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The metric entropy of random dynamical systems in a Banach space: Ruelle inequality

Published online by Cambridge University Press:  15 November 2012

ZHIMING LI  and
LIN SHU
ZHIMING LI
Affiliation:
School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China (email: china-lizhiming@163.com)
LIN SHU
Affiliation:
LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China (email: lshu@math.pku.edu.cn)
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Abstract

Consider a random cocycle Φ on a separable infinite-dimensional Banach space preserving a probability measure, which is supported on a random compact set. We show that if Φ satisfies some mild integrability conditions on the differentials, then Ruelle’s inequality relating entropy and positive Lyapunov exponents holds.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 2 , April 2014 , pp. 594 - 615
Copyright
©2012 Cambridge University Press 

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References

[1]Arnold, L. and Imkeller, P.. On the integrability condition in the multiplicative ergodic theorem for stochastic differential equations. Stoch. Stoch. Rep. 64(3–4) (1998), 195–210.Google Scholar
[2]Bogenschütz, T.. Entropy, pressure, and a variational principle for random dynamical systems. Random Comput. Dyn. 1 (1992/93), 99–116.Google Scholar
[3]Bogenschütz, T.. Equilibrium states for random dynamical systems. PhD Thesis, Institut für Dynamische Systeme, Universität Bremen, 1993.Google Scholar
[4]Brin, M. and Katok, A.. On local entropy. Geometric Dynamics (Rio de Janeiro, 1981) (Lecture Notes in Mathematics, 1007). Springer, Berlin, 1983, pp. 30–38.Google Scholar
[5]Crauel, H. and Flandoli, F.. Attractors for random dynamical systems. Probab. Theory Related Fields 100(3) (1994), 365–393.Google Scholar
[6]Crauel, H.. Random Probability Measures on Polish Spaces (Stochastics Monographs, 11). Taylor and Francis, London, 2002.Google Scholar
[7]Eckmann, J.-P. and Ruelle, D.. Ergodic theory of chaos and strange attractors. Rev. Modern Phys. 57(3) (1985), 617–656.Google Scholar
[8]Froyland, G., Lloyd, S. and Quas, A.. Coherent structures and isolated spectrum for Perron–Frobenius cocycles. Ergod. Th. & Dynam. Sys. 30(3) (2010), 729–756.CrossRefGoogle Scholar
[9]Froyland, G., Lloyd, S. and Santitissadeekorn, N.. Coherent sets for nonautonomous dynamical systems. Phys. D 239(16) (2010), 1527–1541.CrossRefGoogle Scholar
[10]Katok, A., Strelcyn, J.-M., Ledrappier, F. and Przytycki, F.. Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities (Lecture Notes in Mathematics, 1222). Springer, Berlin, 1986.CrossRefGoogle Scholar
[11]Ledrappier, F.. Some properties of absolutely continuous invariant measures on an interval. Ergod. Th. & Dynam. Sys. 1(1) (1981), 77–93.Google Scholar
[12]Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula. Ann. of Math. (2) 122(3) (1985), 509–539.CrossRefGoogle Scholar
[13]Ledrappier, F. and Young, L.-S.. The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension. Ann. of Math. (2) 122(3) (1985), 540–574.CrossRefGoogle Scholar
[14]Li, Z.-M. and Shu, L.. The metric entropy of random dynamical systems in a Hilbert space: characterization of measures satisfying Pesin’s entropy formula. Preprint.Google Scholar
