# American Institute of Mathematical Sciences

September  2013, 33(9): 4123-4155. doi: 10.3934/dcds.2013.33.4123

## The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula

 1 School of Mathematical Sciences, Peking University, Beijing 100871, China 2 BICMR and LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China

Received  October 2010 Revised  December 2012 Published  March 2013

Consider a random cocycle $\Phi$ on a separable infinite-dimensional Hilbert space preserving a probability measure $\mu$, which is supported on a random compact set $K$. We show that if $\Phi$ is $C^2$ (over $K$) and satisfies some mild integrable conditions of the differentials, then Pesin's entropy formula holds if $\mu$ has absolutely continuous conditional measures on the unstable manifolds. The converse is also true under an additional condition on $K$ when the system has no zero Lyapunov exponent.
Citation: Zhiming Li, Lin Shu. The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123
##### References:
 [1] L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. [2] J. Bahnmüller and P.-D. Liu, Characterization of measures satisfying the Pesin entropy formula for random dynamical systems, J. Dynam. Differential Equations, 10 (1998), 425-448. doi: 10.1023/A:1022653229891. [3] T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems,, Random Comput. Dynam., 1 (): 99. [4] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. [5] H. Crauel, "Random Probability Measures on Polish Spaces," Stochastics Monographs, 11, Taylor & Francis, London, 2002. [6] J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys., 57 (1985), 617-656. doi: 10.1103/RevModPhys.57.617. [7] G. Froyland, S. Lloyd and A. Quas, Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory Dynam. Systems, 30 (2010), 729-756. doi: 10.1017/S0143385709000339. [8] A. Katok and J.-M. Strelcyn, "Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities," with the collab. of F. Ledrappier and F. Przytycki, Lecture Notes in Mathematics, 1222, Springer-Verlag, Berlin, 1986. [9] F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, (French) [Ergodic properties of Sinaï measures], Inst. Hautes Études Sci. Publ. Math., No. 59 (1984), 163-188. [10] F. Ledrappier and J.-M. Strelcyn, A proof of the estimation from below in Pesin's entropy formula, Ergodic Theory Dynam. Systems, 2 (1982), 203-219. doi: 10.1017/S0143385700001528. [11] F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math. (2), 122 (1985), 509-539; The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. of Math. (2), 122 (1985), 540-574. [12] F. Ledrappier and L.-S. Young, Entropy formula for random transformations, Probab. Theory Related Fields, 80 (1988), 217-240. doi: 10.1007/BF00356103. [13] Z.-M. Li and L. Shu, The metric entropy of random dynamical systems in a Banach space: Ruelle inequality,, to appear in Ergodic Theory Dynam. Systems., (). [14] Z. Lian and K.-N. Lu, "Lyapunov Exponents and Invariant Manifolds for Random Dynamical Systems in a Banach Space," Memoirs of AMS, 2009. [15] P.-D. Liu and M. Qian, "Smooth Ergodic Theory of Random Dynamical Systems," Lecture Notes in Mathematics, 1606, Springer-Verlag, Berlin, 1995. [16] P.-D. Liu, Pesin's entropy formula for endomorphisms, Nagoya Math. J., 150 (1998), 197-209. [17] R. Mañé, A proof of Pesin's formula, Ergodic Theory Dynam. Systems, 1 (1981), 95-102. [18] R. Mañé, Lyapounov exponents and stable manifolds for compact transformations, in "Geometric Dynamics" (Rio de Janeiro, 1981), Lecture Notes in Mathematics, 1007, Springer, Berlin, (1983), 522-577. doi: 10.1007/BFb0061433. [19] V.-I. Oseledeč, A multiplicative ergodic theorem. Characteristic Lyapunov numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-221. [20] R.-R. Phelps, "Lectures on Choquet's Theorem," D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. [21] M. Qian and Z. S. Zhang, Ergodic theory for Axiom A endomorphisms, Ergodic Theory Dynam. Systems, 15 (1995), 161-174. doi: 10.1017/S0143385700008294. [22] M. Qian and S. Zhu, SRB measures and Pesin's entropy formula for endomorphisms, Trans. Amer. Math. Soc., 354 (2002), 1453-1471. doi: 10.1090/S0002-9947-01-02792-1. [23] D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2), 115 (1982), 243-290. doi: 10.2307/1971392. [24] D. Ruelle, Characteristic exponents for a viscous fluid subjected to time dependent forces, Comm. Math. Phys., 93 (1984), 285-300. [25] K.-U. Schaumlöffel, "Zufällige Evolutionsoperatoren Für Stochastische Partielle Differentialgleichungen," Dissertation, Universität Bremen, 1990. [26] K.-U. Schaumlöffel and F. Flandoli, A multiplicative ergodic theorem with applications to a first order stochastic hyperbolic equation in a bounded domain, Stochastics Stochastics Rep., 34 (1991), 241-255. [27] P. Thieullen, Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension, (French) [Asymptotically compact dynamic bundles. Lyapunov exponents. Entropy. Dimension], Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 49-97. [28] P. Thieullen, Entropy and the Hausdorff dimension for infinite-dimensional dynamical systems, J. Dynam. Differential Equations, 4 (1992), 127-159. doi: 10.1007/BF01048158. [29] P. Thieullen, Fibres dynamiques. Entropie et dimension, (French) [Dynamical bundles. Entropy and dimension], Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 119-146. [30] P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.

