Abstract
We show that partially hyperbolic diffeomorphisms of \(d\)-dimensional tori isotopic to an Anosov diffeomorphism, where the isotopy is contained in the set of partially hyperbolic diffeomorphisms, are dynamically coherent. Moreover, we show a global stability result, i.e. every partially hyperbolic diffeomorphism as above is leaf-conjugate to the linear one. As a consequence, we obtain intrinsic ergodicity and measure equivalence for partially hyperbolic diffeomorphisms with one-dimensional center direction that are isotopic to Anosov diffeomorphisms through such a path.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
For absolute partial hyperbolicity it is required that the inequalities hold for unit vectors that may belong to the bundles of different points.
This can also be expressed as: \((H_f^\sigma )^{-1}(B^\sigma _R (H_f(x))) \cap \widetilde{\mathcal {W}}^\sigma _f(x) \subset D^\sigma _{R',f}(x)\).
References
Artigue, A., Brum, J., Potrie, R.: Local product structure for expansive homeomorphisms. Topol. Appl. 156(4), 674–685 (2009)
Bonatti, C., Díaz, L., Viana, M.: Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective. In: Encyclopaedia of Mathematical Sciences 102. Mathematical Physics III. Springer, Berlin (2005)
Brin, M.: On dynamical coherence. Ergod. Theory Dyn. Syst. 23, 395–401 (2003)
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)
Brin, M., Burago, D., Ivanov, S.: Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. J. Mod. Dyn. 3, 1–11 (2009)
Burns, K., Wilkinson, A.: Dynamical coherence and center bunching. Discrete Cont. Dyn. Syst. A (Pesin birthday issue) 22, 89–100 (2008)
Burns, K., Wilkinson, A.: On the ergodicity of partially hyperbolic systems. Ann. Math. 171, 451–489 (2010)
Buzzi, J., Fisher, T.: Entropic stability beyond partial hyperbolicity. To appear in, J. Mod. Dyn. arXiv:1103.2707
Buzzi, J., Fisher, T., Sambarino, M., Vasquez, C.: Maximal entropy measures for certain partially hyperbolic, derived from Anosov systems. Ergod. Theory Dyn. Syst. 32(1), 63–79 (2012)
Farrel, F.T., Gogolev, A.: The space of Anosov diffeomorphisms. To appear in J. Lond. Math. Soc. arXiv:1201.3595
Franks, J.: Anosov diffeomorphisms, 1970 global analysis. In: Proceedings of Symposium Pure Mathematics, Berkeley, California, vol. 14, pp. 61–93. American Mathematical Society, Providence, RI (1968)
Ghys, E.: Flots d’Anosov sur les 3-variétés fibrées en cercles. Ergod. Theory Dyn. Syst. 4, 67–80 (1984)
Gourmelon, N.: Adapted metrics for dominated splittings. Ergod. Theory Dyn. Syst. 27, 1839–1849 (2007)
Gromov, M.: Three remarks on geodesic dynamics and the fundamental group. Enseign Math. 46(2), 391–402 (2000)
Hammerlindl, A.: Leaf conjugacies in the torus. Ergod. Theory Dyn. Syst. 33(3), 896–933 (2013)
Hammerlindl, A.: Quasi-isometry and plaque expansiveness. Can. Math. Bull. 54(4), 676–679 (2011)
Hammerlindl, A., Potrie, R.: Pointwise partial hyperbolicity in 3-dimensional nilmanifolds. To appear in J. Lond. Math. Soc. arXiv:1302.0543
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant manifolds. In: Lecture Notes in Mathematics, vol. 583, ii+149 pp. Springer, Berlin, New York (1977)
Mañé, R.: Contributions to the stability conjecture. Topology 17, 383–396 (1978)
Mañé, R.: Ergodic Theory and Differentiable Dynamics. Springer, Berlin (1983)
Newhouse, S., Young, L.-S.: Dynamics of Certain Skew Products, vol. 1007 of Lecture Notes in Mathematics, pp. 611–629. Springer, Berlin (1983)
Potrie, R.: Partial hyperbolicity and foliations in \({\mathbb{T}}^3\). Preprint. arXiv:1206.2860
Rodriguez Hertz, F., Rodriguez Hertz, J., Ures, R.: A non-dynamically coherent example in \({\mathbb{T}^3}\). Preprint
Ures, R.: Intrinsic ergodicity of partially hyperbolic diffeomorphisms with hyperbolic linear part. Proc. AMS 140, 1973–1985 (2012)
Author information
Authors and Affiliations
Corresponding author
Additional information
T.F. was partially supported by the Simons Foundation Grant No. 239708. R.P. and M.S. were partially supported by CSIC group 618 and Balzan’s research project of J.Palis. R.P. was also partially supported by FCE-3-2011-1-6749.
Rights and permissions
About this article
Cite this article
Fisher, T., Potrie, R. & Sambarino, M. Dynamical coherence of partially hyperbolic diffeomorphisms of tori isotopic to Anosov. Math. Z. 278, 149–168 (2014). https://doi.org/10.1007/s00209-014-1310-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-014-1310-x