[15]Lian, Z. and Lu, K.-N.. Lyapunov exponents and invariant manifolds for random dyanmical systems in a Banach space. Mem. Amer. Math. Soc. 206 (2010), no. 967.Google Scholar
[16]Lian, Z. and Young, L.-S.. Lyapunov exponents, periodic orbits and horseshoes for mappings of Hilbert spaces. Ann. Henri Poincaré 12 (2011), 1081–1108.CrossRefGoogle Scholar
[17]Liu, P.-D. and Qian, M.. Smooth Ergodic Theory of Random Dynamical Systems (Lecture Notes in Mathematics, 1606). Springer, Berlin, 1995.Google Scholar
[18]Liu, P.-D.. Ruelle inequality relating entropy, folding entropy and negative Lyapunov exponents. Comm. Math. Phys. 240(3) (2003), 531–538.Google Scholar
[19]Mané, R.. A proof of Pesin’s formula. Ergod. Th. & Dynam. Sys. 1(1) (1981), 95–102.Google Scholar
[20]Mané, R.. Lyapounov exponents and stable manifolds for compact transformations. Geometric Dynamics (Rio de Janeiro, 1981) (Lecture Notes in Mathematics, 1007). Springer, Berlin, 1983, pp. 522–577.CrossRefGoogle Scholar
[21]Oseledeč, V.-I.. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197–221.Google Scholar
[22]Pesin, J.-B.. Characteristic Lyapunov exponents, and smooth ergodic theory. Uspehi Mat. Nauk 32(4 (196)) (1977), 55–112, 287 (in Russian).Google Scholar
[23]Phelps, R.-R.. Lectures on Choquet’s Theorem. Van Nostrand, Princeton, NJ, 1966.Google Scholar
[24]Qian, M. and Zhang, Z.-S.. Ergodic theory for axiom A endomorphisms. Ergod. Th. & Dynam. Sys. 15 (1995), 161–174.CrossRefGoogle Scholar
[25]Qian, M., Xie, J.-S. and Zhu, S.. Smooth Ergodic Theory for Endomorphisms (Lecture Notes in Mathematics, 1978). Springer, Berlin, 2009.CrossRefGoogle Scholar
[26]Ruelle, D.. An inequality for the entropy of differentiable maps. Bol. Soc. Brasil. Mat. 9(1) (1978), 83–87.Google Scholar
[27]Ruelle, D.. Characteristic exponents and invariant manifolds in Hilbert space. Ann. of Math. (2) 115(2) (1982), 243–290.Google Scholar
[28]Ruelle, D.. Characteristic exponents for a viscous fluid subjected to time dependent forces. Comm. Math. Phys. 93(3) (1984), 285–300.Google Scholar
[29]Schaumlöffel, K. -U.. Zufällige Evolutionsoperatoren für stochastische partielle Differentialgleichungen. Dissertation, Universität Bremen, 1990.Google Scholar
[30]Schaumlöffel, K.-U. and Flandoli, F.. A multiplicative ergodic theorem with applications to a first order stochastic hyperbolic equation in a bounded domain. Stoch. Stoch. Rep. 34(3–4) (1991), 241–255.Google Scholar
[31]Shu, L.. The Hausdorff dimension of horseshoes under random perturbations. Stoch. Dyn. 9(1) (2009), 101–120.Google Scholar
[32]Shu, L.. The metric entropy of endomorphisms. Commun. Math. Phys. 291 (2009), 491–512.Google Scholar
[33]Thieullen, P.. Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension [Asymptotically compact dynamic bundles. Lyapunov exponents. Entropy. Dimension]. Ann. Inst. H. Poincaré Anal. Non Linéaire 4(1) (1987), 49–97 (in French).CrossRefGoogle Scholar
[34]Thieullen, P.. Entropy and the Hausdorff dimension for infinite-dimensional dynamical systems. J. Dynam. Differential Equations 4(1) (1992), 127–159.Google Scholar
[35]Thieullen, P.. Fibres dynamiques. Entropie et dimension [Dynamical bundles. Entropy and dimension]. Ann. Inst. H. Poincaré Anal. Non Linéaire 9(2) (1992), 119–146 (in French).Google Scholar
[36]Young, L.-S.. Chaotic phenomena in three settings: large, noisy and out of equilibrium. Nonlinearity 21(11) (2008), T245–T252.Google Scholar