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October  2013, 7(4): 605-618. doi: 10.3934/jmd.2013.7.605

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

Christian Bonatti 1, , Sylvain Crovisier 2, and  Katsutoshi Shinohara 3,

1. 

Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon 21004
2. 

Laboratoire de Mathématiques d’Orsay CNRS - UMR 8628 Université Paris-Sud 11 Orsay 91405, France
3. 

FIRST, Aihara Innovative Mathematical Modelling Project, JST, Institute of Industrial Science, University of Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo 153-8505, Japan

Received  June 2013 Published  March 2014

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.
Keywords: wild diffeomorphism, dominated splitting, Pesin theory, Lyapunov exponents..
Mathematics Subject Classification: Primary: 37C20, 37D25, 37D30, 37G30; Secondary: 37C29, 37C40, 37G15, 37G2.
Citation: Christian Bonatti, Sylvain Crovisier, Katsutoshi Shinohara. The $C^{1+\alpha }$ hypothesis in Pesin Theory revisited. Journal of Modern Dynamics, 2013, 7 (4) : 605-618. doi: 10.3934/jmd.2013.7.605
References:
[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.  Google Scholar

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

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

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

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

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

N. Gourmelon, An isotopic perturbation lemma along periodic orbits,, , ().   Google Scholar

[12]

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.  Google Scholar

[13]

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

[14]

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.  Google Scholar

[15]

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

[16]

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

show all references

References:
[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.  Google Scholar

[2]

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.  Google Scholar

[3]

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.  Google Scholar

[4]

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

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

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

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

N. Gourmelon, An isotopic perturbation lemma along periodic orbits,, , ().   Google Scholar

[12]

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.  Google Scholar

[13]

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

[14]

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.  Google Scholar

[15]

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

[16]

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

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