Anomalous partially hyperbolic diffeomorphisms I: dynamically coherent examples

TL;DR

This paper constructs a novel example of a non-transitive, dynamically coherent partially hyperbolic diffeomorphism on a closed 3-manifold, challenging existing conjectures by combining Anosov flow maps with Dehn twists.

Contribution

It provides the first explicit example of such a diffeomorphism with exponential fundamental group growth, contradicting previous conjectures.

Findings

Example of a non-transitive, dynamically coherent partially hyperbolic diffeomorphism

Counterexample to a conjecture in the field

Demonstrates the use of Dehn twists with Anosov flow maps

Abstract

We build an example of a non-transitive, dynamically coherent partially hyperbolic diffeomorphism $f$ on a closed $3$-manifold with exponential growth in its fundamental group such that $f^n$ is not isotopic to the identity for all $n\neq 0$. This example contradicts a conjecture in \cite{HHU}. The main idea is to consider a well-understood time-$t$ map of a non-transitive Anosov flow and then carefully compose with a Dehn twist.