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 - A, 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. 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. doi: 10.1017/S0143385704000975. Google Scholar [3] A. Ávila, On the regularization of conservative maps,, Acta Mathematica, 205 (2010), 5. 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. 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. 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. 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. doi: 10.1017/S0143385702001773. Google Scholar [9] J. Bochi, Genericity of zero Lyapunov exponents,, Erg. Th. & Dyn. Sys., 22 (2002), 1667. 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. doi: 10.1017/S1474748009000061. Google Scholar [11] J. Bochi and M. Viana, Lyapunov exponents: How frequently are dynamical systems hyperbolic?, in, (2004), 271. Google Scholar [12] J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps,, Ann. Math. (2), 161 (2005), 1423. doi: 10.4007/annals.2005.161.1423. Google Scholar [13] C. Bonatti and S. Crovisier, Récurrence et généricité,, Invent. Math., 158 (2004), 33. 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. 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. 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. Google Scholar [17] D. Dolgopyat and A. Wilkinson, Stable accessibility is $C^1$ dense,, Astérisque, 287 (2003), 33. 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. doi: 10.2307/2007021. Google Scholar [20] R. Mañé, Oseledec's theorem from the generic viewpoint,, in, (1983). 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 (1987). Google Scholar [22] R. Mañé, The Lyapunov exponents of generic area preserving diffeomorphisms,, in, 362 (1996), 110. Google Scholar [23] V. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 197. Google Scholar [24] J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity,, Ann. of Math. (2), 142 (1941), 874. Google Scholar [25] C. Pugh and M. Shub, Stable ergodicity and partial hyperbolicity,, in, 362 (1996), 182. Google Scholar [26] C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality,, J. Eur. Math. Soc., 2 (2000), 1. doi: 10.1007/s100970050013. Google Scholar [27] J. Rodriguez Hertz, Genericity of non-uniform hyperbolicity in dimension 3,, J. Modern Dyn., 6 (2012), 121. 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. 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. 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. 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. Google Scholar [33] A. Tahzibi, Stably ergodic diffeomorphisms which are not partially hyperbolic,, Israel J. Math., 142 (2004), 315. doi: 10.1007/BF02771539. Google Scholar [34] E. Zehnder, Note on smoothing symplectic and volume preserving diffeomorphisms,, in, 597 (1977), 828. 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] M.-C. Arnaud, C. Bonatti and S. Crovisier, Dynamiques symplectiques génériques,, Ergod. Theory Dynam. Sys., 25 (2005), 1401. doi: 10.1017/S0143385704000975. Google Scholar [3] A. Ávila, On the regularization of conservative maps,, Acta Mathematica, 205 (2010), 5. 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. 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. 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. 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. doi: 10.1017/S0143385702001773. Google Scholar [9] J. Bochi, Genericity of zero Lyapunov exponents,, Erg. Th. & Dyn. Sys., 22 (2002), 1667. 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. doi: 10.1017/S1474748009000061. Google Scholar [11] J. Bochi and M. Viana, Lyapunov exponents: How frequently are dynamical systems hyperbolic?, in, (2004), 271. Google Scholar [12] J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps,, Ann. Math. (2), 161 (2005), 1423. doi: 10.4007/annals.2005.161.1423. Google Scholar [13] C. Bonatti and S. Crovisier, Récurrence et généricité,, Invent. Math., 158 (2004), 33. 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. 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. 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. Google Scholar [17] D. Dolgopyat and A. Wilkinson, Stable accessibility is $C^1$ dense,, Astérisque, 287 (2003), 33. 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. doi: 10.2307/2007021. Google Scholar [20] R. Mañé, Oseledec's theorem from the generic viewpoint,, in, (1983). 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 (1987). Google Scholar [22] R. Mañé, The Lyapunov exponents of generic area preserving diffeomorphisms,, in, 362 (1996), 110. Google Scholar [23] V. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 197. Google Scholar [24] J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity,, Ann. of Math. (2), 142 (1941), 874. Google Scholar [25] C. Pugh and M. Shub, Stable ergodicity and partial hyperbolicity,, in, 362 (1996), 182. Google Scholar [26] C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality,, J. Eur. Math. Soc., 2 (2000), 1. doi: 10.1007/s100970050013. Google Scholar [27] J. Rodriguez Hertz, Genericity of non-uniform hyperbolicity in dimension 3,, J. Modern Dyn., 6 (2012), 121. 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. 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. 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. 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. Google Scholar [33] A. Tahzibi, Stably ergodic diffeomorphisms which are not partially hyperbolic,, Israel J. Math., 142 (2004), 315. doi: 10.1007/BF02771539. Google Scholar [34] E. Zehnder, Note on smoothing symplectic and volume preserving diffeomorphisms,, in, 597 (1977), 828. Google Scholar
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