Anomalous partially hyperbolic diffeomorphisms I: dynamically coherent examples

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 $n ≠ 0$. This example contradicts a conjecture in [F. RODRIGUEZ HERTZ, M. A. RODRIGUEZ HERTZ, R. URES, Partial hyperbolicity in 3-manifolds] The main idea is to consider a well-understood time-t map of a non-transitive Anosov ßow and then carefully compose with a Dehn twist.