# 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 & 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).

show all references

##### 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).
 [1] Kening Lu, Alexandra Neamţu, Björn Schmalfuss. On the Oseledets-splitting for infinite-dimensional random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1219-1242. doi: 10.3934/dcdsb.2018149 [2] Markus Böhm, Björn Schmalfuss. Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3115-3138. doi: 10.3934/dcdsb.2018303 [3] Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281 [4] J. C. Robinson. A topological time-delay embedding theorem for infinite-dimensional cocycle dynamical systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 731-741. doi: 10.3934/dcdsb.2008.9.731 [5] Brendan Weickert. Infinite-dimensional complex dynamics: A quantum random walk. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 517-524. doi: 10.3934/dcds.2001.7.517 [6] Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069 [7] Radu Ioan Boţ, Sorin-Mihai Grad. On linear vector optimization duality in infinite-dimensional spaces. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 407-415. doi: 10.3934/naco.2011.1.407 [8] Yujun Zhu. Preimage entropy for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 829-851. doi: 10.3934/dcds.2007.18.829 [9] Björn Augner, Birgit Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations & Control Theory, 2014, 3 (2) : 207-229. doi: 10.3934/eect.2014.3.207 [10] Chris Guiver, Mark R. Opmeer. Bounded real and positive real balanced truncation for infinite-dimensional systems. Mathematical Control & Related Fields, 2013, 3 (1) : 83-119. doi: 10.3934/mcrf.2013.3.83 [11] Satoshi Ito, Soon-Yi Wu, Ting-Jang Shiu, Kok Lay Teo. A numerical approach to infinite-dimensional linear programming in $L_1$ spaces. Journal of Industrial & Management Optimization, 2010, 6 (1) : 15-28. doi: 10.3934/jimo.2010.6.15 [12] Eleonora Bardelli, Andrea Carlo Giuseppe Mennucci. Probability measures on infinite-dimensional Stiefel manifolds. Journal of Geometric Mechanics, 2017, 9 (3) : 291-316. doi: 10.3934/jgm.2017012 [13] Didier Georges. Infinite-dimensional nonlinear predictive control design for open-channel hydraulic systems. Networks & Heterogeneous Media, 2009, 4 (2) : 267-285. doi: 10.3934/nhm.2009.4.267 [14] Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-26. doi: 10.3934/dcdsb.2019122 [15] Xiongping Dai, Yunping Jiang. Distance entropy of dynamical systems on noncompact-phase spaces. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 313-333. doi: 10.3934/dcds.2008.20.313 [16] Wen-Guei Hu, Song-Sun Lin. On spatial entropy of multi-dimensional symbolic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3705-3717. doi: 10.3934/dcds.2016.36.3705 [17] María J. Garrido-Atienza, Oleksiy V. Kapustyan, José Valero. Preface to the special issue "Finite and infinite dimensional multivalued dynamical systems". Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : ⅰ-ⅳ. doi: 10.3934/dcdsb.201705i [18] Xavier Cabré, Amadeu Delshams, Marian Gidea, Chongchun Zeng. Preface of Llavefest: A broad perspective on finite and infinite dimensional dynamical systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : ⅰ-ⅲ. doi: 10.3934/dcds.201812i [19] Tapio Helin. On infinite-dimensional hierarchical probability models in statistical inverse problems. Inverse Problems & Imaging, 2009, 3 (4) : 567-597. doi: 10.3934/ipi.2009.3.567 [20] Qing Xu. Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5379-5412. doi: 10.3934/dcds.2015.35.5379

2018 Impact Factor: 1.143