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Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence

Christian Bonatti, Andrey Gogolev, Andy Hammerlindl and Rafael Potrie

Geometry & Topology 24 (2020) 1751–1790

Let M be a closed 3–manifold which admits an Anosov flow. We develop a technique for constructing partially hyperbolic representatives in many mapping classes of M. We apply this technique both in the setting of geodesic flows on closed hyperbolic surfaces and for Anosov flows which admit transverse tori. We emphasize the similarity of both constructions through the concept of h–transversality, a tool which allows us to compose different mapping classes while retaining partial hyperbolicity.

In the case of the geodesic flow of a closed hyperbolic surface S we build stably ergodic, partially hyperbolic diffeomorphisms whose mapping classes form a subgroup of the mapping class group (T1S) which is isomorphic to (S). At the same time we show that the totality of mapping classes which can be realized by partially hyperbolic diffeomorphisms does not form a subgroup of (T1S).

Finally, some of the examples on T1S are absolutely partially hyperbolic, stably ergodic and robustly nondynamically coherent, disproving a conjecture of Rodriguez Hertz, Rodriguez Hertz and Ures  (Ann. Inst. H Poincaré Anal. Non Linéaire 33 (2016) 1023–1032).

partially hyperbolic diffeomorphisms, dynamical coherence, stable ergodicity
Mathematical Subject Classification 2010
Primary: 37C15, 37D30
Received: 17 August 2018
Revised: 9 October 2019
Accepted: 9 November 2019
Published: 10 November 2020
Proposed: Dmitri Burago
Seconded: Leonid Polterovich, David M Fisher
Christian Bonatti
Institut de Mathématiques de Bourgogne
CNRS - URM 5584
Université de Bourgogne
Andrey Gogolev
Department of Mathematics
The Ohio State University
Columbus, OH
United States
Andy Hammerlindl
School of Mathematical Sciences
Monash University
Clayton VIC
Rafael Potrie
Facultad de Ciencias - Centro de Matemática
Universidad de la República
Institute for Advanced Study
Princeton, NJ
United States