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Let be a closed –manifold which admits an Anosov flow. We develop a technique for constructing partially hyperbolic representatives in many mapping classes of . 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 –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 we build stably ergodic, partially hyperbolic diffeomorphisms whose mapping classes form a subgroup of the mapping class group which is isomorphic to . 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 .
Finally, some of the examples on 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).
Bonatti, Christian 1 ; Gogolev, Andrey 2 ; Hammerlindl, Andy 3 ; Potrie, Rafael 4
@article{GT_2020_24_4_a3, author = {Bonatti, Christian and Gogolev, Andrey and Hammerlindl, Andy and Potrie, Rafael}, title = {Anomalous partially hyperbolic diffeomorphisms {III:} {Abundance} and incoherence}, journal = {Geometry & topology}, pages = {1751--1790}, publisher = {mathdoc}, volume = {24}, number = {4}, year = {2020}, doi = {10.2140/gt.2020.24.1751}, url = {http://geodesic.mathdoc.fr/articles/10.2140/gt.2020.24.1751/} }
TY - JOUR AU - Bonatti, Christian AU - Gogolev, Andrey AU - Hammerlindl, Andy AU - Potrie, Rafael TI - Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence JO - Geometry & topology PY - 2020 SP - 1751 EP - 1790 VL - 24 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.2140/gt.2020.24.1751/ DO - 10.2140/gt.2020.24.1751 ID - GT_2020_24_4_a3 ER -
%0 Journal Article %A Bonatti, Christian %A Gogolev, Andrey %A Hammerlindl, Andy %A Potrie, Rafael %T Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence %J Geometry & topology %D 2020 %P 1751-1790 %V 24 %N 4 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.2140/gt.2020.24.1751/ %R 10.2140/gt.2020.24.1751 %F GT_2020_24_4_a3
Bonatti, Christian; Gogolev, Andrey; Hammerlindl, Andy; Potrie, Rafael. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geometry & topology, Tome 24 (2020) no. 4, pp. 1751-1790. doi : 10.2140/gt.2020.24.1751. http://geodesic.mathdoc.fr/articles/10.2140/gt.2020.24.1751/
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