American Institute of Mathematical Sciences

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

Center Lyapunov exponents in partially hyperbolic dynamics

Andrey Gogolev 1, and  Ali Tahzibi 2,

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.
Keywords: Center Lyapunov exponent, Pesin manifold., partially hyperbolic.
Mathematics Subject Classification: Primary: 37D25; Secondary: 37D30, 37D3.
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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[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. Google Scholar

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[17]

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

[18]

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

[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. Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

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

[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. Google Scholar

[26]

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

[27]

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

[28]

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

[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. Google Scholar

[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. Google Scholar

[31]

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

[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. Google Scholar

[33]

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

[34]

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

[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. Google Scholar

[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. Google Scholar

[37]

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

[38]

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

[39]

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

[40]

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

[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. Google Scholar

[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. Google Scholar

[43]

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

[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. Google Scholar

[45]

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

[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. Google Scholar

[47]

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

[48]

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

[49]

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

[50]

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

[51]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[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. Google Scholar

[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. Google Scholar

[11]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[16]

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

[17]

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

[18]

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

[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. Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

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

[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. Google Scholar

[26]

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

[27]

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

[28]

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

[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. Google Scholar

[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. Google Scholar

[31]

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

[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. Google Scholar

[33]

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

[34]

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

[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. Google Scholar

[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. Google Scholar

[37]

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

[38]

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

[39]

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

[40]

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

[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. Google Scholar

[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. Google Scholar

[43]

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

[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. Google Scholar

[45]

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

[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. Google Scholar

[47]

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

[48]

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

[49]

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

[50]

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

[51]

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

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