Advertisement

Inventiones mathematicae

, Volume 206, Issue 3, pp 801–836 | Cite as

Anomalous partially hyperbolic diffeomorphisms II: stably ergodic examples

  • Christian Bonatti
  • Andrey Gogolev
  • Rafael PotrieEmail author
Article

Abstract

We construct examples of robustly transitive and stably ergodic partially hyperbolic diffeomorphisms f on compact 3-manifolds with fundamental groups of exponential growth such that \(f^n\) is not homotopic to identity for all \(n>0\). These provide counterexamples to a classification conjecture of Pujals.

Mathematical Subject Classification

Primary 37D30 37C15 

Notes

Acknowledgments

The second author would like to thank Dmitry Scheglov for many enlightening discussions and Anton Petrunin for a very useful communication. The third author thanks the hospitality of the Institut de Matématiques de Bourgogne, Dijon.

References

  1. 1.
    Barbot, T.: Generalizations of the Bonatti-Langevin example of Anosov flow and their classification up to topological equivalence. Commun. Anal. Geom. 6(4), 749–798 (1998)Google Scholar
  2. 2.
    Béguin, F., Bonatti, C., Yu, B.: Building Anosov flows on 3-manifolds, Preprint. arXiv:1408.3951
  3. 3.
    Bonatti, C., Díaz, L.: Persistent transitive diffeomorphisms. Ann. Math. 143(2), 357–396 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bonatti, C., Díaz, L., Ures, R.: Minimality of strong stable and strong unstable foliations for partially hyperbolic diffeomorphisms. J. Inst. Math. Jussieu 1(4), 513–541 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bonatti, C., Díaz, L., Viana, M.: Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective. Encyclopaedia of mathematical sciences, vol. 102. Mathematical physics III, vol. 102, xviii+384 pp. Springer, Berlin (2005)Google Scholar
  6. 6.
    Bonatti, C., Langevin, R.: Un exemple de flot d’Anosov transitif transverse à un tore et non conjugué à une suspension. Ergod. Theory Dyn. Syst. 14, 633–643 (1994)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bonatti, C., Matheus, C., Viana, M., Wilkinson, A.: Abundance of stable ergodicity. Comment. Math. Helv. 79, 753–757 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bonatti, C., Parwani, K., Potrie, R.: Anomalous partially hyperbolic diffeomorphisms I: dynamically coherent examples. Ann. Sci. l’ENS. arXiv:1411.1221 (2016)
  9. 9.
    Bonatti, C., Wilkinson, A.: Transitive partially hyperbolic diffeomorphisms on 3-manifolds. Topology 44(3), 475–508 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brin, M., Burago, D., Ivanov, S.: On partially hyperbolic diffeomorphisms of 3-manifolds with commutative fundamental group. Modern dynamical systems and applications. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  11. 11.
    Burago, D., Ivanov, S.: Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups. J. Modern Dyn. 2, 541–580 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Brunella, M.: Separating the basic sets of a nontransitive Anosov flow. Bull. Lond. Math. Soc. 25(5), 487–490 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Carrasco, P., Rodriguez Hertz, F., Rodriguez Hertz, M.A., Ures, R.: Partially hyperbolic dynamics in dimension 3, Preprint. arXiv:1501.00932
  14. 14.
    Fathi, A., Laudenbach, F., Poénaru, V.: Travaux de Thurston sur les surfaces. Astérisque 66–67, 284 (1979)Google Scholar
  15. 15.
    Farb, B., Margalit, D.: A primer on mapping class groups. Princeton Mathematical Series, vol. 49, p. xiv+472. Princeton University Press, Princeton (2012)zbMATHGoogle Scholar
  16. 16.
    Farrell, F.T., Gogolev, A.: On bundles that admit fiberwise hyperbolic dynamics. Math. Ann. 364(1), 401–438 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hammerlindl, A., Potrie, R.: Pointwise partial hyperbolicity in three dimensional nilmanifolds. J. Lond. Math. Soc. 89(3), 853–875 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hammerlindl, A., Potrie, R.: Classification of partially hyperbolic diffeomorphisms in three dimensional manifolds with solvable fundamental group. J. Topol. 8(3), 842–870 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Handel, M., Thurston, W.: Anosov flows on new three manifolds. Invent. Math. 59(2), 95–103 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hasselblatt, B., Pesin, Y.: Partially hyperbolic dynamics. In: Hasselblatt, B., Katok, A. (eds.) Handbook of dynamical systems, vol. 1B. Elsevier, Amsterdam (2005)Google Scholar
  21. 21.
    Hirsch, M., Pugh, C., Shub, M.: Invariant manifolds. Springer Lecture Notes in Mathematics, vol. 583, ii+149 pp. Springer, Berlin, NY (1977)Google Scholar
  22. 22.
    Johannson, K.: Homotopy equivalences of 3-manifolds with boundary. Lecture Notes in Mathematics, vol. 761, ii+303 pp. Springer, Berlin (1979)Google Scholar
  23. 23.
    McCullough, D.: Virtually geometrically finite mapping class groups of 3-manifolds. J. Differ. Geom. 33, 1–65 (1991)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Parwani, K.: On 3-manifolds that support partially hyperbolic diffeomorphisms. Nonlinearity 23, 589–606 (2010)Google Scholar
  25. 25.
    Rodriguez Hertz, F., Rodriguez Hertz, M.A., Ures, R.: Some results on the integrability of the center bundle for partially hyperbolic diffeomorphisms. In: Forni, G., Lyubich, M., Pugh, C., Shub, M. (eds.) Partially hyperbolic dynamics , laminations and teichmuller flow. Fields Institute Communications vol. 51, pp. 103–112. AMS, Providence, RI (2007)Google Scholar
  26. 26.
    Rodriguez Hertz, F., Rodriguez Hertz, M.A., Ures, R.: Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1d-center bundle. Invent. Math. 172, 353–381 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rodriguez Hertz, F., Rodriguez Hertz, M.A., Tahzibi, A., Ures, R.: Creation of blenders in the conservative setting. Nonlinearity 23(2), 211–223 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Rodriguez Hertz, F., Rodriguez Hertz, M.A., Ures, R.: A non-dynamically coherent example \({\mathbb{T}}^3\). Ann l’IHP (C) Non linear Anal. (2015). doi: 10.1016/j.anihpc.2015.03.003
  29. 29.
    Waldhausen, F.: On irreducible 3-manifolds which are sufficiently large. Ann. Math (2) 87, 56–88 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wilkinson, A.: Conservative partially hyperbolic dynamics. In: 2010 ICM Proceedings (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Christian Bonatti
    • 1
  • Andrey Gogolev
    • 2
  • Rafael Potrie
    • 3
    Email author
  1. 1.Institut de Mathématiques de Bourgogne CNRS-URM 5584Université de BourgogneDijonFrance
  2. 2.Department of Mathematical Sciences, Binghamton UniversityBinghamtonUSA
  3. 3.CMAT, Facultad de CienciasUniversidad de la RepúblicaMontevideoUruguay

Personalised recommendations