2014, 8(3&4): 549-576. doi: 10.3934/jmd.2014.8.549

Center Lyapunov exponents in partially hyperbolic dynamics

1. 

Department of Mathematical Sciences, Binghamton University, P. O. Box 6000, Binghamton, NY 13902, United States

2. 

Departamento de Matemática, ICMC-USP São Carlos, Caixa Postal 668, 13560-970 São Carlos-SP, Brazil

Received  October 2013 Revised  June 2014 Published  April 2015

In this survey, we discuss the problem of removing zero Lyapunov exponents of smooth invariant measures along the center direction of a partially hyperbolic diffeomorphism and various related questions. In particular, we discuss disintegration of a smooth invariant measure along the center foliation. We also simplify the proofs of some known results and include new questions and conjectures.
Citation: Andrey Gogolev, Ali Tahzibi. Center Lyapunov exponents in partially hyperbolic dynamics. Journal of Modern Dynamics, 2014, 8 (3&4) : 549-576. doi: 10.3934/jmd.2014.8.549
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.

[2]

J. Alves, V. Araújo and B. Saussol, On the uniform hyperbolicity of some nonuniformly hyperbolic systems,, Proc. Amer. Math. Soc., 131 (2003), 1303. doi: 10.1090/S0002-9939-02-06857-0.

[3]

J. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding,, Invent. Math., 140 (2000), 351. doi: 10.1007/s002220000057.

[4]

A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications,, Invent. Math., 181 (2010), 115. doi: 10.1007/s00222-010-0243-1.

[5]

A. Avila, J. Santamaria and M. Viana, Cocycles over partially hyperbolic maps,, preprint, (2008).

[6]

A. Avila, M. Viana and A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows,, preprint, (2011).

[7]

A. Baraviera and C. Bonatti, Removing zero Lyapunov exponents,, Ergodic Theory Dynam. Systems, 23 (2003), 1655. doi: 10.1017/S0143385702001773.

[8]

J. Bochi, Genericity of zero Lyapunov exponents,, Ergodic Theory Dynam. Systems, 22 (2002), 1667. doi: 10.1017/S0143385702001165.

[9]

J. Bochi, Ch. Bonatti and L. J. Díaz, Robust vanishing of all Lyapunov exponents for iterated function systems,, Mathematische Zeitschrift, 276 (2014), 469. doi: 10.1007/s00209-013-1209-y.

[10]

C. Bonatti, L. J. Díaz and A. Gorodetski, Non-hyperbolic ergodic measures with large support,, Nonlinearity, 23 (2010), 687. doi: 10.1088/0951-7715/23/3/015.

[11]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,, Encyclopaedia of Mathematical Sciences, (2005).

[12]

D. Burago and S. Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups,, J. Mod. Dyn., 2 (2008), 541. doi: 10.3934/jmd.2008.2.541.

[13]

K. Burns and A. Wilkinson, Stable ergodicity of skew products,, Ann. Sci. École Norm. Sup. (4), 32 (1999), 859. doi: 10.1016/S0012-9593(00)87721-6.

[14]

C. Q. Cheng and Y. S. Sun, Existence of invariant tori in three-dimensional measure-preserving mappings,, Celestial Mech. Dynam. Astronom., 47 (): 275. doi: 10.1007/BF00053456.

[15]

L. J. Díaz and A. Gorodetski, Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes,, Ergodic Theory Dynam. Systems, 29 (2009), 1479. doi: 10.1017/S0143385708000849.

[16]

D. Dolgopyat, On dynamics of mostly contracting diffeomorphisms,, Comm. Math. Phys., 213 (2000), 181. doi: 10.1007/s002200000238.

[17]

D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems,, Invent. Math., 155 (2004), 389. doi: 10.1007/s00222-003-0324-5.

