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Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

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

Barthelmé, T., Fenley, S., Frankel, S. and Potrie, R.. Research announcement: partially hyperbolic diffeomorphisms homotopic to the identity on 3-manifolds. 2018 MATRIX Annals. Ed. Wood, D.R., de Gier, J., Praeger, C.E. and Tao, T.. Springer, Cham, 2020.Google 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
Barthelmé, T., Fenley, S., Frankel, S. and Potrie, R.. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv:2008.04871.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., Gogolev, A., Hammerlindl, A. and Potrie, R.. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear.Google Scholar
Bonatti, C. and Wilkinson, A.. Transitive partially hyperbolic diffeomorphisms on $3$ -manifolds. Topology 44(3) (2005), 475–508.CrossRefGoogle Scholar
Burago, D. and Ivanov, S.. Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups. J. Mod. Dyn. 2(4) (2008), 541–580.CrossRefGoogle Scholar
Burns, K. and Wilkinson, A.. Dynamical coherence and center bunching. Discrete Contin. Dyn. Syst. 22(1–2) (2008), 89–100.Google Scholar
Calegari, D.. The geometry of $\mathsf{R}$ -covered foliations. Geom. Topol. 4(2000), 457–515.CrossRefGoogle Scholar
Calegari, D.. Foliations and the Geometry of 3-Manifolds (Oxford Mathematical Monographs). Oxford University Press, Oxford, 2007.Google Scholar
Casson, A. and Bleiler, S.. Automorphisms of Surfaces After Nielsen and Thurston (London Mathematical Society Student Texts, 9). Cambridge University Press, Cambridge, 1988.CrossRefGoogle Scholar
Fathi, A.. Homotopical stability of pseudo-Anosov diffeomorphisms. Ergod. Th. & Dynam. Sys. 10(2) (1990) 287–294.CrossRefGoogle Scholar
Fathi, A., Laudenbach, F. and Poénaru, V.. Travaux de Thurston sur les surfaces (Astérisque, 66–67). Société Mathématique de France, Paris, 1979.Google Scholar
Fenley, S.. Foliations, topology and geometry of 3-manifolds: $\mathbb{R}$ -covered foliations and transverse pseudo-Anosov flows. Comment. Math. Helv. 77(3) (2002), 415–490.CrossRefGoogle Scholar
Gilman, J.. On the Nielsen type and the classification for the mapping class group. Adv. in Math. 40(1) (1981), 68–96.CrossRefGoogle Scholar
Hammerlindl, A. and Potrie, R.. Partial hyperbolicity and classification: a survey. Ergod. Th. & Dynam. Sys. 38(2) (2018), 401–443.CrossRefGoogle Scholar
Hammerlindl, A., Potrie, R. and Shannon, M.. Seifert manifolds admitting partially hyperbolic diffeomorphisms. J. Mod. Dyn. 12(2018), 193–222.CrossRefGoogle Scholar
Handel, M. and Thurston, W.. New proofs of some results of Nielsen. Adv. Math. 56(2) (1985), 173–191.CrossRefGoogle Scholar
Hempel, J.. $\textit{3}$ -Manifolds (Annals of Mathematical Studies, 86) Princeton University Press, Princeton, NJ, 1976.Google Scholar
Hertz, F. R., Hertz, J. R. and Ures, R.. A non-dynamically coherent example on ${T}^3$ . Ann. Inst. H. Poincaré Anal. Non Linéaire 33(4) (2016), 1023–1032.10.1016/j.anihpc.2015.03.003CrossRefGoogle Scholar
Hirsch, M., Pugh, C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, Berlin, 1977.CrossRefGoogle Scholar
Miller, R.. Geodesic laminations from Nielsen’s viewpoint. Adv. Math. 45(2) (1982), 189–212.CrossRefGoogle Scholar
Potrie, R.. Robust dynamics, invariant geometric structures and topological classification. Proceedings of the International Congress of Mathematicians. Vol. 2. World Scientific, Singapore, 2018, pp. 2057–2080.Google Scholar
Thurston, W.. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19(2) (1988), 417–431.CrossRefGoogle Scholar
Thurston, W.. Three-manifolds, foliations and circles, I. Preprint, 1997, arXiv:math/9712268.Google Scholar