# Anomalous partially hyperbolic diffeomorphisms III: abundance and incoherence

## TL;DR

This paper develops techniques to construct partially hyperbolic diffeomorphisms on 3-manifolds with Anosov flows, demonstrating their abundance, properties, and counterexamples to existing conjectures in dynamical systems.

## Contribution

It introduces a new method using $h$-transversality to build partially hyperbolic diffeomorphisms across various mapping classes, including stable ergodicity and non-coherence.

## Key findings

- Constructed stably ergodic, partially hyperbolic diffeomorphisms on hyperbolic surfaces.
- Showed the set of realizable mapping classes is not a subgroup.
- Provided examples that are absolutely partially hyperbolic and non-dynamically coherent.

## Abstract

Let $M$ be a closed 3-manifold which admits an Anosov flow. In this paper we develop a technique for constructing partially hyperbolic representatives in many mapping classes of $M$. We apply this technique both in the setting of geodesic flows on closed hyperbolic surfaces and for Anosov flows which admit transverse tori. We emphasize the similarity of both constructions through the concept of $h$-transversality, a tool which allows us to compose different mapping classes while retaining partial hyperbolicity. In the case of the geodesic flow of a closed hyperbolic surface $S$ we build stably ergodic, partially hyperbolic diffeomorphisms whose mapping classes form a subgroup of the mapping class group $\mathcal{M}(T^1S)$ which is isomorphic to $\mathcal{M}(S)$. At the same time we show that the totality of mapping classes which can be realized by partially hyperbolic diffeomorphisms does not form a subgroup of $\mathcal{M}(T^1S)$. Finally, some of the examples on $T^1S$ are absolutely partially hyperbolic, stably ergodic and robustly non-dynamically coherent, disproving a conjecture by F. Rodriguez Hertz, J. Rodriguez Hertz and R. Ures.