[18]

A. Gogolev, Smooth conjugacy in hyperbolic dynamics,, preprint., ().

[19]

A. Gogolev, How typical are pathological foliations in partially hyperbolic dynamics: An example,, Israel J. Math., 187 (2012), 493. doi: 10.1007/s11856-011-0088-3.

[20]

A. S. Gorodetskiĭ, Regularity of central leaves of partially hyperbolic sets and applications,, Izv. Math., 70 (2006), 1093. doi: 10.1070/IM2006v070n06ABEH002340.

[21]

A. S. Gorodetskiĭ and Yu. S. Il'yashenko, Some new robust properties of invariant sets and attractors of dynamical systems,, Funktsional. Anal. i Prilozhen., 33 (1999), 16. doi: 10.1007/BF02465190.

[22]

A. S. Gorodetskiĭ and Yu. S. Il'yashenko, Some properties of skew products over a horseshoe and a solenoid,, Proc. Steklov Inst. Math., 231 (2000), 90.

[23]

A. S. Gorodetskiĭ, Yu. S. Il'yashenko, V. A. Kleptsyn and M. B. Nal'skiĭ, Nonremovability of zero Lyapunov exponents,, (Russian) Funktsional. Anal. i Prilozhen., 39 (2005), 27. doi: 10.1007/s10688-005-0014-8.

[24]

A. Hammerlindl, Leaf conjugacies on the torus,, Ergodic Theory Dynam. Systems, 33 (2013), 896. doi: 10.1017/etds.2012.171.

[25]

B. Hasselblatt and Ya. Pesin, Partially hyperbolic dynamical systems,, in Handbook of Dynamical Systems. Vol. 1B, (2006), 1. doi: 10.1016/S1874-575X(06)80026-3.

[26]

M. Hirayama and Ya. Pesin, Non-absolutely continuous foliations,, Israel J. Math., 160 (2007), 173. doi: 10.1007/s11856-007-0060-4.

[27]

M. Herman, Stabilité topologique des systèmes dynamiques conservatifs,, preprint, (1990).

[28]

A. J. Homburg, Atomic disintegrations for partially hyperbolic diffeomorphisms,, preprint, (2010).

[29]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, With a supplementary chapter by Katok and L. Mendoza, (1995). doi: 10.1017/CBO9780511809187.

[30]

V. Kleptsyn and M. Nal'skiĭ, Robustness of nonhyperbolic measures for $C^1$-diffeomorphisms,, Funct. Anal. Appl., 41 (2007), 271. doi: 10.1007/s10688-007-0025-8.

[31]

F. Ledrappier, Positivity of the exponent for stationary sequences of matrices,, in Lyapunov Exponents (Bremen, (1984), 56. doi: 10.1007/BFb0076833.

[32]

C. Liang, W. Sun and J. Yang, Some results on perturbations to Lyapunov exponents,, Discrete Contin. Dyn. Syst., 32 (2012), 4287. doi: 10.3934/dcds.2012.32.4287.

[33]

R. de la Llave, Invariants for smooth conjugacy of hyperbolic dynamical systems. II,, Commun. Math. Phys., 109 (1987), 369. doi: 10.1007/BF01206141.

[34]

J. Milnor, Fubini foiled: Katok's paradoxical example in measure theory,, Math. Intelligencer, 19 (1997), 30. doi: 10.1007/BF03024428.

[35]

J. M. Marco and R. Moriyón, Invariants for smooth conjugacy of hyperbolic dynamical systems. III,, Comm. Math. Phys., 112 (1987), 317. doi: 10.1007/BF01217815.

[36]

F. Micena and A. Tahzibi, Regularity of foliations and Lyapunov exponents for partially hyperbolic dynamics on 3-torus,, Nonlinearity, 26 (2013), 1071. doi: 10.1088/0951-7715/26/4/1071.

[37]

M. Nal'skiĭ, Non-hyporbolic invariant measures on the maximal attractor (in Russian),, , ().

[38]

Ya. Pesin, Characteristic Lyapunov exponents, and smooth ergodic theory,, Russian Math. Surveys, 32 (1977), 55. doi: 10.1070/RM1977v032n04ABEH001639.

