Partially hyperbolic dynamics in dimension three

Abstract

Partial hyperbolicity appeared in the sixties as a natural generalization of hyperbolicity. In the last 20 years in this area there has been great activity.Here we survey the state of the art in some topics, focusing especially in partial hyperbolicity in dimension 3. The reason for this is not only that it is the smallest dimension in which non-degenerate partial hyperbolicity can occur, but also that the topology of 3-manifolds influences this dynamics in revealing ways.1991 Mathematics Subject Classification. 37D30,57R30.

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

“…This resolves a classification conjecture in [CRRU15] for hyperbolic 3-manifolds. Note that this theorem does not assume that f is homotopic to the identity, since any homeomorphism on a closed hyperbolic 3-manifold has a finite power that is homotopic to the identity.…”

“…Major results. Most of the existing progress towards classifying partially hyperbolic diffeomorphisms takes an outside-in approach, restricting attention to particular classes of manifolds, and comparing to an a priori known model partially hyperbolic (see [CRRU15,HP18] for recent surveys). In particular, partially hyperbolic diffeomorphisms have been completely classified in manifolds with solvable or virtually solvable fundamental group [HP14,HP15].…”

Anomalous partially hyperbolic diffeomorphisms II: stably ergodic examples

“…Assuming absolute partial hyperbolicity, geometric conditions on the strong foliations (always satisfied on the torus T 3 ) are known to imply dynamical coherence (see [Bri, BBI]). For certain families of 3-manifolds, exact conditions for dynamial coherence are known (see [CHHU,HP 2 ] and references therein). However, there is no known general criterion for deciding whether the center bundle is integrable.…”

“…Partially hyperbolic diffeomorphisms appear naturally in many contexts and provide natural classes to study phenomena such as robust transitivity and stable ergodicity (see e.g., [BDV,Wi,CHHU,HP 2 ]). For a long time, the known examples in dimension three were rather restrictive, and so many efforts were made to try to show that the behavior seen in that restricted family of examples was indeed general: that is mostly the case on manifolds with solvable fundamental group (see [HP 1 ] BPP, BGP]).…”

Seifert manifolds admitting partially hyperbolic diffeomorphisms

“…This program was initiated by the works of Bonatti-Wilkinson ([BoW]) and Brin-Burago-Ivanov ([BBI]). These results were also motivated by an informal conjecture, due to Pujals, which suggested that there were no new transitive partially hyperbolic examples to discover in dimension 3 (see [BoW,CHHU,HP 3 ]).…”