# American Institute of Mathematical Sciences

April  2011, 30(1): 1-16. doi: 10.3934/dcds.2011.30.1

## Recurrence for random dynamical systems

 1 Université d'Aix-Marseille, Centre de physique théorique, UMR 6206 CNRS, Campus de Luminy, Case 907, 13288 Marseille cedex 9 and Université du Sud, Toulon-Var, France 2 Université Européenne de Bretagne, Université de Brest, Laboratoire de Mathématiques CNRS UMR 6205,6 avenue Victor le Gorgeu, CS93837, F-29238 Brest Cedex 3, France

Received  December 2009 Revised  May 2010 Published  February 2011

This paper is a first step in the study of the recurrence behaviour in random dynamical systems and randomly perturbed dynamical systems. In particular we define a concept of quenched and annealed return times for systems generated by the composition of random maps. We moreover prove that for super-polynomially mixing systems, the random recurrence rate is equal to the local dimension of the stationary measure.
Citation: Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1
##### References:
 [1] L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.  Google Scholar [2] A. Ayyer and M.Stenlund, Exponential decay of correlations for randomly chosen hyperbolic toral automorphisms, Chaos, 17 (2007), 043116. doi: 10.1063/1.2785145.  Google Scholar [3] V. Baladi, "Positive Transfer Operators and Decay of Correlations," Advances Series in Nonlinear Dynamics, 16, World Scientific Publishing Co. Inc., River Edge, NJ, 2000.  Google Scholar [4] V. Baladi and L-S. Young, On the spectra of randomly perturbed expanding maps, Comm. Math. Phys., 156 (1993), 355-385. doi: 10.1007/BF02098487.  Google Scholar [5] L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence, Comm. Math. Phys, 219 (2001), 443-463. doi: 10.1007/s002200100427.  Google Scholar [6] R. Bhattacharya and O. Lee, Asymptotics of a class of Markov processes which are not in general irreducible, Ann. Probab., 16 (1988), 1333-1347. doi: 10.1214/aop/1176991694.  Google Scholar [7] R. Bhattacharya and M. Majumdar, "Random Dynamical Systems: Theory and Applications," Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511618628.  Google Scholar [8] M. Boshernitzan, Quantitative recurrence results, Invent. Math., 113 (1993), 617-631. doi: 10.1007/BF01244320.  Google Scholar [9] P. Diaconis and D. Freedman, Iterated random functions, SIAM Rev., 41 (1999), 45-76. doi: 10.1137/S0036144598338446.  Google Scholar [10] K. Falconer, "Techniques in Fractal Geometry," John Wiley & Sons Ltd., Chichester, 1997.  Google Scholar [11] G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141-152.  Google Scholar [12] Y. Kifer, "Ergodic Theory of Random Transformations," Progress in Probability and Statistics, 10, Birkhäuser Boston Inc., Boston, MA, 1986.  Google Scholar [13] Y. Kifer and P.-D. Liu, Random dynamics, Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, (2006), 379-499. doi: 10.1016/S1874-575X(06)80030-5.  Google Scholar [14] P.-D. Liu, Dynamics of random transformations: Smooth ergodic theory, Ergodic Theory Dynam. Systems, 21 (2001), 1279-1319. doi: 10.1017/S0143385701001614.  Google Scholar [15] T. Ohno, Asymptotic behaviours of dynamical systems with random parameters, Publ. Res. Inst. Math. Sci, 19 (1983), 83-98. doi: 10.2977/prims/1195182976.  Google Scholar [16] M. Pollicott and M. Yuri, "Dynamical Systems and Ergodic Theory," London Mathematical Society Student Texts, 40, Cambridge University Press, Cambridge, 1998.  Google Scholar [17] J. Rousseau and B. Saussol, Poincaré recurrence for observations, Trans. Amer. Math. Soc., 362 (2010), 5845-5859. doi: 10.1090/S0002-9947-2010-05078-0.  Google Scholar [18] B. Saussol, Recurrence rate in rapidly mixing dynamical systems, Discrete Contin. Dyn. Syst., 15 (2006), 259-267. doi: 10.3934/dcds.2006.15.259.  Google Scholar [19] M. Viana, "Stochastic Dynamics of Deterministic Systems," Brazilian Math. Colloquium, IMPA, 1997. Google Scholar

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##### References:
 [1] L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.  Google Scholar [2] A. Ayyer and M.Stenlund, Exponential decay of correlations for randomly chosen hyperbolic toral automorphisms, Chaos, 17 (2007), 043116. doi: 10.1063/1.2785145.  Google Scholar [3] V. Baladi, "Positive Transfer Operators and Decay of Correlations," Advances Series in Nonlinear Dynamics, 16, World Scientific Publishing Co. Inc., River Edge, NJ, 2000.  Google Scholar [4] V. Baladi and L-S. Young, On the spectra of randomly perturbed expanding maps, Comm. Math. Phys., 156 (1993), 355-385. doi: 10.1007/BF02098487.  Google Scholar [5] L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence, Comm. Math. Phys, 219 (2001), 443-463. doi: 10.1007/s002200100427.  Google Scholar [6] R. Bhattacharya and O. Lee, Asymptotics of a class of Markov processes which are not in general irreducible, Ann. Probab., 16 (1988), 1333-1347. doi: 10.1214/aop/1176991694.  Google Scholar [7] R. Bhattacharya and M. Majumdar, "Random Dynamical Systems: Theory and Applications," Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511618628.  Google Scholar [8] M. Boshernitzan, Quantitative recurrence results, Invent. Math., 113 (1993), 617-631. doi: 10.1007/BF01244320.  Google Scholar [9] P. Diaconis and D. Freedman, Iterated random functions, SIAM Rev., 41 (1999), 45-76. doi: 10.1137/S0036144598338446.  Google Scholar [10] K. Falconer, "Techniques in Fractal Geometry," John Wiley & Sons Ltd., Chichester, 1997.  Google Scholar [11] G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141-152.  Google Scholar [12] Y. Kifer, "Ergodic Theory of Random Transformations," Progress in Probability and Statistics, 10, Birkhäuser Boston Inc., Boston, MA, 1986.  Google Scholar [13] Y. Kifer and P.-D. Liu, Random dynamics, Handbook of Dynamical Systems, Vol. 1B, Elsevier B. V., Amsterdam, (2006), 379-499. doi: 10.1016/S1874-575X(06)80030-5.  Google Scholar [14] P.-D. Liu, Dynamics of random transformations: Smooth ergodic theory, Ergodic Theory Dynam. Systems, 21 (2001), 1279-1319. doi: 10.1017/S0143385701001614.  Google Scholar [15] T. Ohno, Asymptotic behaviours of dynamical systems with random parameters, Publ. Res. Inst. Math. Sci, 19 (1983), 83-98. doi: 10.2977/prims/1195182976.  Google Scholar [16] M. Pollicott and M. Yuri, "Dynamical Systems and Ergodic Theory," London Mathematical Society Student Texts, 40, Cambridge University Press, Cambridge, 1998.  Google Scholar [17] J. Rousseau and B. Saussol, Poincaré recurrence for observations, Trans. Amer. Math. Soc., 362 (2010), 5845-5859. doi: 10.1090/S0002-9947-2010-05078-0.  Google Scholar [18] B. Saussol, Recurrence rate in rapidly mixing dynamical systems, Discrete Contin. Dyn. Syst., 15 (2006), 259-267. doi: 10.3934/dcds.2006.15.259.  Google Scholar [19] M. Viana, "Stochastic Dynamics of Deterministic Systems," Brazilian Math. Colloquium, IMPA, 1997. Google Scholar
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