American Institute of Mathematical Sciences

Advanced
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.  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

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.  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

[1]

Wenxiang Sun, Xueting Tian. Dominated splitting and Pesin's entropy formula. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1421-1434. doi: 10.3934/dcds.2012.32.1421
[2]

Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469
[3]

Dante Carrasco-Olivera, Bernardo San Martín. Robust attractors without dominated splitting on manifolds with boundary. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4555-4563. doi: 10.3934/dcds.2014.34.4555
[4]

Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006
[5]

Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91
[6]

Edson de Faria, Pablo Guarino. Real bounds and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1957-1982. doi: 10.3934/dcds.2016.36.1957
[7]

Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107
[8]

Sebastian J. Schreiber. Expansion rates and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 433-438. doi: 10.3934/dcds.1997.3.433
[9]

Zoltán Buczolich, Gabriella Keszthelyi. Isentropes and Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2020, 40 (4) : 1989-2009. doi: 10.3934/dcds.2020102
[10]

Xinsheng Wang, Lin Wang, Yujun Zhu. Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2125-2140. doi: 10.3934/dcds.2018087
[11]

Pedro Duarte, Silvius Klein. Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5379-5387. doi: 10.3934/dcds.2018237
[12]

Chao Liang, Wenxiang Sun, Jiagang Yang. Some results on perturbations of Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4287-4305. doi: 10.3934/dcds.2012.32.4287
[13]

Shrihari Sridharan, Atma Ram Tiwari. The dependence of Lyapunov exponents of polynomials on their coefficients. Journal of Computational Dynamics, 2019, 6 (1) : 95-109. doi: 10.3934/jcd.2019004
[14]

Nguyen Dinh Cong, Thai Son Doan, Stefan Siegmund. On Lyapunov exponents of difference equations with random delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 861-874. doi: 10.3934/dcdsb.2015.20.861
[15]

Wilhelm Schlag. Regularity and convergence rates for the Lyapunov exponents of linear cocycles. Journal of Modern Dynamics, 2013, 7 (4) : 619-637. doi: 10.3934/jmd.2013.7.619
[16]

Jianyu Chen. On essential coexistence of zero and nonzero Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4149-4170. doi: 10.3934/dcds.2012.32.4149
[17]

Paul L. Salceanu, H. L. Smith. Lyapunov exponents and persistence in discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 187-203. doi: 10.3934/dcdsb.2009.12.187
[18]

Andrey Gogolev, Ali Tahzibi. Center Lyapunov exponents in partially hyperbolic dynamics. Journal of Modern Dynamics, 2014, 8 (3&4) : 549-576. doi: 10.3934/jmd.2014.8.549
[19]

Fei Yu, Kang Zuo. Weierstrass filtration on Teichmüller curves and Lyapunov exponents. Journal of Modern Dynamics, 2013, 7 (2) : 209-237. doi: 10.3934/jmd.2013.7.209
[20]

Lucas Backes, Aaron Brown, Clark Butler. Continuity of Lyapunov exponents for cocycles with invariant holonomies. Journal of Modern Dynamics, 2018, 12: 223-260. doi: 10.3934/jmd.2018009

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences