Partially Hyperbolic Dynamics with Quasi-isometric Center

TL;DR

This paper proves ergodicity and classifies certain partially hyperbolic diffeomorphisms on 3-manifolds with quasi-isometric center, confirming a conjecture and revealing their structure as skew products or discretized Anosov flows.

Contribution

It establishes ergodicity for volume-preserving cases without $su$-tori and classifies these diffeomorphisms into two main types, advancing understanding of their dynamics.

Findings

Volume-preserving diffeomorphisms are ergodic without $su$-tori.

Existence of transitive Anosov flows with quasi-isometric center.

Classification into skew products and discretized Anosov flows.

Abstract

We consider the class of partially hyperbolic diffeomorphisms on a closed 3-manifold with quasi-isometric center. Under the non-wandering condition, we prove that the diffeomorphisms are accessible if there is no $su$-torus. As a consequence, volume-preserving diffeomorphisms in this context are ergodic in the absence of $su$-tori, thereby confirming the Hertz-Hertz-Ures Ergodicity Conjecture for this class. We show the existence of transitive Anosov flows on a closed 3-manifold admitting a non-wandering partially hyperbolic diffeomorphism with quasi-isometric center and fundamental group of exponential growth. Furthermore, we provide a complete classification of these diffeomorphisms, showing they fall into two categories: skew products and discretized Anosov flows.