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

Anomalous partially hyperbolic diffeomorphisms I: dynamically coherent examples