September  2013, 33(9): 4323-4339. doi: 10.3934/dcds.2013.33.4323

Some advances on generic properties of the Oseledets splitting

1. 

Universidad de la Republica, Uruguay

Received  November 2010 Published  March 2013

In his foundational paper [20] , Mañé suggested that some aspects of the Oseledets splitting could be improved if one worked under $C^1$-generic conditions. He announced some powerful theorems, and suggested some lines to follow. Here we survey the state of the art and some recent advances in these directions.
Citation: Jana Rodriguez Hertz. Some advances on generic properties of the Oseledets splitting. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4323-4339. doi: 10.3934/dcds.2013.33.4323
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.

[2]

M.-C. Arnaud, C. Bonatti and S. Crovisier, Dynamiques symplectiques génériques, Ergod. Theory Dynam. Sys., 25 (2005), 1401-1436. doi: 10.1017/S0143385704000975.

[3]

A. Ávila, On the regularization of conservative maps, Acta Mathematica, 205 (2010), 5-18. doi: 10.1007/s11511-010-0050-y.

[4]

A. Ávila and J. Bochi, A generic $C^1$ map has no absolutely continuous invariant probability measure, Nonlinearity, 19 (2006), 2717-2725. doi: 10.1088/0951-7715/19/11/011.

[5]

A. Ávila and J. Bochi, Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms, Transactions of that AMS, 364 (2012), 2883-2907. doi: 10.1090/S0002-9947-2012-05423-7.

[6]

A. Ávila, J. Bochi and A. Wilkinson, Nonuniform center bunching, and the genericity of ergodicity among $C^1$ partially hyperbolic symplectomorphisms, Ann. Scien. Ec. Norm. Sup. (4), 42 (2009), 931-979.

[7]

A. Avila, S. Crovisier and A. Wilkinson, The general case, announcement.

[8]

A. Baraviera and C. Bonatti, Removing zero Lyapunov exponents, Ergod. Th. & Dynam. Sys., 23 (2003), 1655-1670. doi: 10.1017/S0143385702001773.

[9]

J. Bochi, Genericity of zero Lyapunov exponents, Erg. Th. & Dyn. Sys., 22 (2002), 1667-1696. doi: 10.1017/S0143385702001165.

[10]

J. Bochi, $C^1$-generic symplectic diffeomorphisms: Partial hyperbolicity and zero centre Lyapunov exponents, Journal of the Inst. Math. Jussieu, 9 (2010), 49-93. doi: 10.1017/S1474748009000061.

[11]

J. Bochi and M. Viana, Lyapunov exponents: How frequently are dynamical systems hyperbolic? in "Modern Dynamical Systems and Applications" (ed. M. Brin, B. Hasselblatt and Y. Pesin), Cambridge University Press, Cambridge, (2004), 271-297.

[12]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Ann. Math. (2), 161 (2005), 1423-1485. doi: 10.4007/annals.2005.161.1423.

[13]

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

[14]

C. Bonatti, C. Matheus, M. Viana and A. Wilkinson, Abundance of stable ergodicity, Comment. Math. Helv., 79 (2004), 753-757. doi: 10.1007/s00014-004-0819-8.

[15]

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.

[16]

M. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Funkcional. Anal. i Priložen., 9 (1975), 9-19.

[17]

D. Dolgopyat and A. Wilkinson, Stable accessibility is $C^1$ dense, Astérisque, 287 (2003), 33-60.

[18]

E. Grin, Genericity of diffeomorphisms with vanishing Lyapunov exponents almost everywhere, preprint.

[19]

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

[20]

R. Mañé, Oseledec's theorem from the generic viewpoint, in "Proc. Internat. Congress of Mathematicians, Vol. 1, 2" (Warsaw, 1983), PWN, Warsaw, 1984.

[21]

R. Mañé, "Ergodic Theory and Differentiable Dynamics," Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 8, Springer-Verlag, Berlin, 1987.

