Advanced Search
Article Contents
Article Contents

The $C^{1+\alpha }$ hypothesis in Pesin Theory revisited

Abstract Related Papers Cited by
  • We show that for every compact $3$-manifold $M$ there exists an open subset of $Diff^1(M)$ in which every generic diffeomorphism admits uncountably many ergodic probability measures that are hyperbolic while their supports are disjoint and admit a basis of attracting neighborhoods and a basis of repelling neighborhoods. As a consequence, the points in the support of these measures have no stable and no unstable manifolds. This contrasts with the higher-regularity case, where Pesin Theory gives stable and unstable manifolds with complementary dimensions at almost every point. We also give such an example in dimension two, without local genericity.
    Mathematics Subject Classification: Primary: 37C20, 37D25, 37D30, 37G30; Secondary: 37C29, 37C40, 37G15, 37G25.


    \begin{equation} \\ \end{equation}
  • [1]

    F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity for $C^1$-generic diffeomorphisms, Israel J. Math., 183 (2011), 1-60.doi: 10.1007/s11856-011-0041-5.


    F. Abdenur, C. Bonatti, S. Crovisier, L. Díaz and L. Wen, Periodic points and homoclinic classes, Erg. Th. Dyn. Sys., 27 (2007), 1-22.doi: 10.1017/S0143385706000538.


    C. Bonatti, Towards a global view of dynamical systems, for the $C^1$-topology, Erg. Th. Dyn. Sys., 31 (2011), 959-993.doi: 10.1017/S0143385710000891.


    C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104.doi: 10.1007/s00222-004-0368-1.


    C. Bonatti, S. Crovisier, L. Díaz and N. Gourmelon, Internal perturbations of homoclinic classes: Non-domination, cycles, and self-replication, Erg. Th. Dyn. Sys., 33 (2013), 739-776.doi: 10.1017/S0143385712000028.


    C. Bonatti and L. Díaz, On maximal transitive sets of generic diffeomorphisms, Publ. Math. Inst. Hautes Études Sci., 96 (2002), 171-197.doi: 10.1007/s10240-003-0008-0.


    C. Bonatti, L. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2004.


    J. Buescu and I. Stewart, Liapunov stability and adding machines, Erg. Th. Dyn. Sys., 15 (1995), 271-290.doi: 10.1017/S0143385700008373.


    L. Barreira and C. Valls, Existence of stable manifolds for nonuniformly hyperbolic $C^1$ dynamics, Discrete Contin. Dyn. Syst., 16 (2006), 307-327.doi: 10.3934/dcds.2006.16.307.


    N. Gourmelon, Generation of homoclinic tangencies by $C^1$-perturbations, Discrete Contin. Dyn. Syst., 26 (2010), 1-42.doi: 10.3934/dcds.2010.26.1.


    N. Gourmelon, An isotopic perturbation lemma along periodic orbits, arXiv:1212.6638.


    F. Ledrappier, Quelques propriétés des exposants caractéristiques, in École d'été de Probabilités de Saint-Flour, XII-1982, Lecture Notes in Math., 1097, Springer, Berlin, 1984, 305-396.doi: 10.1007/BFb0099434.


    R. Mañé, An ergodic closing lemma, Ann. of Math. (2), 116 (1982), 503-540.doi: 10.2307/2007021.


    L. Markus and K. Meyer, Periodic orbits and solenoids in generic Hamiltonian dynamical systems, Amer. J. Math., 102 (1980), 25-92.doi: 10.2307/2374171.


    Ja. B. Pesin, Families of invariant manifolds that correspond to nonzero characteristic exponents, Math. USSR-Izv., 10 (1976), 1261-1305.


    C. Pugh, The $C^{1+\alpha }$ hypothesis in Pesin theory, Inst. Hautes Études Sci. Publ. Math., 59 (1984), 143-161.

  • 加载中

Article Metrics

HTML views() PDF downloads(140) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint