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Classification of partially hyperbolic diffeomorphisms under some rigid conditions

Part of: Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  22 October 2020

PABLO D. CARRASCO Open the ORCID record for PABLO D. CARRASCO [Opens in a new window] ,
ENRIQUE PUJALS  and
FEDERICO RODRIGUEZ-HERTZ
PABLO D. CARRASCO
Affiliation:
ICEx-UFMG, Avda. Presidente Antonio Carlos 6627, Belo Horizonte – MG, BR 31270-90, Brazil (e-mail: [email protected])
ENRIQUE PUJALS
Affiliation:
CUNY Graduate Center, Room 4208, 365 Fifth Avenue, New York, NY 10016, USA (e-mail: [email protected])
FEDERICO RODRIGUEZ-HERTZ
Affiliation:
Penn State, 227 McAllister Building, University Park, State College, PA 16802, USA (e-mail: [email protected])
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Abstract

Consider a three-dimensional partially hyperbolic diffeomorphism. It is proved that under some rigid hypothesis on the tangent bundle dynamics, the map is (modulo finite covers and iterates) an Anosov diffeomorphism, a (generalized) skew-product or the time-one map of an Anosov flow, thus recovering a well-known classification conjecture of the second author to this restricted setting.

Keywords

partial hyperbolicityrigiditythree manifolds

MSC classification

Primary: 37D30: Partially hyperbolic systems and dominated splittings
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 9 , September 2021 , pp. 2770 - 2781
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Avila, A., Viana, M. and Wilkinson, A.. Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows. J. Eur. Math. Soc. 17(6) (2015), 1435–1462.CrossRefGoogle Scholar
Barthelmé, T., Fenley, S., Frankel, S. and Potrie, R.. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part I: the dynamically coherent case. Preprint, 2019, arXiv:1908.06227.CrossRefGoogle Scholar
Bonatti, C., Gogolev, A. and Potrie, R.. Anomalous partially hyperbolic diffeomorphisms II: stably ergodic examples. Invent. Math. 206(3) (2016), 801–836.CrossRefGoogle Scholar
Bonatti, C., Parwani, K. and Potrie, R.. Anomalous partially hyperbolic diffeomorphisms I: dynamically coherent examples. Ann. Sci. Éc. Norm. Supér. 44(6) (2016), 1387–1402.CrossRefGoogle Scholar
Bonatti, C. and Wilkinson, A.. Transitive partially hyperbolic diffeomorphisms on 3-manifolds. Topology 44(3) (2005), 475–508.CrossRefGoogle Scholar
Bonatti, C. and Zhang, J.. Transitive partially hyperbolic diffeomorphisms with one-dimensional neutral center. Sci. China Math. 63 (2020), 1647–1670.CrossRefGoogle Scholar
Carrasco, P. D., Rodriguez-Hertz, F., Rodriguez-Hertz, J. and Ures, R.. Partially hyperbolic dynamics in dimension three. Ergod. Th. & Dynam. Sys. 38(8) (2017), 2801–2837.CrossRefGoogle Scholar
Franks, J.. Anosov diffeomorphisms. Amer. Math. Soc. Global Anal. 14 (1968), 61–93.Google Scholar
Ghys, E.. Flots d’Anosov dont les feuilletages stables sont différentiables. Ann. Sci. Éc. Norm. Supér. 20(2) (1987), 251–270.CrossRefGoogle Scholar
Gogolev, A., Kalinin, B. and Sadovskaya, V.. Center foliation rigidity for partially hyperbolic toral diffeomorphisms. Preprint, 2019, arXiv:1908.03177.Google Scholar
Gogolev, A.. Bootstrap for local rigidity of Anosov automorphisms on the 3-torus. Comm. Math. Phys. 352(2) (2017), 439–455.CrossRefGoogle Scholar
Gogolev, A.. Surgery for partially hyperbolic dynamical systems I. Blow-ups of invariant submanifolds. Geom. Topol. 22(4) (2018), 2219–2252.CrossRefGoogle Scholar
Hammerlindl, A. and Potrie, R.. Pointwise partial hyperbolicity in three-dimensional nilmanifolds. J. Lond. Math. Soc. 89(3) (2014), 853–875.CrossRefGoogle Scholar
Hammerlindl, A. and Potrie, R.. Classification of partially hyperbolic diffeomorphisms in 3-manifolds with solvable fundamental group. J. Topol. 8(3) (2015), 842–870.CrossRefGoogle Scholar
Hammerlindl, A. and Potrie, R.. Partial hyperbolicity and classification: a survey. Ergod. Th. & Dynam. Sys. 38(2) (2016), 401–443.CrossRefGoogle Scholar
Potrie, R.. Robust dynamics, invariant structures and topological classification. Proc. Int. Congr. Mathematicians (ICM 2018). Vol 3. World Scientific, Singapore, 2018, pp. 2063–2085.Google Scholar
Rodriguez-Hertz, F., Rodriguez-Hertz, J., Tahzibi, A. and Ures, R.. Maximizing measures for partially hyperbolic systems with compact center leaves. Ergod. Th. & Dynam. Sys. 32(2) (2012), 825–839.CrossRefGoogle Scholar
Sagin, R. and Yang, J.. Lyapunov exponents and rigidity of Anosov automorphisms and skew products. Adv. Math. 355 (2019), 106764.CrossRefGoogle Scholar
Varao, R.. Rigidity for partially hyperbolic diffeomorphisms. Ergod. Th. & Dynam. Sys. 38(8) (2018), 3188–3200.CrossRefGoogle Scholar