# American Institue of Mathematical Sciences

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. [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. [3] A. Baraviera and C. Bonatti, Removing zero Lyapunov exponents,, Ergod. Th. & Dynam. Sys., 23 (2003), 1655. [4] J. Bochi, Genericity of zero Lyapunov exponents,, Erg. Th. & Dyn. Sys., 22 (2002), 1667. [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. [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. [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. [8] E. Grin, "Genericity of Diffeomorphisms with Zero Lyapunov Exponents Almost Everywhere,", Msc. Thesis, (2010). [9] A. Haefliger, Variétés feuilletées,, (French), 16 (1962), 367. [10] G. Hector and U. Hirsch, "Introduction to the Geometry of Foliations. Part B. Foliations of Codimension One,", Second edition, E3 (1987). [11] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,", Lect. Notes Math., 583 (1977). [12] J.-L. Journé, A regularity lemma for functions of several variables,, Rev. Mat. Iberoamericana, 4 (1988), 187. [13] R. Mañé, An ergodic closing lemma,, Ann. of Math. (2), 116 (1982), 503. [14] R. Mañé, Oseledec's theorem from the generic viewpoint,, in, (1984), 1269. [15] V. Oseledec, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 197. [16] J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity,, Ann. of Math. (2), 42 (1941), 874. doi: 10.2307/1968772. [17] Y. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory,, Russian Math. Surveys, 32 (1977), 55. doi: 10.1070/RM1977v032n04ABEH001639. [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. [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. [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. [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.

show all references

##### References:
 [1] A. Avila, On the regularization of conservative maps,, Acta Mathematica, 205 (2010), 5. doi: 10.1007/s11511-010-0050-y. [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. [3] A. Baraviera and C. Bonatti, Removing zero Lyapunov exponents,, Ergod. Th. & Dynam. Sys., 23 (2003), 1655. [4] J. Bochi, Genericity of zero Lyapunov exponents,, Erg. Th. & Dyn. Sys., 22 (2002), 1667. [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. [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. [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. [8] E. Grin, "Genericity of Diffeomorphisms with Zero Lyapunov Exponents Almost Everywhere,", Msc. Thesis, (2010). [9] A. Haefliger, Variétés feuilletées,, (French), 16 (1962), 367. [10] G. Hector and U. Hirsch, "Introduction to the Geometry of Foliations. Part B. Foliations of Codimension One,", Second edition, E3 (1987). [11] M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,", Lect. Notes Math., 583 (1977). [12] J.-L. Journé, A regularity lemma for functions of several variables,, Rev. Mat. Iberoamericana, 4 (1988), 187. [13] R. Mañé, An ergodic closing lemma,, Ann. of Math. (2), 116 (1982), 503. [14] R. Mañé, Oseledec's theorem from the generic viewpoint,, in, (1984), 1269. [15] V. Oseledec, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 197. [16] J. Oxtoby and S. Ulam, Measure-preserving homeomorphisms and metrical transitivity,, Ann. of Math. (2), 42 (1941), 874. doi: 10.2307/1968772. [17] Y. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory,, Russian Math. Surveys, 32 (1977), 55. doi: 10.1070/RM1977v032n04ABEH001639. [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. [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. [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. [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.
 [1] Pengfei Zhang. Partially hyperbolic sets with positive measure and $ACIP$ for partially hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1435-1447. doi: 10.3934/dcds.2012.32.1435 [2] Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006 [3] Luis Barreira, Claudia Valls. Growth rates and nonuniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 509-528. doi: 10.3934/dcds.2008.22.509 [4] Zhihong Xia. Hyperbolic invariant sets with positive measures. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 811-818. doi: 10.3934/dcds.2006.15.811 [5] 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 [6] Enrique R. Pujals. Density of hyperbolicity and homoclinic bifurcations for attracting topologically hyperbolic sets. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 335-405. doi: 10.3934/dcds.2008.20.335 [7] Yakov Pesin. On the work of Dolgopyat on partial and nonuniform hyperbolicity. Journal of Modern Dynamics, 2010, 4 (2) : 227-241. doi: 10.3934/jmd.2010.4.227 [8] Todd Young. Partially hyperbolic sets from a co-dimension one bifurcation. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 253-275. doi: 10.3934/dcds.1995.1.253 [9] Carlos H. Vásquez. Stable ergodicity for partially hyperbolic attractors with positive central Lyapunov exponents. Journal of Modern Dynamics, 2009, 3 (2) : 233-251. doi: 10.3934/jmd.2009.3.233 [10] Luis Barreira, Claudia Valls. Regularity of center manifolds under nonuniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 55-76. doi: 10.3934/dcds.2011.30.55 [11] Rasul Shafikov, Christian Wolf. Stable sets, hyperbolicity and dimension. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 403-412. doi: 10.3934/dcds.2005.12.403 [12] Boris Kalinin, Anatole Katok and Federico Rodriguez Hertz. New progress in nonuniform measure and cocycle rigidity. Electronic Research Announcements, 2008, 15: 79-92. doi: 10.3934/era.2008.15.79 [13] Giovanni Forni. A geometric criterion for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle. Journal of Modern Dynamics, 2011, 5 (2) : 355-395. doi: 10.3934/jmd.2011.5.355 [14] Rafael Potrie. Partially hyperbolic diffeomorphisms with a trapping property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5037-5054. doi: 10.3934/dcds.2015.35.5037 [15] Lorenzo J. Díaz, Todd Fisher. Symbolic extensions and partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1419-1441. doi: 10.3934/dcds.2011.29.1419 [16] Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Journal of Modern Dynamics, 2010, 4 (2) : 271-327. doi: 10.3934/jmd.2010.4.271 [17] Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68 [18] Paula Kemp. Characterizations of conditionally complete partially ordered sets. Conference Publications, 2005, 2005 (Special) : 505-509. doi: 10.3934/proc.2005.2005.505 [19] Boris Kalinin, Anatole Katok. Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori. Journal of Modern Dynamics, 2007, 1 (1) : 123-146. doi: 10.3934/jmd.2007.1.123 [20] Todd Fisher. Hyperbolic sets with nonempty interior. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 433-446. doi: 10.3934/dcds.2006.15.433

2016 Impact Factor: 0.706

## Metrics

• PDF downloads (0)
• HTML views (0)
• Cited by (6)

[Back to Top]