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

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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

Abstract
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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.

Keywords: Hilbert spaces, Entropy formula, infinite-dimensional random dynamical systems and SRB property..

Mathematics Subject Classification: Primary: 37A, 37H; Secondary: 28.

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 & Continuous Dynamical Systems - A, 2013, 33 (9) : 4123-4155. doi: 10.3934/dcds.2013.33.4123

References:

[1]	L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (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. 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. doi: 10.1007/BF01193705.
[5]	H. Crauel, "Random Probability Measures on Polish Spaces,", Stochastics Monographs, 11 (2002).
[6]	J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors,, Rev. Modern Phys., 57 (1985), 617. 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. 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, 1222 (1986).
[9]	F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, (French) [Ergodic properties of Sinaï measures],, Inst. Hautes Études Sci. Publ. Math., (1984), 163.
[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. 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.
[12]	F. Ledrappier and L.-S. Young, Entropy formula for random transformations,, Probab. Theory Related Fields, 80 (1988), 217. 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 (1995).
[16]	P.-D. Liu, Pesin's entropy formula for endomorphisms,, Nagoya Math. J., 150 (1998), 197.
[17]	R. Mañé, A proof of Pesin's formula,, Ergodic Theory Dynam. Systems, 1 (1981), 95.
[18]	R. Mañé, Lyapounov exponents and stable manifolds for compact transformations,, in, 1007 (1983), 522. 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.
[20]	R.-R. Phelps, "Lectures on Choquet's Theorem,", D. Van Nostrand Co., (1966).
[21]	M. Qian and Z. S. Zhang, Ergodic theory for Axiom A endomorphisms,, Ergodic Theory Dynam. Systems, 15 (1995), 161. 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. 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. doi: 10.2307/1971392.
[24]	D. Ruelle, Characteristic exponents for a viscous fluid subjected to time dependent forces,, Comm. Math. Phys., 93 (1984), 285.
[25]	K.-U. Schaumlöffel, "Zufällige Evolutionsoperatoren Für Stochastische Partielle Differentialgleichungen,", Dissertation, (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.
[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.
[28]	P. Thieullen, Entropy and the Hausdorff dimension for infinite-dimensional dynamical systems,, J. Dynam. Differential Equations, 4 (1992), 127. 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.
[30]	P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982).

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References:

[1]	L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (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. 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. doi: 10.1007/BF01193705.
[5]	H. Crauel, "Random Probability Measures on Polish Spaces,", Stochastics Monographs, 11 (2002).
[6]	J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors,, Rev. Modern Phys., 57 (1985), 617. 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. 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, 1222 (1986).
[9]	F. Ledrappier, Propriétés ergodiques des mesures de Sinaï, (French) [Ergodic properties of Sinaï measures],, Inst. Hautes Études Sci. Publ. Math., (1984), 163.
[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. 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.
[12]	F. Ledrappier and L.-S. Young, Entropy formula for random transformations,, Probab. Theory Related Fields, 80 (1988), 217. 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 (1995).
[16]	P.-D. Liu, Pesin's entropy formula for endomorphisms,, Nagoya Math. J., 150 (1998), 197.
[17]	R. Mañé, A proof of Pesin's formula,, Ergodic Theory Dynam. Systems, 1 (1981), 95.
[18]	R. Mañé, Lyapounov exponents and stable manifolds for compact transformations,, in, 1007 (1983), 522. 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.
[20]	R.-R. Phelps, "Lectures on Choquet's Theorem,", D. Van Nostrand Co., (1966).
[21]	M. Qian and Z. S. Zhang, Ergodic theory for Axiom A endomorphisms,, Ergodic Theory Dynam. Systems, 15 (1995), 161. 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. 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. doi: 10.2307/1971392.
[24]	D. Ruelle, Characteristic exponents for a viscous fluid subjected to time dependent forces,, Comm. Math. Phys., 93 (1984), 285.
[25]	K.-U. Schaumlöffel, "Zufällige Evolutionsoperatoren Für Stochastische Partielle Differentialgleichungen,", Dissertation, (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.
[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.
[28]	P. Thieullen, Entropy and the Hausdorff dimension for infinite-dimensional dynamical systems,, J. Dynam. Differential Equations, 4 (1992), 127. 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.
[30]	P. Walters, "An Introduction to Ergodic Theory,", Graduate Texts in Mathematics, 79 (1982).

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