[39]

Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity,, Zurich Lectures in Advanced Mathematics, (2004). doi: 10.4171/003.

[40]

Ya. Pesin, Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents,, Regul. Chaotic Dyn., 12 (2007), 476. doi: 10.1134/S1560354707050024.

[41]

G. Ponce and A. Tahzibi, Central Lyapunov exponent of partially hyperbolic diffeomorphisms of $\mathbbT^3$,, Proc. Amer. Math. Soc., 142 (2014), 3193. doi: 10.1090/S0002-9939-2014-12063-6.

[42]

G. Ponce, A. Tahzibi and R. Varão, Minimal yet measurable foliations,, J. Mod. Dyn., 8 (2014), 93. doi: 10.3934/jmd.2014.8.93.

[43]

F. Rodriguez Hertz, J. Rodriguez Hertz and R. Ures, Partially Hyperbolic Dynamics,, IMPA Mathematical Publications, (2011).

[44]

D. Ruelle, Perturbation theory for Lyapunov exponents of a toral map: Extension of a result of Shub and Wilkinson,, Israel J. Math., 134 (2003), 345. doi: 10.1007/BF02787412.

[45]

D. Ruelle and A. Wilkinson, Absolutely singular dynamical foliations,, Comm. Math. Phys., 219 (2001), 481. doi: 10.1007/s002200100420.

[46]

R. Saghin and Zh. Xia, Geometric expansion, Lyapunov exponents and foliations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 689. doi: 10.1016/j.anihpc.2008.07.001.

[47]

M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents,, Invent. Math., 139 (2000), 495. doi: 10.1007/s002229900035.

[48]

K. Sigmund, On the connectedness of ergodic systems,, Manuscripta Math., 22 (1977), 27. doi: 10.1007/BF01182064.

[49]

R. Varão, Center foliation: Absolute continuity, disintegration and rigidity,, Ergod. Th. Dynam. Syst., (2014). doi: 10.1017/etds.2014.53.

[50]

Z. Xia, Existence of invariant tori in volume-preserving diffeomorphisms,, Ergod. Th. Dynam. Syst., 12 (1992), 621. doi: 10.1017/S0143385700006969.

[51]

J.-C. Yoccoz, Travaux de Herman sur les tores invariants,, Séminaire Bourbaki, (1992), 311.

show all references

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.

[2]

J. Alves, V. Araújo and B. Saussol, On the uniform hyperbolicity of some nonuniformly hyperbolic systems,, Proc. Amer. Math. Soc., 131 (2003), 1303. doi: 10.1090/S0002-9939-02-06857-0.

[3]

J. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding,, Invent. Math., 140 (2000), 351. doi: 10.1007/s002220000057.

[4]

A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications,, Invent. Math., 181 (2010), 115. doi: 10.1007/s00222-010-0243-1.

[5]

A. Avila, J. Santamaria and M. Viana, Cocycles over partially hyperbolic maps,, preprint, (2008).

[6]

A. Avila, M. Viana and A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows,, preprint, (2011).

[7]

A. Baraviera and C. Bonatti, Removing zero Lyapunov exponents,, Ergodic Theory Dynam. Systems, 23 (2003), 1655. doi: 10.1017/S0143385702001773.

[8]

J. Bochi, Genericity of zero Lyapunov exponents,, Ergodic Theory Dynam. Systems, 22 (2002), 1667. doi: 10.1017/S0143385702001165.

[9]

J. Bochi, Ch. Bonatti and L. J. Díaz, Robust vanishing of all Lyapunov exponents for iterated function systems,, Mathematische Zeitschrift, 276 (2014), 469. doi: 10.1007/s00209-013-1209-y.

