A semi-invertible Oseledets Theorem with applications to transfer operator cocycles

Gary Froyland; Simon Lloyd; Anthony Quas

doi:10.3934/dcds.2013.33.3835

Article Contents

2013, Volume 33, Issue 9: 3835-3860. Doi: 10.3934/dcds.2013.33.3835

This issue Previous Article Next Article On the non-homogeneous boundary value problem for Schrödinger equations

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
3.
Department of Mathematics and Statistics, University of Victoria, P.O. Box 3060 STN CSC, Victoria, B.C., V8W 3R4

Received: November 30, 2011

Revised: January 31, 2013

Published: August 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 37H15; Secondary: 37L55, 37A30.

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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 (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.