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A semi-invertible Oseledets Theorem with applications to transfer operator cocycles
1. | School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052 |
2. | Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo SP, Brazil |
3. | Department of Mathematics and Statistics, University of Victoria, P.O. Box 3060 STN CSC, Victoria, B.C., V8W 3R4 |
References:
[1] |
A. Arbieto, C. Matheus and K. Oliveira, Equilibrium states for random non-uniformly expanding maps, Nonlinearity, 17 (2004), 581-593.
doi: 10.1088/0951-7715/17/2/013. |
[2] |
L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. |
[3] |
V. Baladi, "Positive Transfer Operators and Decay of Correlations," Advanced Series in Nonlinear Dynamics, 16, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.
doi: 10.1142/9789812813633. |
[4] |
L. Barreira and C. Silva, Lyapunov exponents for continuous transformations and dimension theory, Discrete Contin. Dyn. Syst., 13 (2005), 469-490.
doi: 10.3934/dcds.2005.13.469. |
[5] |
B. Bollobás, "Linear Analysis. An Introductory Course," Second edition, Cambridge University Press, Cambridge, 1999. |
[6] |
J. Buzzi, Absolutely continuous S.R.B. measures for random Lasota-Yorke maps, Trans. Amer. Math. Soc., 352 (2000), 3289-3303.
doi: 10.1090/S0002-9947-00-02607-6. |
[7] |
M. Dellnitz, G. Froyland, C. Horenkamp, K. Padberg-Gehle and A. Sen Gupta, Seasonal variability of the subpolar gyres in the southern ocean: A numerical investigation based on transfer operators, Nonlinear Processes in Geophysics, 16 (2009), 655-663. |
[8] |
M. Dellnitz, O. Junge, W.S. Koon, F. Lekien, M.W. Lo, J.E. Marsden, K. Padberg, R. Preis, S.D. Ross and B. Thiere, Transport in dynamical astronomy and multibody problems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 699-727.
doi: 10.1142/S0218127405012545. |
[9] |
D. H. Fremlin, Measurable functions and almost continuous functions, Manuscripta Math., 33 (1980/81), 387-405.
doi: 10.1007/BF01798235. |
[10] |
G. Froyland, S. Lloyd and A. Quas, Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory and Dynam. Systems, 30 (2010), 729-756.
doi: 10.1017/S0143385709000339. |
[11] |
G. Froyland, S. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems, Phys. D, 239 (2010), 1527-1541.
doi: 10.1016/j.physd.2010.03.009. |
[12] |
G. Froyland, K. Padberg, M.H. England and A.M. Treguier, Detection of coherent oceanic structures via transfer operators, Phys. Rev. Lett., 98 (2007). |
[13] |
F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z., 180 (1982), 119-140.
doi: 10.1007/BF01215004. |
[14] |
G. Keller, On the rate of convergence to equilibrium in one-dimensional systems, Comm. Math. Phys., 96 (1984), 181-193. |
[15] |
Y. Kifer and P.D. Liu, Random dynamics, in "Handbook of Dynamical Systems," Vol. 1B, Elsevier B. V., Amsterdam, (2006), 379-499.
doi: 10.1016/S1874-575X(06)80030-5. |
[16] |
A. Lasota and J.A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488. |
[17] |
Z. Lian, "Lyapunov Exponents and Invariant Manifolds for Random Dynamical Systems in a Banach Space," Ph.D thesis, Brigham Young University, 2008. |
[18] |
C. Liverani, Decay of correlations, Ann. of Math. (2), 142 (1995), 239-301.
doi: 10.2307/2118636. |
[19] |
R. Mañé, Lyapounov exponents and stable manifolds for compact transformations, in "Geometric Dynamics" (Rio de Janeiro, 1981), Lecture Notes in Math., 1007, Springer, Berlin, (1983), 522-577.
doi: 10.1007/BFb0061433. |
[20] |
T. Morita, Random iteration of one-dimensional transformations, Osaka J. Math., 22 (1985), 489-518. |
[21] |
I. Morris, The generalized Berger-Wang formula and the spectral radius of linear cocycles, J. Func. Anal., 262 (2012), 811-824.
doi: 10.1016/j.jfa.2011.09.021. |
[22] |
V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210. |
[23] |
S. Pelikan, Invariant densities for random maps of the interval, Trans. Amer. Math. Soc., 281 (1984), 813-825.
doi: 10.2307/2000087. |
[24] |
D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2), 115 (1982), 243-290.
doi: 10.2307/1971392. |
[25] |
M. Rychlik, Bounded variation and invariant measures, Studia Math., 76 (1983), 69-80. |
[26] |
Ch. Schütte, W. Huisinga and P. Deuflhard, Transfer operator approach to conformational dynamics in biomolecular systems, in "Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems," Springer, Berlin, (2001), 191-223. |
[27] |
P. Thieullen, Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 49-97. |
show all references
References:
[1] |
A. Arbieto, C. Matheus and K. Oliveira, Equilibrium states for random non-uniformly expanding maps, Nonlinearity, 17 (2004), 581-593.