[10]

C. Bonatti, L. J. Díaz and A. Gorodetski, Non-hyperbolic ergodic measures with large support,, Nonlinearity, 23 (2010), 687. doi: 10.1088/0951-7715/23/3/015.

[11]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective,, Encyclopaedia of Mathematical Sciences, (2005).

[12]

D. Burago and S. Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups,, J. Mod. Dyn., 2 (2008), 541. doi: 10.3934/jmd.2008.2.541.

[13]

K. Burns and A. Wilkinson, Stable ergodicity of skew products,, Ann. Sci. École Norm. Sup. (4), 32 (1999), 859. doi: 10.1016/S0012-9593(00)87721-6.

[14]

C. Q. Cheng and Y. S. Sun, Existence of invariant tori in three-dimensional measure-preserving mappings,, Celestial Mech. Dynam. Astronom., 47 (): 275. doi: 10.1007/BF00053456.

[15]

L. J. Díaz and A. Gorodetski, Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes,, Ergodic Theory Dynam. Systems, 29 (2009), 1479. doi: 10.1017/S0143385708000849.

[16]

D. Dolgopyat, On dynamics of mostly contracting diffeomorphisms,, Comm. Math. Phys., 213 (2000), 181. doi: 10.1007/s002200000238.

[17]

D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems,, Invent. Math., 155 (2004), 389. doi: 10.1007/s00222-003-0324-5.

[18]

A. Gogolev, Smooth conjugacy in hyperbolic dynamics,, preprint., ().

[19]

A. Gogolev, How typical are pathological foliations in partially hyperbolic dynamics: An example,, Israel J. Math., 187 (2012), 493. doi: 10.1007/s11856-011-0088-3.

[20]

A. S. Gorodetskiĭ, Regularity of central leaves of partially hyperbolic sets and applications,, Izv. Math., 70 (2006), 1093. doi: 10.1070/IM2006v070n06ABEH002340.

[21]

A. S. Gorodetskiĭ and Yu. S. Il'yashenko, Some new robust properties of invariant sets and attractors of dynamical systems,, Funktsional. Anal. i Prilozhen., 33 (1999), 16. doi: 10.1007/BF02465190.

[22]

A. S. Gorodetskiĭ and Yu. S. Il'yashenko, Some properties of skew products over a horseshoe and a solenoid,, Proc. Steklov Inst. Math., 231 (2000), 90.

[23]

A. S. Gorodetskiĭ, Yu. S. Il'yashenko, V. A. Kleptsyn and M. B. Nal'skiĭ, Nonremovability of zero Lyapunov exponents,, (Russian) Funktsional. Anal. i Prilozhen., 39 (2005), 27. doi: 10.1007/s10688-005-0014-8.

[24]

A. Hammerlindl, Leaf conjugacies on the torus,, Ergodic Theory Dynam. Systems, 33 (2013), 896. doi: 10.1017/etds.2012.171.

[25]

B. Hasselblatt and Ya. Pesin, Partially hyperbolic dynamical systems,, in Handbook of Dynamical Systems. Vol. 1B, (2006), 1. doi: 10.1016/S1874-575X(06)80026-3.

[26]

M. Hirayama and Ya. Pesin, Non-absolutely continuous foliations,, Israel J. Math., 160 (2007), 173. doi: 10.1007/s11856-007-0060-4.

[27]

M. Herman, Stabilité topologique des systèmes dynamiques conservatifs,, preprint, (1990).

[28]

A. J. Homburg, Atomic disintegrations for partially hyperbolic diffeomorphisms,, preprint, (2010).

[29]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, With a supplementary chapter by Katok and L. Mendoza, (1995). doi: 10.1017/CBO9780511809187.

[30]

V. Kleptsyn and M. Nal'skiĭ, Robustness of nonhyperbolic measures for $C^1$-diffeomorphisms,, Funct. Anal. Appl., 41 (2007), 271. doi: 10.1007/s10688-007-0025-8.

