# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [2] M.-C. Arnaud, C. Bonatti and S. Crovisier, Dynamiques symplectiques génériques, Ergod. Theory Dynam. Sys., 25 (2005), 1401-1436. doi: 10.1017/S0143385704000975.  Google Scholar [3] A. Ávila, On the regularization of conservative maps, Acta Mathematica, 205 (2010), 5-18. doi: 10.1007/s11511-010-0050-y.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] A. Avila, S. Crovisier and A. Wilkinson, The general case,, announcement., ().   Google Scholar [8] A. Baraviera and C. Bonatti, Removing zero Lyapunov exponents, Ergod. Th. & Dynam. Sys., 23 (2003), 1655-1670. doi: 10.1017/S0143385702001773.  Google Scholar [9] J. Bochi, Genericity of zero Lyapunov exponents, Erg. Th. & Dyn. Sys., 22 (2002), 1667-1696. doi: 10.1017/S0143385702001165.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104. doi: 10.1007/s00222-004-0368-1.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] D. Dolgopyat and A. Wilkinson, Stable accessibility is $C^1$ dense, Astérisque, 287 (2003), 33-60.  Google Scholar [18] E. Grin, Genericity of diffeomorphisms with vanishing Lyapunov exponents almost everywhere,, preprint., ().   Google Scholar [19] R. Mañé, An ergodic closing lemma, Ann. of Math. (2), 116 (1982), 503-540. doi: 10.2307/2007021.  Google Scholar [20] R. Mañé, Oseledec's theorem from the generic viewpoint, in "Proc. Internat. Congress of Mathematicians, Vol. 1, 2" (Warsaw, 1983), PWN, Warsaw, 1984.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] V. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-231. Google Scholar [24] J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2), 142 (1941), 874-920.  Google Scholar [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.  Google Scholar [26] C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc., 2 (2000), 1-52. doi: 10.1007/s100970050013.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] F. Rodriguez Hertz, M. Rodriguez Hertz, A. Tahzibi and R. Ures, New criteria for ergodicity and non-uniform hyperbolicity,, submitted, ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [32] K. Sigmund, Generic properties of invarient measures for Axiom A diffeomorphisms, Invent. Math., 11 (1970), 99-109.  Google Scholar [33] A. Tahzibi, Stably ergodic diffeomorphisms which are not partially hyperbolic, Israel J. Math., 142 (2004), 315-344. doi: 10.1007/BF02771539.  Google Scholar [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.  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), 1-60. doi: 10.1007/s11856-011-0041-5.  Google Scholar [2] M.-C. Arnaud, C. Bonatti and S. Crovisier, Dynamiques symplectiques génériques, Ergod. Theory Dynam. Sys., 25 (2005), 1401-1436. doi: 10.1017/S0143385704000975.  Google Scholar [3] A. Ávila, On the regularization of conservative maps, Acta Mathematica, 205 (2010), 5-18. doi: 10.1007/s11511-010-0050-y.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] A. Avila, S. Crovisier and A. Wilkinson, The general case,, announcement., ().   Google Scholar [8] A. Baraviera and C. Bonatti, Removing zero Lyapunov exponents, Ergod. Th. & Dynam. Sys., 23 (2003), 1655-1670. doi: 10.1017/S0143385702001773.  Google Scholar [9] J. Bochi, Genericity of zero Lyapunov exponents, Erg. Th. & Dyn. Sys., 22 (2002), 1667-1696. doi: 10.1017/S0143385702001165.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104. doi: 10.1007/s00222-004-0368-1.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] D. Dolgopyat and A. Wilkinson, Stable accessibility is $C^1$ dense, Astérisque, 287 (2003), 33-60.  Google Scholar [18] E. Grin, Genericity of diffeomorphisms with vanishing Lyapunov exponents almost everywhere,, preprint., ().   Google Scholar [19] R. Mañé, An ergodic closing lemma, Ann. of Math. (2), 116 (1982), 503-540. doi: 10.2307/2007021.  Google Scholar [20] R. Mañé, Oseledec's theorem from the generic viewpoint, in "Proc. Internat. Congress of Mathematicians, Vol. 1, 2" (Warsaw, 1983), PWN, Warsaw, 1984.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] V. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-231. Google Scholar [24] J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2), 142 (1941), 874-920.  Google Scholar [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.  Google Scholar [26] C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc., 2 (2000), 1-52. doi: 10.1007/s100970050013.  Google Scholar [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.  Google Scholar [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.  Google Scholar [29] F. Rodriguez Hertz, M. Rodriguez Hertz, A. Tahzibi and R. Ures, New criteria for ergodicity and non-uniform hyperbolicity,, submitted, ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [32] K. Sigmund, Generic properties of invarient measures for Axiom A diffeomorphisms, Invent. Math., 11 (1970), 99-109.  Google Scholar [33] A. Tahzibi, Stably ergodic diffeomorphisms which are not partially hyperbolic, Israel J. Math., 142 (2004), 315-344. doi: 10.1007/BF02771539.  Google Scholar [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.  Google Scholar
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