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##### References:
 [1] L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. [2] J. Bahnmüller and P.-D. Liu, Characterization of measures satisfying the Pesin entropy formula for random dynamical systems, J. Dynam. Differential Equations, 10 (1998), 425-448. doi: 10.1023/A:1022653229891. [3] T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems,, Random Comput. Dynam., 1 (): 99. [4] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. [5] H. Crauel, "Random Probability Measures on Polish Spaces," Stochastics Monographs, 11, Taylor & Francis, London, 2002. [6] J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Modern Phys., 57 (1985), 617-656. doi: 10.1103/RevModPhys.57.617. [7] G. Froyland, S. Lloyd and A. Quas, Coherent structures and isolated spectrum for Perron-Frobenius cocycles, Ergodic Theory Dynam. Systems, 30 (2010), 729-756. doi: 10.1017/S0143385709000339. [8] A. Katok and J.-M. Strelcyn, "Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities," with the collab. of F. Ledrappier and F. Przytycki, Lecture Notes in Mathematics, 1222, Springer-Verlag, Berlin, 1986. [9] F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, (French) [Ergodic properties of Sinaï measures], Inst. Hautes Études Sci. Publ. Math., No. 59 (1984), 163-188. [10] F. Ledrappier and J.-M. Strelcyn, A proof of the estimation from below in Pesin's entropy formula, Ergodic Theory Dynam. Systems, 2 (1982), 203-219. doi: 10.1017/S0143385700001528. [11] F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin's entropy formula, Ann. of Math. (2), 122 (1985), 509-539; The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. of Math. (2), 122 (1985), 540-574. [12] F. Ledrappier and L.-S. Young, Entropy formula for random transformations, Probab. Theory Related Fields, 80 (1988), 217-240. doi: 10.1007/BF00356103. [13] Z.-M. Li and L. Shu, The metric entropy of random dynamical systems in a Banach space: Ruelle inequality,, to appear in Ergodic Theory Dynam. Systems., (). [14] Z. Lian and K.-N. Lu, "Lyapunov Exponents and Invariant Manifolds for Random Dynamical Systems in a Banach Space," Memoirs of AMS, 2009. [15] P.-D. Liu and M. Qian, "Smooth Ergodic Theory of Random Dynamical Systems," Lecture Notes in Mathematics, 1606, Springer-Verlag, Berlin, 1995. [16] P.-D. Liu, Pesin's entropy formula for endomorphisms, Nagoya Math. J., 150 (1998), 197-209. [17] R. Mañé, A proof of Pesin's formula, Ergodic Theory Dynam. Systems, 1 (1981), 95-102. [18] R. Mañé, Lyapounov exponents and stable manifolds for compact transformations, in "Geometric Dynamics" (Rio de Janeiro, 1981), Lecture Notes in Mathematics, 1007, Springer, Berlin, (1983), 522-577. doi: 10.1007/BFb0061433. [19] V.-I. Oseledeč, A multiplicative ergodic theorem. Characteristic Lyapunov numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-221. [20] R.-R. Phelps, "Lectures on Choquet's Theorem," D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. [21] M. Qian and Z. S. Zhang, Ergodic theory for Axiom A endomorphisms, Ergodic Theory Dynam. Systems, 15 (1995), 161-174. doi: 10.1017/S0143385700008294. [22] M. Qian and S. Zhu, SRB measures and Pesin's entropy formula for endomorphisms, Trans. Amer. Math. Soc., 354 (2002), 1453-1471. doi: 10.1090/S0002-9947-01-02792-1. [23] D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2), 115 (1982), 243-290. doi: 10.2307/1971392. [24] D. Ruelle, Characteristic exponents for a viscous fluid subjected to time dependent forces, Comm. Math. Phys., 93 (1984), 285-300. [25] K.-U. Schaumlöffel, "Zufällige Evolutionsoperatoren Für Stochastische Partielle Differentialgleichungen," Dissertation, Universität Bremen, 1990. [26] K.-U. Schaumlöffel and F. Flandoli, A multiplicative ergodic theorem with applications to a first order stochastic hyperbolic equation in a bounded domain, Stochastics Stochastics Rep., 34 (1991), 241-255. [27] P. Thieullen, Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension, (French) [Asymptotically compact dynamic bundles. Lyapunov exponents. Entropy. Dimension], Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 49-97. [28] P. Thieullen, Entropy and the Hausdorff dimension for infinite-dimensional dynamical systems, J. Dynam. Differential Equations, 4 (1992), 127-159. doi: 10.1007/BF01048158. [29] P. Thieullen, Fibres dynamiques. Entropie et dimension, (French) [Dynamical bundles. Entropy and dimension], Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 119-146. [30] P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.
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