.

Findings

Center leaves have uniformly bounded volume in the given setting.

The result applies to dynamically coherent partially hyperbolic diffeomorphisms with one-dimensional unstable bundle.

Compactness of leaves is a key assumption for the volume bound.

Abstract

Let $f:M\to M$ be a dynamically coherent partially hyperbolic diffeomorphism whose center foliation has all its leaves compact. We prove that if the unstable bundle of $f$ is one-dimensional, then the volume of center leaves must be bounded in $M$.