[22]

R. Mañé, The Lyapunov exponents of generic area preserving diffeomorphisms, in "Intl. Conference on Dynamical Systems" (ed. F. Ledrappier, J. Lewowicz and S. Newhouse) (Montevideo, 1995), Pitman Research Notes Math. Ser., 362, Longman, Harlow, (1996), 110-119.

[23]

V. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-231.

[24]

J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2), 142 (1941), 874-920.

[25]

C. Pugh and M. Shub, Stable ergodicity and partial hyperbolicity, in "Intl. Conference on Dynamical Systems" (ed. F. Ledrappier, J. Lewowicz, S. Newhouse) (Montevideo, 1995), Pitman Research Notes Math. Ser., 362, Longman, Harlow, (1996), 182-187.

[26]

C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc., 2 (2000), 1-52. doi: 10.1007/s100970050013.

[27]

J. Rodriguez Hertz, Genericity of non-uniform hyperbolicity in dimension 3, J. Modern Dyn., 6 (2012), 121-138. doi: 10.3934/jmd.2012.6.121.

[28]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math., 172 (2008), 353-381. doi: 10.1007/s00222-007-0100-z.

[29]

F. Rodriguez Hertz, M. Rodriguez Hertz, A. Tahzibi and R. Ures, New criteria for ergodicity and non-uniform hyperbolicity, submitted, arXiv:0907.4539.

[30]

F. Rodriguez Hertz, M. Rodriguez Hertz, A. Tahzibi and R. Ures, Creation of blenders in the conservative setting, Nonlinearity, 23 (2010), 211-223. doi: 10.1088/0951-7715/23/2/001.

[31]

R. Saghin and Z. Xia, Partial hyperbolicity or dense elliptic periodic points for $C^1$-generic symplectic diffeomorphisms, Trans. Amer. Math. Soc., 358 (2006), 5119-5138. doi: 10.1090/S0002-9947-06-04171-7.

[32]

K. Sigmund, Generic properties of invarient measures for Axiom A diffeomorphisms, Invent. Math., 11 (1970), 99-109.

[33]

A. Tahzibi, Stably ergodic diffeomorphisms which are not partially hyperbolic, Israel J. Math., 142 (2004), 315-344. doi: 10.1007/BF02771539.

[34]

E. Zehnder, Note on smoothing symplectic and volume preserving diffeomorphisms, in "Geometry and Topology" (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), Lect. Notes in Math., 597, Springer, Berlin, (1977), 828-854.

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.

[2]

M.-C. Arnaud, C. Bonatti and S. Crovisier, Dynamiques symplectiques génériques, Ergod. Theory Dynam. Sys., 25 (2005), 1401-1436. doi: 10.1017/S0143385704000975.

[3]

A. Ávila, On the regularization of conservative maps, Acta Mathematica, 205 (2010), 5-18. doi: 10.1007/s11511-010-0050-y.

[4]

A. Ávila and J. Bochi, A generic $C^1$ map has no absolutely continuous invariant probability measure, Nonlinearity, 19 (2006), 2717-2725. doi: 10.1088/0951-7715/19/11/011.

[5]

A. Ávila and J. Bochi, Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms, Transactions of that AMS, 364 (2012), 2883-2907. doi: 10.1090/S0002-9947-2012-05423-7.

[6]

A. Ávila, J. Bochi and A. Wilkinson, Nonuniform center bunching, and the genericity of ergodicity among $C^1$ partially hyperbolic symplectomorphisms, Ann. Scien. Ec. Norm. Sup. (4), 42 (2009), 931-979.

[7]

A. Avila, S. Crovisier and A. Wilkinson, The general case, announcement.

[8]

A. Baraviera and C. Bonatti, Removing zero Lyapunov exponents, Ergod. Th. & Dynam. Sys., 23 (2003), 1655-1670. doi: 10.1017/S0143385702001773.

[9]

J. Bochi, Genericity of zero Lyapunov exponents, Erg. Th. & Dyn. Sys., 22 (2002), 1667-1696. doi: 10.1017/S0143385702001165.

[10]

J. Bochi, $C^1$-generic symplectic diffeomorphisms: Partial hyperbolicity and zero centre Lyapunov exponents, Journal of the Inst. Math. Jussieu, 9 (2010), 49-93. doi: 10.1017/S1474748009000061.

