Center Lyapunov exponents in partially hyperbolic dynamics

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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

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), 1-60. 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), 1303-1309. 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), 351-398. doi: 10.1007/s002220000057.
[4]	A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications, Invent. Math., 181 (2010), 115-189. doi: 10.1007/s00222-010-0243-1.
[5]	A. Avila, J. Santamaria and M. Viana, Cocycles over partially hyperbolic maps, preprint, 2008. Available from: http://www.preprint.impa.br.
[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), 1655-1670. doi: 10.1017/S0143385702001773.
[8]	J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems, 22 (2002), 1667-1696. 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), 469-503. 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), 687-705. 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, 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2005.
[12]	D. Burago and S. Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups, J. Mod. Dyn., 2 (2008), 541-580. 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), 859-889. 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), 1479-1513. doi: 10.1017/S0143385708000849.
[16]	D. Dolgopyat, On dynamics of mostly contracting diffeomorphisms, Comm. Math. Phys., 213 (2000), 181-201. doi: 10.1007/s002200000238.
[17]	D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems, Invent. Math., 155 (2004), 389-449. 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), 493-507. 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), 1093-1116. 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), 16-30. 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), 90-112.
[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), 27-38, 95; translation in Funct. Anal. Appl., 39 (2005), 21-30. doi: 10.1007/s10688-005-0014-8.
[24]	A. Hammerlindl, Leaf conjugacies on the torus, Ergodic Theory Dynam. Systems, 33 (2013), 896-933. doi: 10.1017/etds.2012.171.
[25]	B. Hasselblatt and Ya. Pesin, Partially hyperbolic dynamical systems, in Handbook of Dynamical Systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, 1-55. doi: 10.1016/S1874-575X(06)80026-3.
[26]	M. Hirayama and Ya. Pesin, Non-absolutely continuous foliations, Israel J. Math., 160 (2007), 173-187. 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, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 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), 271-283. doi: 10.1007/s10688-007-0025-8.
[31]	F. Ledrappier, Positivity of the exponent for stationary sequences of matrices, in Lyapunov Exponents (Bremen, 1984), Lect. Notes Math., 1186, Springer, Berlin, 1986, 56-73. 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), 4287-4305. 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), 369-378. doi: 10.1007/BF01206141.
[34]	J. Milnor, Fubini foiled: Katok's paradoxical example in measure theory, Math. Intelligencer, 19 (1997), 30-32. 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), 317-333. 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), 1071-1082. 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), 55-114. doi: 10.1070/RM1977v032n04ABEH001639.
[39]	Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2004. doi: 10.4171/003.
[40]	Ya. Pesin, Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents, Regul. Chaotic Dyn., 12 (2007), 476-489. 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), 3193-3205. 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), 93-107. doi: 10.3934/jmd.2014.8.93.
[43]	F. Rodriguez Hertz, J. Rodriguez Hertz and R. Ures, Partially Hyperbolic Dynamics, IMPA Mathematical Publications, 28th Brazilian Mathematics Colloquium, Rio de Janeiro, 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), 345-361. doi: 10.1007/BF02787412.
[45]	D. Ruelle and A. Wilkinson, Absolutely singular dynamical foliations, Comm. Math. Phys., 219 (2001), 481-487. 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), 689-704. doi: 10.1016/j.anihpc.2008.07.001.
[47]	M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents, Invent. Math., 139 (2000), 495-508. doi: 10.1007/s002229900035.
[48]	K. Sigmund, On the connectedness of ergodic systems, Manuscripta Math., 22 (1977), 27-32. doi: 10.1007/BF01182064.
[49]	R. Varão, Center foliation: Absolute continuity, disintegration and rigidity, Ergod. Th. Dynam. Syst., (2014), 20pp. doi: 10.1017/etds.2014.53.
[50]	Z. Xia, Existence of invariant tori in volume-preserving diffeomorphisms, Ergod. Th. Dynam. Syst., 12 (1992), 621-631. doi: 10.1017/S0143385700006969.
[51]	J.-C. Yoccoz, Travaux de Herman sur les tores invariants, Séminaire Bourbaki, Vol. 1991/92, Exp. No. 754, Astérisque, No. 206 (1992), 311-344.

show all references

References:

