\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Partially hyperbolic sets with a dynamically minimal lamination

Abstract Full Text(HTML) Related Papers Cited by
  • We study partially hyperbolic sets of $C^1$-diffeomorphisms. For these sets there are defined the strong stable and strong unstable laminations.A lamination is called dynamically minimal when the orbit of each leaf intersects the set densely.

    We prove that partially hyperbolic sets having a dynamically minimal lamination have empty interior. We also study the Lebesgue measure and the spectral decomposition of these sets. These results can be applied to $C^1$-generic/robustly transitive attractors with one-dimensional center bundle.

    Mathematics Subject Classification: 37B20, 37B29, 37C20, 37C70, 37D10, 37D30.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] F. Abdenur, Attractors of generic diffeomorphisms are persistent, Nonlinearity, 16 (2003), 301-311.  doi: 10.1088/0951-7715/16/1/318.
    [2] F. AbdenurC. Bonatti and L. Díaz, Non-wandering sets with non-empty interior, Nonlinearity, 17 (2004), 175-191.  doi: 10.1088/0951-7715/17/1/011.
    [3] F. Abdenur and S. Crovisier, Transitivity and topological mixing for C1 diffeomorphisms, Essays in Mathematics and Its Applications, Springer, Heidelberg, (2012), 1-16.
    [4] J. Alves and V. Pinheiro, Topological structure of partially hyperbolic sets with positive volume, Trans. Amer. Math. Soc., 360 (2008), 5551-5569.  doi: 10.1090/S0002-9947-08-04484-X.
    [5] C. Bonatti and S. Crovisier, Recurrence et genéricité, Invent. Math., 1 (2002), 513-541. 
    [6] C. BonattiL. Díaz and R. Ures, Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms, J. Inst. Math. Jussieu, 1 (2002), 513-541. 
    [7] C. BonattiL. Díaz and  M. VianaDynamics Beyond Uniform Hyperbolicity, Springer-Verlag, Berlin, 2005. 
    [8] C. BonattiS. Gan and L. Wen, On the existence of non-trivial homoclinic classes, Ergod. Th. & Dynam. Sys., 27 (2007), 1473-1508. 
    [9] C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly contracting, Israel J. Math., 115 (2000), 157-193.  doi: 10.1007/BF02810585.
    [10] R. Bowen, A horseshoe with positive measure, Invent. Math., 29 (1975), 203-204.  doi: 10.1007/BF01389849.
    [11] R. BowenEquilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Springer-Verlag, Berlin-New York, 1975. 
    [12] M. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Funkcional. Anal. i Prilozen, 9 (1975), 9-19. 
    [13] C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38, Amer. Math. Soc., Providence, R. I., 1978.
    [14] T. Fisher, Hyperbolic sets with non-empty interior, Discrete Contin. Dyn. Syst., 15 (2006), 433-446.  doi: 10.3934/dcds.2006.15.433.
    [15] F. Hertz, M. Hertz and R. Ures, Some results on the integrability of the center bundle for partially hyperbolic diffeomorphisms, Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, 103-109, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007.
    [16] R. Mañé, An ergodic closing lemma, Ann. of Math. (2), 116 (1982), 503-540.  doi: 10.2307/2007021.
    [17] F. Nobili, Minimality of one invariant foliation for partially hyperbolic attractors, Nonlinearity, 28 (2015), 1897-1918.  doi: 10.1088/0951-7715/28/6/1897.
    [18] R. Potrie, Generic bi-Lyapunov stable homoclinic classes, Nonlinearity, 23 (2010), 1631-1649.  doi: 10.1088/0951-7715/23/7/006.
    [19] S. Smale, Diffeomorphisms with many periodic points, Bull. Am. Math. Soc., 73 (1967), 747-817. 
  • 加载中
SHARE

Article Metrics

HTML views(285) PDF downloads(259) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return