Partially Hyperbolic Dynamics with Quasi-isometric Center

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.