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Partially hyperbolic diffeomorphisms and Lagrangian contact structures

Part of: Dynamical systems with hyperbolic behavior Global differential geometry

Published online by Cambridge University Press:  12 November 2021

MARTIN MION-MOUTON
MARTIN MION-MOUTON*
Affiliation:
IRMA, 7 rue René Descartes, 67084Strasbourg, France
*
e-mail: martin.mionmouton@math.unistra.fr
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Abstract

In this paper, we classify the three-dimensional partially hyperbolic diffeomorphisms whose stable, unstable, and central distributions $E^s$ , $E^u$ , and $E^c$ are smooth, such that $E^s\oplus E^u$ is a contact distribution, and whose non-wandering set equals the whole manifold. We prove that up to a finite quotient or a finite power, they are smoothly conjugated either to a time-map of an algebraic contact-Anosov flow, or to an affine partially hyperbolic automorphism of a nil- ${\mathrm {Heis}}{(3)}$ -manifold. The rigid geometric structure induced by the invariant distributions plays a fundamental part in the proof.

Keywords

partially hyperbolic diffeomorphismsrigid geometric structuresLagrangian contact structuresparabolic Cartan geometries

MSC classification

Primary: 37D30: Partially hyperbolic systems and dominated splittings
Secondary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 53C30: Homogeneous manifolds 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 8 , August 2022 , pp. 2583 - 2629
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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