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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, (1967).   Google Scholar

[2]

L. Barriera and C. Valls, Center manifolds for nonuniformly partially hyperbolic diffeomorphisms,, Journal de Mathématiques Pures et Appliquées, 84 (2005), 1693.  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.   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.  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.  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.   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.  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, (2007), 35.   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.  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, (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)], (1987).   Google Scholar

[12]

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

[13]

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

[14]

F. Rampazzo, Frobenius-type theorems for Lipschitz distributions,, Journal of Differential Equations, 243 (2007), 270.  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.  doi: 10.1090/S0002-9939-96-03423-5.  Google Scholar

[16]

S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747.  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.  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, (1967).   Google Scholar

[2]

L. Barriera and C. Valls, Center manifolds for nonuniformly partially hyperbolic diffeomorphisms,, Journal de Mathématiques Pures et Appliquées, 84 (2005), 1693.  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.   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.  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.  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.   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.  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, (2007), 35.   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.  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, (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)], (1987).   Google Scholar

[12]

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

[13]

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

[14]

F. Rampazzo, Frobenius-type theorems for Lipschitz distributions,, Journal of Differential Equations, 243 (2007), 270.  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.  doi: 10.1090/S0002-9939-96-03423-5.  Google Scholar

[16]

S. Smale, Differentiable dynamical systems,, Bull. Amer. Math. Soc., 73 (1967), 747.  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.  doi: 10.1017/S0143385798117984.  Google Scholar

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