doi: 10.1088/0951-7715/17/2/013. |
[2] |
L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. |
[3] |
V. Baladi, "Positive Transfer Operators and Decay of Correlations," Advanced Series in Nonlinear Dynamics, 16, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.
doi: 10.1142/9789812813633. |
[4] |
L. Barreira and C. Silva, Lyapunov exponents for continuous transformations and dimension theory, Discrete Contin. Dyn. Syst., 13 (2005), 469-490.
doi: 10.3934/dcds.2005.13.469. |
[5] |
B. Bollobás, "Linear Analysis. An Introductory Course," Second edition, Cambridge University Press, Cambridge, 1999. |
[6] |
J. Buzzi, Absolutely continuous S.R.B. measures for random Lasota-Yorke maps, Trans. Amer. Math. Soc., 352 (2000), 3289-3303.
doi: 10.1090/S0002-9947-00-02607-6. |
[7] |
M. Dellnitz, G. Froyland, C. Horenkamp, K. Padberg-Gehle and A. Sen Gupta, Seasonal variability of the subpolar gyres in the southern ocean: A numerical investigation based on transfer operators, Nonlinear Processes in Geophysics, 16 (2009), 655-663. |
[8] |
M. Dellnitz, O. Junge, W.S. Koon, F. Lekien, M.W. Lo, J.E. Marsden, K. Padberg, R. Preis, S.D. Ross and B. Thiere, Transport in dynamical astronomy and multibody problems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 699-727.
doi: 10.1142/S0218127405012545. |
[9] |
D. H. Fremlin, Measurable functions and almost continuous functions, Manuscripta Math., 33 (1980/81), 387-405.
doi: 10.1007/BF01798235. |
[10] |
G. Froyland, S. Lloyd and A. Quas, Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory and Dynam. Systems, 30 (2010), 729-756.
doi: 10.1017/S0143385709000339. |
[11] |
G. Froyland, S. Lloyd and N. Santitissadeekorn, Coherent sets for nonautonomous dynamical systems, Phys. D, 239 (2010), 1527-1541.
doi: 10.1016/j.physd.2010.03.009. |
[12] |
G. Froyland, K. Padberg, M.H. England and A.M. Treguier, Detection of coherent oceanic structures via transfer operators, Phys. Rev. Lett., 98 (2007). |
[13] |
F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z., 180 (1982), 119-140.
doi: 10.1007/BF01215004. |
[14] |
G. Keller, On the rate of convergence to equilibrium in one-dimensional systems, Comm. Math. Phys., 96 (1984), 181-193. |
[15] |
Y. Kifer and P.D. Liu, Random dynamics, in "Handbook of Dynamical Systems," Vol. 1B, Elsevier B. V., Amsterdam, (2006), 379-499.
doi: 10.1016/S1874-575X(06)80030-5. |
[16] |
A. Lasota and J.A. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186 (1973), 481-488. |
[17] |
Z. Lian, "Lyapunov Exponents and Invariant Manifolds for Random Dynamical Systems in a Banach Space," Ph.D thesis, Brigham Young University, 2008. |
[18] |
C. Liverani, Decay of correlations, Ann. of Math. (2), 142 (1995), 239-301.
doi: 10.2307/2118636. |
[19] |
R. Mañé, Lyapounov exponents and stable manifolds for compact transformations, in "Geometric Dynamics" (Rio de Janeiro, 1981), Lecture Notes in Math., 1007, Springer, Berlin, (1983), 522-577.
doi: 10.1007/BFb0061433. |
[20] |
T. Morita, Random iteration of one-dimensional transformations, Osaka J. Math., 22 (1985), 489-518. |
[21] |
I. Morris, The generalized Berger-Wang formula and the spectral radius of linear cocycles, J. Func. Anal., 262 (2012), 811-824.
doi: 10.1016/j.jfa.2011.09.021. |
[22] |
V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč., 19 (1968), 179-210. |
[23] |
S. Pelikan, Invariant densities for random maps of the interval, Trans. Amer. Math. Soc., 281 (1984), 813-825.
doi: 10.2307/2000087. |
[24] |
D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2), 115 (1982), 243-290.
doi: 10.2307/1971392. |
[25] |
M. Rychlik, Bounded variation and invariant measures, Studia Math., 76 (1983), 69-80. |
[26] |
Ch. Schütte, W. Huisinga and P. Deuflhard, Transfer operator approach to conformational dynamics in biomolecular systems, in "Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems," Springer, Berlin, (2001), 191-223. |
[27] |
P. Thieullen, Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 49-97. |
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