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

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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

Abstract
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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. 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. 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. doi: 10.1017/S0143385710000891. Google Scholar
[4]	C. Bonatti and S. Crovisier, Récurrence et généricité,, Invent. Math., 158 (2004), 33. 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. 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. 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, (2004). Google Scholar
[8]	J. Buescu and I. Stewart, Liapunov stability and adding machines,, Erg. Th. Dyn. Sys., 15 (1995), 271. 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. 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. 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, (1097), 305. doi: 10.1007/BFb0099434. Google Scholar
[13]	R. Mañé, An ergodic closing lemma,, Ann. of Math. (2), 116 (1982), 503. 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. 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. Google Scholar
[16]	C. Pugh, The $C^{1+\alpha }$ hypothesis in Pesin theory,, Inst. Hautes Études Sci. Publ. Math., 59 (1984), 143. Google Scholar

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References:

[1]	F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity for $C^1$-generic diffeomorphisms,, Israel J. Math., 183 (2011), 1. 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. 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. doi: 10.1017/S0143385710000891. Google Scholar
[4]	C. Bonatti and S. Crovisier, Récurrence et généricité,, Invent. Math., 158 (2004), 33. 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. 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. 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, (2004). Google Scholar
[8]	J. Buescu and I. Stewart, Liapunov stability and adding machines,, Erg. Th. Dyn. Sys., 15 (1995), 271. 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. 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. 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, (1097), 305. doi: 10.1007/BFb0099434. Google Scholar
[13]	R. Mañé, An ergodic closing lemma,, Ann. of Math. (2), 116 (1982), 503. 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. 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. Google Scholar
[16]	C. Pugh, The $C^{1+\alpha }$ hypothesis in Pesin theory,, Inst. Hautes Études Sci. Publ. Math., 59 (1984), 143. Google Scholar

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