# 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), 5-18. 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), 2883-2907. 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), 1655-1670.  Google Scholar [4] J. Bochi, Genericity of zero Lyapunov exponents, Erg. Th. & Dyn. Sys., 22 (2002), 1667-1696.  Google Scholar [5] J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Ann. Math., 161 (2005), 1423-1485. 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), 753-757. 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), 65-77 (electronic). doi: 10.2140/gt.1998.2.65.  Google Scholar [8] E. Grin, "Genericity of Diffeomorphisms with Zero Lyapunov Exponents Almost Everywhere," Msc. Thesis, Montevideo, 2010. Google Scholar [9] A. Haefliger, Variétés feuilletées, (French), Ann. Scuola Norm. Sup. Pisa (3), 16 (1962), 367-397.  Google Scholar [10] G. Hector and U. Hirsch, "Introduction to the Geometry of Foliations. Part B. Foliations of Codimension One," Second edition, Aspects of Mathematics, E3, Friedr. Vieweg & Sohn, Braunschweig, 1987.  Google Scholar [11] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lect. Notes Math., 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar [12] J.-L. Journé, A regularity lemma for functions of several variables, Rev. Mat. Iberoamericana, 4 (1988), 187-193.  Google Scholar [13] R. Mañé, An ergodic closing lemma, Ann. of Math. (2), 116 (1982), 503-540.  Google Scholar [14] R. Mañé, Oseledec's theorem from the generic viewpoint, in "Proc. Internat. Congress of Mathematicians" (Warsaw, 1983), Vol. 1, 2, PWN, Warsaw, (1984), 1269-1276.  Google Scholar [15] V. Oseledec, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-221.  Google Scholar [16] J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2), 42 (1941), 874-920. doi: 10.2307/1968772.  Google Scholar [17] Y. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-112, 287. 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), 353-381. 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), 187-208. 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), 599-629. 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), 1435-1447. 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), 5-18. 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), 2883-2907. 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), 1655-1670.  Google Scholar [4] J. Bochi, Genericity of zero Lyapunov exponents, Erg. Th. & Dyn. Sys., 22 (2002), 1667-1696.  Google Scholar [5] J. Bochi and M. Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps, Ann. Math., 161 (2005), 1423-1485. 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), 753-757. 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), 65-77 (electronic). doi: 10.2140/gt.1998.2.65.  Google Scholar [8] E. Grin, "Genericity of Diffeomorphisms with Zero Lyapunov Exponents Almost Everywhere," Msc. Thesis, Montevideo, 2010. Google Scholar [9] A. Haefliger, Variétés feuilletées, (French), Ann. Scuola Norm. Sup. Pisa (3), 16 (1962), 367-397.  Google Scholar [10] G. Hector and U. Hirsch, "Introduction to the Geometry of Foliations. Part B. Foliations of Codimension One," Second edition, Aspects of Mathematics, E3, Friedr. Vieweg & Sohn, Braunschweig, 1987.  Google Scholar [11] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lect. Notes Math., 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar [12] J.-L. Journé, A regularity lemma for functions of several variables, Rev. Mat. Iberoamericana, 4 (1988), 187-193.  Google Scholar [13] R. Mañé, An ergodic closing lemma, Ann. of Math. (2), 116 (1982), 503-540.  Google Scholar [14] R. Mañé, Oseledec's theorem from the generic viewpoint, in "Proc. Internat. Congress of Mathematicians" (Warsaw, 1983), Vol. 1, 2, PWN, Warsaw, (1984), 1269-1276.  Google Scholar [15] V. Oseledec, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-221.  Google Scholar [16] J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity, Ann. of Math. (2), 42 (1941), 874-920. doi: 10.2307/1968772.  Google Scholar [17] Y. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surveys, 32 (1977), 55-112, 287. 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), 353-381. 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), 187-208. 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), 599-629. 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), 1435-1447. doi: 10.3934/dcds.2012.32.1435.  Google Scholar
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