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Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

Part of: Smooth dynamical systems: general theory Differential topology Dynamical systems with hyperbolic behavior Low-dimensional topology

Published online by Cambridge University Press:  03 November 2020

THOMAS BARTHELMÉ ,
SERGIO R. FENLEY ,
STEVEN FRANKEL  and
RAFAEL POTRIE
THOMAS BARTHELMÉ
Affiliation:
Queen’s University, Kingston, ON, Canada (e-mail: thomas.barthelme@queensu.ca)
SERGIO R. FENLEY
Affiliation:
Florida State University, Tallahassee, FL32306, USA (e-mail: fenley@math.fsu.edu)
STEVEN FRANKEL
Affiliation:
Washington University in St Louis, St Louis, MO, USA (e-mail: steven.frankel@wustl.edu)
RAFAEL POTRIE*
Affiliation:
Centro de Matemática, Universidad de la República, Montevideo, Uruguay
*
e-mail: rpotrie@cmat.edu.uy
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Abstract

We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.

Keywords

classificationfoliations3-manifold topologypartial hyperbolicity

MSC classification

Primary: 37D30: Partially hyperbolic systems and dominated splittings 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification
Secondary: 57R30: Foliations; geometric theory 57M50: Geometric structures on low-dimensional manifolds 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 11 , November 2021 , pp. 3227 - 3243
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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