Research Announcement: Partially Hyperbolic Diffeomorphisms Homotopic to the Identity on 3-Manifolds

Abstract

We study partially hyperbolic diffeomorphisms homotopic to the identity in 3-manifolds. Under a general minimality condition, we show a dichotomy for the dynamics of the (branching) foliations in the universal cover. This allows us to give a full classification in certain settings: partially hyperbolic diffeomorphisms homotopic to the identity on Seifert fibered manifolds (proving a conjecture of Pujals [BW05] in this setting), and dynamically coherent partially hyperbolic diffeomorphisms on hyperbolic 3-manifolds (proving a classification conjecture of Hertz-Hertz-Ures [CRRU15] in this setting). In both cases, up to iterates we prove that the diffeomorphism is leaf conjugate to the time one map of an Anosov flow. Several other results of independent interest are obtained along the way.

Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

“…We conclude this introduction by saying that, even if independent of [BFFP2,BFFP3], this paper shares several ideas with one case which is treated in those papers, which we call double translation in hyperbolic manifolds (see, in particular, [BFFP2,§8 and §9]). In §6 we give some positive results towards understanding these partially hyperbolic maps along the lines of what is done in [BFFP3,§11].…”

“…Notice that in [BFFP2,BFFP3] (see also [BFFP1]) we have shown that if f is a partially hyperbolic diffeomorphism in a Seifert manifold M which is homotopic to the identity, then f must be dynamically coherent (and we even give a full classification of such maps). In [BGHP] it is shown that if f in a Seifert manifold induces the identity in the base, then it has to be homotopic to the identity.…”

Ergodicity and partial hyperbolicity on Seifert manifolds

“…In this setting of Seifert manifolds, Barthelmé, Fenley, Frankel, and Potrie have announced that any partially hyperbolic diffeomorphism isotopic to the identity is leaf conjugate to a topological Anosov flow [2]. Using this result would simplify some parts of our proof.…”

“…After this paper was announced, Fenley and Potrie, adapting techniques in [2] to the study of accessibility classes, announced that for any closed 3-manifold with hyperbolic geometry and also for the case dealt with in Theorem 1.1, every volume preserving partially hyperbolic diffeomorphism is accessible and ergodic [15]. However, their result strongly relies on results in [2]. The proof of Theorem 1.1 we present here is self-contained and uses completely different techniques.…”

“…As shown by [14,Proposition 5.3], up to replacing g by an iterate, we may assume g has at least four fixed points on the circle ∂O at infinity. Moreover, these fixed points correspond exactly to the prongs of the stable and unstable manifolds of p. Since we are in the topologically Anosov setting (instead of the pseudo-Anosov setting), there are exactly four fixed points on ∂O given by a u (2) implies that A u + and A u − are disjoint and so they must exactly equal these two connected components.…”