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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

Zhiming Li 1, and  Lin Shu 2,

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.
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, 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.  Google Scholar

[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.  Google Scholar

[3]

T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems,, Random Comput. Dynam., 1 (): 99.   Google Scholar

[4]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.  Google Scholar

[5]

H. Crauel, "Random Probability Measures on Polish Spaces," Stochastics Monographs, 11, Taylor & Francis, London, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[12]

F. Ledrappier and L.-S. Young, Entropy formula for random transformations, Probab. Theory Related Fields, 80 (1988), 217-240. doi: 10.1007/BF00356103.  Google Scholar

[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., ().   Google Scholar

[14]

Z. Lian and K.-N. Lu, "Lyapunov Exponents and Invariant Manifolds for Random Dynamical Systems in a Banach Space," Memoirs of AMS, 2009. Google Scholar

[15]

P.-D. Liu and M. Qian, "Smooth Ergodic Theory of Random Dynamical Systems," Lecture Notes in Mathematics, 1606, Springer-Verlag, Berlin, 1995.  Google Scholar

[16]

P.-D. Liu, Pesin's entropy formula for endomorphisms, Nagoya Math. J., 150 (1998), 197-209.  Google Scholar

[17]

R. Mañé, A proof of Pesin's formula, Ergodic Theory Dynam. Systems, 1 (1981), 95-102.  Google Scholar

[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.  Google Scholar

[19]

V.-I. Oseledeč, A multiplicative ergodic theorem. Characteristic Lyapunov numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-221. Google Scholar

[20]

R.-R. Phelps, "Lectures on Choquet's Theorem," D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2), 115 (1982), 243-290. doi: 10.2307/1971392.  Google Scholar

[24]

D. Ruelle, Characteristic exponents for a viscous fluid subjected to time dependent forces, Comm. Math. Phys., 93 (1984), 285-300.  Google Scholar

[25]

K.-U. Schaumlöffel, "Zufällige Evolutionsoperatoren Für Stochastische Partielle Differentialgleichungen," Dissertation, Universität Bremen, 1990. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

show all references

References:
[1]

L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.  Google Scholar

[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.  Google Scholar

[3]

T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems,, Random Comput. Dynam., 1 (): 99.   Google Scholar

[4]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.  Google Scholar

[5]

H. Crauel, "Random Probability Measures on Polish Spaces," Stochastics Monographs, 11, Taylor & Francis, London, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[12]

F. Ledrappier and L.-S. Young, Entropy formula for random transformations, Probab. Theory Related Fields, 80 (1988), 217-240. doi: 10.1007/BF00356103.  Google Scholar

[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., ().   Google Scholar

[14]

Z. Lian and K.-N. Lu, "Lyapunov Exponents and Invariant Manifolds for Random Dynamical Systems in a Banach Space," Memoirs of AMS, 2009. Google Scholar

[15]

P.-D. Liu and M. Qian, "Smooth Ergodic Theory of Random Dynamical Systems," Lecture Notes in Mathematics, 1606, Springer-Verlag, Berlin, 1995.  Google Scholar

[16]

P.-D. Liu, Pesin's entropy formula for endomorphisms, Nagoya Math. J., 150 (1998), 197-209.  Google Scholar

[17]

R. Mañé, A proof of Pesin's formula, Ergodic Theory Dynam. Systems, 1 (1981), 95-102.  Google Scholar

[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.  Google Scholar

[19]

V.-I. Oseledeč, A multiplicative ergodic theorem. Characteristic Lyapunov numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-221. Google Scholar

[20]

R.-R. Phelps, "Lectures on Choquet's Theorem," D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2), 115 (1982), 243-290. doi: 10.2307/1971392.  Google Scholar

[24]

D. Ruelle, Characteristic exponents for a viscous fluid subjected to time dependent forces, Comm. Math. Phys., 93 (1984), 285-300.  Google Scholar

[25]

K.-U. Schaumlöffel, "Zufällige Evolutionsoperatoren Für Stochastische Partielle Differentialgleichungen," Dissertation, Universität Bremen, 1990. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

P. Walters, "An Introduction to Ergodic Theory," Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

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