Collapsed Anosov flows and self orbit equivalences

Abstract

We propose a generalization of the concept of discretized Anosov flows that covers a wide class of partially hyperbolic diffeomorphisms in 3-manifolds, and that we call collapsed Anosov flows. They are related to Anosov flows via a self orbit equivalence of the flow. We show that all the examples from Bonatti, Gogolev, Hammerlindl, and Potrie [Geom. Topol. 24 (2020), 1751–1790] belong to this class, and that it is an open and closed class among partially hyperbolic diffeomorphisms. We provide some equivalent definitions which may be more amenable to analysis and are useful in different situations. Conversely, we describe the isotopy classes of partially hyperbolic diffeomorphisms that are collapsed Anosov flows associated with certain types of Anosov flows.