Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence

Bonatti, Christian; Gogolev, Andrey; Hammerlindl, Andy; Potrie, Rafael

doi:10.2140/gt.2020.24.1751

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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

DOI: 10.2140/gt.2020.24.1751

Abstract

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 $ℳ (T^{1} S)$ which is isomorphic to $ℳ (S)$ . 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 $ℳ (T^{1} S)$ .

Finally, some of the examples on $T^{1} S$ 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).

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Keywords

partially hyperbolic diffeomorphisms, dynamical coherence, stable ergodicity

Mathematical Subject Classification 2010

Primary: 37C15, 37D30

References

Publication

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

Authors

Christian Bonatti
Institut de Mathématiques de Bourgogne CNRS - URM 5584 Université de Bourgogne Dijon France

Andrey Gogolev
Department of Mathematics The Ohio State University Columbus, OH United States

Andy Hammerlindl
School of Mathematical Sciences Monash University Clayton VIC Australia

Rafael Potrie
Facultad de Ciencias - Centro de Matemática Universidad de la República Montevideo Uruguay	Institute for Advanced Study Princeton, NJ United States

Volume 24, issue 4 (2020)