[31]

F. Ledrappier, Positivity of the exponent for stationary sequences of matrices,, in Lyapunov Exponents (Bremen, (1984), 56. doi: 10.1007/BFb0076833.

[32]

C. Liang, W. Sun and J. Yang, Some results on perturbations to Lyapunov exponents,, Discrete Contin. Dyn. Syst., 32 (2012), 4287. doi: 10.3934/dcds.2012.32.4287.

[33]

R. de la Llave, Invariants for smooth conjugacy of hyperbolic dynamical systems. II,, Commun. Math. Phys., 109 (1987), 369. doi: 10.1007/BF01206141.

[34]

J. Milnor, Fubini foiled: Katok's paradoxical example in measure theory,, Math. Intelligencer, 19 (1997), 30. doi: 10.1007/BF03024428.

[35]

J. M. Marco and R. Moriyón, Invariants for smooth conjugacy of hyperbolic dynamical systems. III,, Comm. Math. Phys., 112 (1987), 317. doi: 10.1007/BF01217815.

[36]

F. Micena and A. Tahzibi, Regularity of foliations and Lyapunov exponents for partially hyperbolic dynamics on 3-torus,, Nonlinearity, 26 (2013), 1071. doi: 10.1088/0951-7715/26/4/1071.

[37]

M. Nal'skiĭ, Non-hyporbolic invariant measures on the maximal attractor (in Russian),, , ().

[38]

Ya. Pesin, Characteristic Lyapunov exponents, and smooth ergodic theory,, Russian Math. Surveys, 32 (1977), 55. doi: 10.1070/RM1977v032n04ABEH001639.

[39]

Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity,, Zurich Lectures in Advanced Mathematics, (2004). doi: 10.4171/003.

[40]

Ya. Pesin, Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents,, Regul. Chaotic Dyn., 12 (2007), 476. doi: 10.1134/S1560354707050024.

[41]

G. Ponce and A. Tahzibi, Central Lyapunov exponent of partially hyperbolic diffeomorphisms of $\mathbbT^3$,, Proc. Amer. Math. Soc., 142 (2014), 3193. doi: 10.1090/S0002-9939-2014-12063-6.

[42]

G. Ponce, A. Tahzibi and R. Varão, Minimal yet measurable foliations,, J. Mod. Dyn., 8 (2014), 93. doi: 10.3934/jmd.2014.8.93.

[43]

F. Rodriguez Hertz, J. Rodriguez Hertz and R. Ures, Partially Hyperbolic Dynamics,, IMPA Mathematical Publications, (2011).

[44]

D. Ruelle, Perturbation theory for Lyapunov exponents of a toral map: Extension of a result of Shub and Wilkinson,, Israel J. Math., 134 (2003), 345. doi: 10.1007/BF02787412.

[45]

D. Ruelle and A. Wilkinson, Absolutely singular dynamical foliations,, Comm. Math. Phys., 219 (2001), 481. doi: 10.1007/s002200100420.

[46]

R. Saghin and Zh. Xia, Geometric expansion, Lyapunov exponents and foliations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 689. doi: 10.1016/j.anihpc.2008.07.001.

[47]

M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents,, Invent. Math., 139 (2000), 495. doi: 10.1007/s002229900035.

[48]

K. Sigmund, On the connectedness of ergodic systems,, Manuscripta Math., 22 (1977), 27. doi: 10.1007/BF01182064.

[49]

R. Varão, Center foliation: Absolute continuity, disintegration and rigidity,, Ergod. Th. Dynam. Syst., (2014). doi: 10.1017/etds.2014.53.

[50]

Z. Xia, Existence of invariant tori in volume-preserving diffeomorphisms,, Ergod. Th. Dynam. Syst., 12 (1992), 621. doi: 10.1017/S0143385700006969.

[51]

J.-C. Yoccoz, Travaux de Herman sur les tores invariants,, Séminaire Bourbaki, (1992), 311.

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