Anomalous partially hyperbolic diffeomorphisms. II: Stably ergodic examples (Q2520611)

The authors provide counterexamples to a conjecture of Pujals (see [\textit{B. Hasselblatt} (ed.) and \textit{A. Katok} (ed.), Handbook of dynamical systems. Volume 1B. Amsterdam: Elsevier (2006; Zbl 1081.00006)]). Pujals' conjecture asserts that transitive partially hyperbolic diffeomorphisms on 3-manifolds (up to taking finite lifts and iterates) belong to one of the following classes: (1) deformations of a linear Anosov automorphism of 3-torus; (2) deformations of skew products over a linear Anosov automorphism of 2-torus; (3) deformations of time-one maps of Anosov flows. The authors present two examples of closed orientable 3-manifold \(M\) and absolutely partial hyperbolic diffeomorphism \({f:M\to M}\) satisfying the following properties: \(M\) admits an Anosov flow; \(f\) is volume preserving, robustly transitive and stably ergodic; \(f^n\) is not homotopic to the identity map for all \({n>0}.\) The first construction relies on the time-one map of a geodesic flow on a hyperbolic surface and uses the Dehn twist along some closed geodesic. The second construction is an adaptation of an example from the first part of the paper [Ann. Sci. Éc. Norm. Supér. (4) 49, No. 6, 1387--1402 (2016; Zbl 1375.37093)]. It is based on a transitive Anosov flow which admits a transverse torus disjoint with periodic orbits. The new examples of partially hyperbolic diffeomorphisms motivate the authors to ask several questions which are listed at the beginning of the paper. For Part III, see [the authors, Geom. Topol. 24, No. 4, 1751--1790 (2020; Zbl 1470.37049)].