[1]	F. Abdenur, C. Bonatti and S. Crovisier, Nonuniform hyperbolicity for $C^1$-generic diffeomorphisms, Israel J. Math., 183 (2011), 1-60. 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), 1303-1309. 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), 351-398. doi: 10.1007/s002220000057.
[4]	A. Avila and M. Viana, Extremal Lyapunov exponents: An invariance principle and applications, Invent. Math., 181 (2010), 115-189. doi: 10.1007/s00222-010-0243-1.
[5]	A. Avila, J. Santamaria and M. Viana, Cocycles over partially hyperbolic maps, preprint, 2008. Available from: http://www.preprint.impa.br.
[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), 1655-1670. doi: 10.1017/S0143385702001773.
[8]	J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems, 22 (2002), 1667-1696. 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), 469-503. 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), 687-705. 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, 102, Mathematical Physics, III, Springer-Verlag, Berlin, 2005.
[12]	D. Burago and S. Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelian fundamental groups, J. Mod. Dyn., 2 (2008), 541-580. 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), 859-889. 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), 1479-1513. doi: 10.1017/S0143385708000849.
[16]	D. Dolgopyat, On dynamics of mostly contracting diffeomorphisms, Comm. Math. Phys., 213 (2000), 181-201. doi: 10.1007/s002200000238.
[17]	D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems, Invent. Math., 155 (2004), 389-449. 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), 493-507. 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), 1093-1116. 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), 16-30. 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), 90-112.
[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), 27-38, 95; translation in Funct. Anal. Appl., 39 (2005), 21-30. doi: 10.1007/s10688-005-0014-8.
[24]	A. Hammerlindl, Leaf conjugacies on the torus, Ergodic Theory Dynam. Systems, 33 (2013), 896-933. doi: 10.1017/etds.2012.171.
[25]	B. Hasselblatt and Ya. Pesin, Partially hyperbolic dynamical systems, in Handbook of Dynamical Systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, 1-55. doi: 10.1016/S1874-575X(06)80026-3.
[26]	M. Hirayama and Ya. Pesin, Non-absolutely continuous foliations, Israel J. Math., 160 (2007), 173-187. 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, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 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), 271-283. doi: 10.1007/s10688-007-0025-8.
[31]	F. Ledrappier, Positivity of the exponent for stationary sequences of matrices, in Lyapunov Exponents (Bremen, 1984), Lect. Notes Math., 1186, Springer, Berlin, 1986, 56-73. 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), 4287-4305. 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), 369-378. doi: 10.1007/BF01206141.
[34]	J. Milnor, Fubini foiled: Katok's paradoxical example in measure theory, Math. Intelligencer, 19 (1997), 30-32. 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), 317-333. 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), 1071-1082. 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), 55-114. doi: 10.1070/RM1977v032n04ABEH001639.
[39]	Ya. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2004. doi: 10.4171/003.
[40]	Ya. Pesin, Existence and genericity problems for dynamical systems with nonzero Lyapunov exponents, Regul. Chaotic Dyn., 12 (2007), 476-489. 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), 3193-3205. 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), 93-107. doi: 10.3934/jmd.2014.8.93.
[43]	F. Rodriguez Hertz, J. Rodriguez Hertz and R. Ures, Partially Hyperbolic Dynamics, IMPA Mathematical Publications, 28th Brazilian Mathematics Colloquium, Rio de Janeiro, 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), 345-361. doi: 10.1007/BF02787412.
[45]	D. Ruelle and A. Wilkinson, Absolutely singular dynamical foliations, Comm. Math. Phys., 219 (2001), 481-487. 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), 689-704. doi: 10.1016/j.anihpc.2008.07.001.
[47]	M. Shub and A. Wilkinson, Pathological foliations and removable zero exponents, Invent. Math., 139 (2000), 495-508. doi: 10.1007/s002229900035.
[48]	K. Sigmund, On the connectedness of ergodic systems, Manuscripta Math., 22 (1977), 27-32. doi: 10.1007/BF01182064.
[49]	R. Varão, Center foliation: Absolute continuity, disintegration and rigidity, Ergod. Th. Dynam. Syst., (2014), 20pp. doi: 10.1017/etds.2014.53.
[50]	Z. Xia, Existence of invariant tori in volume-preserving diffeomorphisms, Ergod. Th. Dynam. Syst., 12 (1992), 621-631. doi: 10.1017/S0143385700006969.
[51]	J.-C. Yoccoz, Travaux de Herman sur les tores invariants, Séminaire Bourbaki, Vol. 1991/92, Exp. No. 754, Astérisque, No. 206 (1992), 311-344.

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