# American Institute of Mathematical Sciences

September  2013, 33(9): 3835-3860. doi: 10.3934/dcds.2013.33.3835

## 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

Received  December 2011 Revised  February 2013 Published  March 2013

Oseledets' celebrated Multiplicative Ergodic Theorem (MET) [V.I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179--210.] is concerned with the exponential growth rates of vectors under the action of a linear cocycle on $\mathbb{R}^d$. When the linear actions are invertible, the MET guarantees an almost-everywhere pointwise splitting of $\mathbb{R}^d$ into subspaces of distinct exponential growth rates (called Lyapunov exponents). When the linear actions are non-invertible, Oseledets' MET only yields the existence of a filtration of subspaces, the elements of which contain all vectors that grow no faster than exponential rates given by the Lyapunov exponents. The authors recently demonstrated [G. Froyland, S. Lloyd, and A. Quas, Coherent structures and exceptional spectrum for Perron--Frobenius cocycles, Ergodic Theory and Dynam. Systems 30 (2010), , 729--756.] that a splitting over $\mathbb{R}^d$ is guaranteed without the invertibility assumption on the linear actions. Motivated by applications of the MET to cocycles of (non-invertible) transfer operators arising from random dynamical systems, we demonstrate the existence of an Oseledets splitting for cocycles of quasi-compact non-invertible linear operators on Banach spaces.
Citation: Gary Froyland, Simon Lloyd, Anthony Quas. A semi-invertible Oseledets Theorem with applications to transfer operator cocycles. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 3835-3860. doi: 10.3934/dcds.2013.33.3835
##### 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 (): 387.  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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##### 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 (): 387.  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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