[11]

J. Bochi and M. Viana, Lyapunov exponents: How frequently are dynamical systems hyperbolic? in "Modern Dynamical Systems and Applications" (ed. M. Brin, B. Hasselblatt and Y. Pesin), Cambridge University Press, Cambridge, (2004), 271-297.

[12]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Ann. Math. (2), 161 (2005), 1423-1485. doi: 10.4007/annals.2005.161.1423.

[13]

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

[14]

C. Bonatti, C. Matheus, M. Viana and A. Wilkinson, Abundance of stable ergodicity, Comment. Math. Helv., 79 (2004), 753-757. doi: 10.1007/s00014-004-0819-8.

[15]

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.

[16]

M. Brin, Topological transitivity of a certain class of dynamical systems, and flows of frames on manifolds of negative curvature, Funkcional. Anal. i Priložen., 9 (1975), 9-19.

[17]

D. Dolgopyat and A. Wilkinson, Stable accessibility is $C^1$ dense, Astérisque, 287 (2003), 33-60.

[18]

E. Grin, Genericity of diffeomorphisms with vanishing Lyapunov exponents almost everywhere, preprint.

[19]

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

[20]

R. Mañé, Oseledec's theorem from the generic viewpoint, in "Proc. Internat. Congress of Mathematicians, Vol. 1, 2" (Warsaw, 1983), PWN, Warsaw, 1984.

[21]

R. Mañé, "Ergodic Theory and Differentiable Dynamics," Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 8, Springer-Verlag, Berlin, 1987.

[22]

R. Mañé, The Lyapunov exponents of generic area preserving diffeomorphisms, in "Intl. Conference on Dynamical Systems" (ed. F. Ledrappier, J. Lewowicz and S. Newhouse) (Montevideo, 1995), Pitman Research Notes Math. Ser., 362, Longman, Harlow, (1996), 110-119.

[23]

V. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-231.

[24]

J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2), 142 (1941), 874-920.

[25]

C. Pugh and M. Shub, Stable ergodicity and partial hyperbolicity, in "Intl. Conference on Dynamical Systems" (ed. F. Ledrappier, J. Lewowicz, S. Newhouse) (Montevideo, 1995), Pitman Research Notes Math. Ser., 362, Longman, Harlow, (1996), 182-187.

[26]

C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc., 2 (2000), 1-52. doi: 10.1007/s100970050013.

[27]

J. Rodriguez Hertz, Genericity of non-uniform hyperbolicity in dimension 3, J. Modern Dyn., 6 (2012), 121-138. doi: 10.3934/jmd.2012.6.121.

[28]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math., 172 (2008), 353-381. doi: 10.1007/s00222-007-0100-z.

[29]

F. Rodriguez Hertz, M. Rodriguez Hertz, A. Tahzibi and R. Ures, New criteria for ergodicity and non-uniform hyperbolicity, submitted, arXiv:0907.4539.

[30]

F. Rodriguez Hertz, M. Rodriguez Hertz, A. Tahzibi and R. Ures, Creation of blenders in the conservative setting, Nonlinearity, 23 (2010), 211-223. doi: 10.1088/0951-7715/23/2/001.

[31]

R. Saghin and Z. Xia, Partial hyperbolicity or dense elliptic periodic points for $C^1$-generic symplectic diffeomorphisms, Trans. Amer. Math. Soc., 358 (2006), 5119-5138. doi: 10.1090/S0002-9947-06-04171-7.

[32]

K. Sigmund, Generic properties of invarient measures for Axiom A diffeomorphisms, Invent. Math., 11 (1970), 99-109.

[33]

A. Tahzibi, Stably ergodic diffeomorphisms which are not partially hyperbolic, Israel J. Math., 142 (2004), 315-344. doi: 10.1007/BF02771539.

[34]

E. Zehnder, Note on smoothing symplectic and volume preserving diffeomorphisms, in "Geometry and Topology" (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), Lect. Notes in Math., 597, Springer, Berlin, (1977), 828-854.

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