Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part I: The dynamically coherent case

TL;DR

This paper classifies 3D dynamically coherent partially hyperbolic diffeomorphisms homotopic to the identity on certain 3-manifolds, showing they are leaf conjugate to time-one maps of Anosov flows, confirming a conjecture.

Contribution

It proves a classification conjecture for such diffeomorphisms on hyperbolic and Seifert fibered 3-manifolds, linking them to Anosov flows.

Findings

Dichotomy in the structure of center stable and unstable foliations.

Every such diffeomorphism is leaf conjugate to a time-one map of an Anosov flow.

Results confirm the Hertz-Hertz-Ures classification conjecture.

Abstract

We study 3-dimensional dynamically coherent partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the transverse geometry and topology of the center stable and center unstable foliations, and the dynamics within their leaves. We find a structural dichotomy for these foliations, which we use to show that every such diffeomorphism on a hyperbolic or Seifert fibered 3-manifold is leaf conjugate to the time one map of a (topological) Anosov flow. This proves a classification conjecture of Hertz-Hertz-Ures in hyperbolic 3-manifolds and in the homotopy class of the identity of Seifert manifolds.