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January  2011, 5(1): 107-122. doi: 10.3934/jmd.2011.5.107

Integrability and Lyapunov exponents

Andy Hammerlindl 1,

1. 

Instituto Nacional deMatemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, Brazil

Received  April 2010 Revised  December 2010 Published  April 2011

A smooth distribution, invariant under a dynamical system, integrates to give an invariant foliation, unless certain resonance conditions are present.
Keywords: Integrability, Lyapunov exponents..
Mathematics Subject Classification: Primary: 37D10; Secondary: 37D20, 37D3.
Citation: Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107
References:
[1]

D. V. Anosov, "Geodesic Flows on Closed Riemannian Manifolds with Negative Curvature," Proceedings of the Steklov Institute of Mathematics, No. 90 (1967), Translated from the Russian by S. Feder American Mathematical Society, Providence, R.I. 1969, iv+235 pp.  Google Scholar

[2]

L. Barriera and C. Valls, Center manifolds for nonuniformly partially hyperbolic diffeomorphisms, Journal de Mathématiques Pures et Appliquées, 84 (2005), 1693-1715. doi: 10.1016/j.matpur.2005.07.005.  Google Scholar

[3]

M. Brin and Ja. Pesin, Partially hyperbolic dynamical systems, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.  Google Scholar

[4]

K. Burns, F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Talitskaya and R. Ures, Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center, Discrete and Continuous Dynamical Systems, 22 (2008), 75-88. doi: 10.3934/dcds.2008.22.75.  Google Scholar

[5]

K. Burns and A. Wilkinson, Dynamical coherence and center bunching, Discrete and Continuous Dynamical Systems, 22 (2008), 89-100. doi: 10.3934/dcds.2008.22.89.  Google Scholar

[6]

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Annals of Math., 171 (2010), 451-489.  Google Scholar

[7]

X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds I: Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), 283-328. doi: 10.1512/iumj.2003.52.2245.  Google Scholar

[8]

F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics, Partially Hyperbolic Dynamics, Lamnations, and Teichmüller Flow, 35-87, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007.  Google Scholar

[9]

F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math., 172 (2008), 353-381. doi: 10.1007/s00222-007-0100-z.  Google Scholar

[10]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," volume 583 of "Lecture Notes in Mathematics," Vol. 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[11]

R. Mañé, "Ergodic Theory and Differentiable Dynamics," Translated from the Portuguese by Silvio Levy. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 8, Springer-Verlag, Berlin, 1987.  Google Scholar

[12]

V. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-210.  Google Scholar

[13]

V. Oseledets, Oseledets theorem, Scholarpedia, 3 (2008), 1846. doi: 10.4249/scholarpedia.1846.  Google Scholar

[14]

F. Rampazzo, Frobenius-type theorems for Lipschitz distributions, Journal of Differential Equations, 243 (2007), 270-300. doi: 10.1016/j.jde.2007.05.040.  Google Scholar

[15]

S. Simić, Lipschitz distributions and Anosov flows, Proc. of the Amer. Math. Soc., 124 (1996), 1869-1877. doi: 10.1090/S0002-9939-96-03423-5.  Google Scholar

[16]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[17]

A. Wilkinson, Stable ergodicity of the time-one map of a geodesic flow, Ergod. Th. and Dynam. Sys., 18 (1998), 1545-1588. doi: 10.1017/S0143385798117984.  Google Scholar

show all references

References:
[1]

D. V. Anosov, "Geodesic Flows on Closed Riemannian Manifolds with Negative Curvature," Proceedings of the Steklov Institute of Mathematics, No. 90 (1967), Translated from the Russian by S. Feder American Mathematical Society, Providence, R.I. 1969, iv+235 pp.  Google Scholar

[2]

L. Barriera and C. Valls, Center manifolds for nonuniformly partially hyperbolic diffeomorphisms, Journal de Mathématiques Pures et Appliquées, 84 (2005), 1693-1715. doi: 10.1016/j.matpur.2005.07.005.  Google Scholar

[3]

M. Brin and Ja. Pesin, Partially hyperbolic dynamical systems, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.  Google Scholar

[4]

K. Burns, F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Talitskaya and R. Ures, Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center, Discrete and Continuous Dynamical Systems, 22 (2008), 75-88. doi: 10.3934/dcds.2008.22.75.  Google Scholar

[5]

K. Burns and A. Wilkinson, Dynamical coherence and center bunching, Discrete and Continuous Dynamical Systems, 22 (2008), 89-100. doi: 10.3934/dcds.2008.22.89.  Google Scholar

[6]

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Annals of Math., 171 (2010), 451-489.  Google Scholar

[7]

X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds I: Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), 283-328. doi: 10.1512/iumj.2003.52.2245.  Google Scholar

[8]

F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, A survey of partially hyperbolic dynamics, Partially Hyperbolic Dynamics, Lamnations, and Teichmüller Flow, 35-87, Fields Inst. Commun., 51, Amer. Math. Soc., Providence, RI, 2007.  Google Scholar

[9]

F. Rodriguez Hertz, M. A. Rodriguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math., 172 (2008), 353-381. doi: 10.1007/s00222-007-0100-z.  Google Scholar

[10]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," volume 583 of "Lecture Notes in Mathematics," Vol. 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[11]

R. Mañé, "Ergodic Theory and Differentiable Dynamics," Translated from the Portuguese by Silvio Levy. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 8, Springer-Verlag, Berlin, 1987.  Google Scholar

[12]

V. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-210.  Google Scholar

[13]

V. Oseledets, Oseledets theorem, Scholarpedia, 3 (2008), 1846. doi: 10.4249/scholarpedia.1846.  Google Scholar

[14]

F. Rampazzo, Frobenius-type theorems for Lipschitz distributions, Journal of Differential Equations, 243 (2007), 270-300. doi: 10.1016/j.jde.2007.05.040.  Google Scholar

[15]

S. Simić, Lipschitz distributions and Anosov flows, Proc. of the Amer. Math. Soc., 124 (1996), 1869-1877. doi: 10.1090/S0002-9939-96-03423-5.  Google Scholar

[16]

S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1.  Google Scholar

[17]

A. Wilkinson, Stable ergodicity of the time-one map of a geodesic flow, Ergod. Th. and Dynam. Sys., 18 (1998), 1545-1588. doi: 10.1017/S0143385798117984.  Google Scholar

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