Centralizers of partially hyperbolic diffeomorphisms in dimension 3

TL;DR

This paper classifies the centralizers of volume-preserving, partially hyperbolic diffeomorphisms homotopic to identity on certain 3-manifolds, extending previous classifications in hyperbolic geometry.

Contribution

It provides a classification of centralizers for a broad class of partially hyperbolic diffeomorphisms in dimension 3, building on recent manifold classification results.

Findings

Centralizers are classified for volume-preserving partially hyperbolic diffeomorphisms.

Results extend previous classifications from geodesic flows to more general 3-manifolds.

The approach relies on recent advances in 3-manifold topology and dynamics.

Abstract

In this note we describe centralizers of volume preserving partially hyperbolic diffeomorphisms which are homotopic to identity on Seifert fibered and hyperbolic 3-manifolds. Our proof follows the strategy of Damjanovic, Wilkinson and Xu (arXiv:1902.05201) who recently classified the centralizer for perturbations of time-$1$ maps of geodesic flows in negative curvature. We strongly rely on recent classification results in dimension 3 established in (arXiv:1908.06227).