# American Institute of Mathematical Sciences

January  2012, 6(1): 121-138. doi: 10.3934/jmd.2012.6.121

## Genericity of nonuniform hyperbolicity in dimension 3

 1 IMERL-Facultad de Ingeniería, Universidad de la República, CC 30 Montevideo, Uruguay

Received  March 2012 Published  May 2012

For a generic conservative diffeomorphism of a closed connected 3-manifold $M$, the Oseledets splitting is a globally dominated splitting. Moreover, either all Lyapunov exponents vanish almost everywhere, or else the system is nonuniformly hyperbolic and ergodic.
This is the 3-dimensional version of the well-known result by Mañé-Bochi [14, 4], stating that a generic conservative surface diffeomorphism is either Anosov or all Lyapunov exponents vanish almost everywhere. This result was inspired by and answers in the positive in dimension 3 a conjecture by Avila-Bochi [2].
Citation: Jana Rodriguez Hertz. Genericity of nonuniform hyperbolicity in dimension 3. Journal of Modern Dynamics, 2012, 6 (1) : 121-138. doi: 10.3934/jmd.2012.6.121
##### References:
 [1] A. Avila, On the regularization of conservative maps,, Acta Mathematica, 205 (2010), 5.  doi: 10.1007/s11511-010-0050-y.  Google Scholar [2] A. Avila and J. Bochi, Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms,, Transactions AMS, 364 (2012), 2883.  doi: 10.1090/S0002-9947-2012-05423-7.  Google Scholar [3] A. Baraviera and C. Bonatti, Removing zero Lyapunov exponents,, Ergod. Th. & Dynam. Sys., 23 (2003), 1655.   Google Scholar [4] J. Bochi, Genericity of zero Lyapunov exponents,, Erg. Th. & Dyn. Sys., 22 (2002), 1667.   Google Scholar [5] J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps,, Ann. Math., 161 (2005), 1423.  doi: 10.4007/annals.2005.161.1423.  Google Scholar [6] 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 [7] D. Gabai and W. Kazez, Group negative curvature for 3-manifolds with genuine laminations,, Geom. Topol., 2 (1998), 65.  doi: 10.2140/gt.1998.2.65.  Google Scholar [8] E. Grin, "Genericity of Diffeomorphisms with Zero Lyapunov Exponents Almost Everywhere,", Msc. Thesis, (2010).   Google Scholar [9] A. Haefliger, Variétés feuilletées,, (French), 16 (1962), 367.   Google Scholar [10] G. Hector and U. Hirsch, "Introduction to the Geometry of Foliations. Part B. Foliations of Codimension One,", Second edition, E3 (1987).   Google Scholar [11] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,", Lect. Notes Math., 583 (1977).   Google Scholar [12] J.-L. Journé, A regularity lemma for functions of several variables,, Rev. Mat. Iberoamericana, 4 (1988), 187.   Google Scholar [13] R. Mañé, An ergodic closing lemma,, Ann. of Math. (2), 116 (1982), 503.   Google Scholar [14] R. Mañé, Oseledec's theorem from the generic viewpoint,, in, (1984), 1269.   Google Scholar [15] V. Oseledec, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 197.   Google Scholar [16] J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity,, Ann. of Math. (2), 42 (1941), 874.  doi: 10.2307/1968772.  Google Scholar [17] Y. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory,, Russian Math. Surveys, 32 (1977), 55.  doi: 10.1070/RM1977v032n04ABEH001639.  Google Scholar [18] 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 [19] F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, Partial hyperbolicity and ergodicity in dimension three,, Journal of Modern Dynamics, 2 (2008), 187.  doi: 10.3934/jmd.2008.2.187.  Google Scholar [20] F. Rodriguez Hertz, M. Rodriguez Hertz, A. Tahzibi and R. Ures, New criteria for ergodicity and nonuniform hyperbolicity,, Duke Math. Journal, 160 (2011), 599.  doi: 10.1215/00127094-1444314.  Google Scholar [21] P. Zhang, Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems,, Disc. Cont. Dyn. Sys., 32 (2012), 1435.  doi: 10.3934/dcds.2012.32.1435.  Google Scholar

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##### References:
 [1] A. Avila, On the regularization of conservative maps,, Acta Mathematica, 205 (2010), 5.  doi: 10.1007/s11511-010-0050-y.  Google Scholar [2] A. Avila and J. Bochi, Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms,, Transactions AMS, 364 (2012), 2883.  doi: 10.1090/S0002-9947-2012-05423-7.  Google Scholar [3] A. Baraviera and C. Bonatti, Removing zero Lyapunov exponents,, Ergod. Th. & Dynam. Sys., 23 (2003), 1655.   Google Scholar [4] J. Bochi, Genericity of zero Lyapunov exponents,, Erg. Th. & Dyn. Sys., 22 (2002), 1667.   Google Scholar [5] J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps,, Ann. Math., 161 (2005), 1423.  doi: 10.4007/annals.2005.161.1423.  Google Scholar [6] 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 [7] D. Gabai and W. Kazez, Group negative curvature for 3-manifolds with genuine laminations,, Geom. Topol., 2 (1998), 65.  doi: 10.2140/gt.1998.2.65.  Google Scholar [8] E. Grin, "Genericity of Diffeomorphisms with Zero Lyapunov Exponents Almost Everywhere,", Msc. Thesis, (2010).   Google Scholar [9] A. Haefliger, Variétés feuilletées,, (French), 16 (1962), 367.   Google Scholar [10] G. Hector and U. Hirsch, "Introduction to the Geometry of Foliations. Part B. Foliations of Codimension One,", Second edition, E3 (1987).   Google Scholar [11] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,", Lect. Notes Math., 583 (1977).   Google Scholar [12] J.-L. Journé, A regularity lemma for functions of several variables,, Rev. Mat. Iberoamericana, 4 (1988), 187.   Google Scholar [13] R. Mañé, An ergodic closing lemma,, Ann. of Math. (2), 116 (1982), 503.   Google Scholar [14] R. Mañé, Oseledec's theorem from the generic viewpoint,, in, (1984), 1269.   Google Scholar [15] V. Oseledec, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 197.   Google Scholar [16] J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity,, Ann. of Math. (2), 42 (1941), 874.  doi: 10.2307/1968772.  Google Scholar [17] Y. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory,, Russian Math. Surveys, 32 (1977), 55.  doi: 10.1070/RM1977v032n04ABEH001639.  Google Scholar [18] 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 [19] F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, Partial hyperbolicity and ergodicity in dimension three,, Journal of Modern Dynamics, 2 (2008), 187.  doi: 10.3934/jmd.2008.2.187.  Google Scholar [20] F. Rodriguez Hertz, M. Rodriguez Hertz, A. Tahzibi and R. Ures, New criteria for ergodicity and nonuniform hyperbolicity,, Duke Math. Journal, 160 (2011), 599.  doi: 10.1215/00127094-1444314.  Google Scholar [21] P. Zhang, Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems,, Disc. Cont. Dyn. Sys., 32 (2012), 1435.  doi: 10.3934/dcds.2012.32.1435.  Google Scholar
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