Center Lyapunov exponents in partially hyperbolic dynamics

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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

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

Abstract
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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
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[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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