Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence

Bonatti, Christian¹ ; Gogolev, Andrey² ; Hammerlindl, Andy³ ; Potrie, Rafael⁴

¹ Institut de Mathématiques de Bourgogne, CNRS - URM 5584, Université de Bourgogne, Dijon, France
² Department of Mathematics, The Ohio State University, Columbus, OH, United States
³ School of Mathematical Sciences, Monash University, Clayton VIC, Australia
⁴ Facultad de Ciencias - Centro de Matemática, Universidad de la República, Montevideo, Uruguay, Institute for Advanced Study, Princeton, NJ, United States

Geometry & topology, Tome 24 (2020) no. 4, pp. 1751-1790.

Résumé

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 $\mathcal{ℳ}\left(T^{1}S\right)$ which is isomorphic to $\mathcal{ℳ}\left(S\right)$. 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 $\mathcal{ℳ}\left(T^{1}S\right)$.

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).

Détail

DOI : 10.2140/gt.2020.24.1751

Classification : 37C15, 37D30
Keywords: partially hyperbolic diffeomorphisms, dynamical coherence, stable ergodicity

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