On hyperbolic and Seifert fibered 3-manifolds, dynamically coherent partially hyperbolic diffeomorphisms homotopic to the identity are, up to iterate, leaf conjugate to time-one maps of topological Anosov flows.
A dichotomy for measures of maximal entropy near time-one maps of transitive Anosov flows
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abstract
We show that time-one maps of transitive Anosov flows of compact manifolds are accumulated by diffeomorphisms robustly satisfying the following dichotomy: either all of the measures of maximal entropy are non-hyperbolic, or there are exactly two ergodic measures of maximal entropy, one with a positive central exponent and the other with a negative central exponent. We establish this dichotomy for certain partially hyperbolic diffeomorphisms isotopic to the identity whenever both of their strong foliations are minimal. Our proof builds on the approach developed by Margulis for Anosov flows where he constructs suitable families of measures on the dynamical foliations.
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Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part I: The dynamically coherent case
On hyperbolic and Seifert fibered 3-manifolds, dynamically coherent partially hyperbolic diffeomorphisms homotopic to the identity are, up to iterate, leaf conjugate to time-one maps of topological Anosov flows.