Center Lyapunov exponents in partially hyperbolic dynamics

Previous Article
Rigidity of Julia sets for Hénon type maps
JMD Home
This Issue
Next Article

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
Related Papers

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.
[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.

[1]	Lin Wang, Yujun Zhu. Center specification property and entropy for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 469-479. doi: 10.3934/dcds.2016.36.469
[2]	Andrey Gogolev. Partially hyperbolic diffeomorphisms with compact center foliations. Journal of Modern Dynamics, 2011, 5 (4) : 747-769. doi: 10.3934/jmd.2011.5.747
[3]	Keith Burns, Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Anna Talitskaya, Raúl Ures. Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 75-88. doi: 10.3934/dcds.2008.22.75
[4]	Doris Bohnet. Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation. Journal of Modern Dynamics, 2013, 7 (4) : 565-604. doi: 10.3934/jmd.2013.7.565
[5]	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
[6]	Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249
[7]	David Burguet, Todd Fisher. Symbolic extensionsfor partially hyperbolic dynamical systems with 2-dimensional center bundle. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2253-2270. doi: 10.3934/dcds.2013.33.2253
[8]	Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839
[9]	Mauricio Poletti. Stably positive Lyapunov exponents for symplectic linear cocycles over partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5163-5188. doi: 10.3934/dcds.2018228
[10]	Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873
[11]	Stefano Bianchini, Alberto Bressan. A center manifold technique for tracing viscous waves. Communications on Pure & Applied Analysis, 2002, 1 (2) : 161-190. doi: 10.3934/cpaa.2002.1.161
[12]	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
[13]	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
[14]	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
[15]	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
[16]	Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68
[17]	Janusz Mierczyński, Wenxian Shen. Formulas for generalized principal Lyapunov exponent for parabolic PDEs. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1189-1199. doi: 10.3934/dcdss.2016048
[18]	Pedro Duarte, Silvius Klein, Manuel Santos. A random cocycle with non Hölder Lyapunov exponent. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4841-4861. doi: 10.3934/dcds.2019197
[19]	A. Carati. Center manifold of unstable periodic orbits of helium atom: numerical evidence. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 97-104. doi: 10.3934/dcdsb.2003.3.97
[20]	Luiz Felipe Nobili França. Partially hyperbolic sets with a dynamically minimal lamination. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2717-2729. doi: 10.3934/dcds.2018114

2018 Impact Factor: 0.295

American Institute of Mathematical Sciences