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 - A, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1
References:
[1]

L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (1998).   Google Scholar

[2]

A. Ayyer and M.Stenlund, Exponential decay of correlations for randomly chosen hyperbolic toral automorphisms,, Chaos, 17 (2007).  doi: 10.1063/1.2785145.  Google Scholar

[3]

V. Baladi, "Positive Transfer Operators and Decay of Correlations,", Advances Series in Nonlinear Dynamics, 16 (2000).   Google Scholar

[4]

V. Baladi and L-S. Young, On the spectra of randomly perturbed expanding maps,, Comm. Math. Phys., 156 (1993), 355.  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.  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.  doi: 10.1214/aop/1176991694.  Google Scholar

[7]

R. Bhattacharya and M. Majumdar, "Random Dynamical Systems: Theory and Applications,", Cambridge University Press, (2007).  doi: 10.1017/CBO9780511618628.  Google Scholar

[8]

M. Boshernitzan, Quantitative recurrence results,, Invent. Math., 113 (1993), 617.  doi: 10.1007/BF01244320.  Google Scholar

[9]

P. Diaconis and D. Freedman, Iterated random functions,, SIAM Rev., 41 (1999), 45.  doi: 10.1137/S0036144598338446.  Google Scholar

[10]

K. Falconer, "Techniques in Fractal Geometry,", John Wiley & Sons Ltd., (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.   Google Scholar

[12]

Y. Kifer, "Ergodic Theory of Random Transformations,", Progress in Probability and Statistics, (1986).   Google Scholar

[13]

Y. Kifer and P.-D. Liu, Random dynamics,, Handbook of Dynamical Systems, 1B (2006), 379.  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.  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.  doi: 10.2977/prims/1195182976.  Google Scholar

[16]

M. Pollicott and M. Yuri, "Dynamical Systems and Ergodic Theory,", London Mathematical Society Student Texts, (1998).   Google Scholar

[17]

J. Rousseau and B. Saussol, Poincaré recurrence for observations,, Trans. Amer. Math. Soc., 362 (2010), 5845.  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.  doi: 10.3934/dcds.2006.15.259.  Google Scholar

[19]

M. Viana, "Stochastic Dynamics of Deterministic Systems,", Brazilian Math. Colloquium, (1997).   Google Scholar

show all references

References:
[1]

L. Arnold, "Random Dynamical Systems,", Springer Monographs in Mathematics, (1998).   Google Scholar

[2]

A. Ayyer and M.Stenlund, Exponential decay of correlations for randomly chosen hyperbolic toral automorphisms,, Chaos, 17 (2007).  doi: 10.1063/1.2785145.  Google Scholar

[3]

V. Baladi, "Positive Transfer Operators and Decay of Correlations,", Advances Series in Nonlinear Dynamics, 16 (2000).   Google Scholar

[4]

V. Baladi and L-S. Young, On the spectra of randomly perturbed expanding maps,, Comm. Math. Phys., 156 (1993), 355.  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.  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.  doi: 10.1214/aop/1176991694.  Google Scholar

[7]

R. Bhattacharya and M. Majumdar, "Random Dynamical Systems: Theory and Applications,", Cambridge University Press, (2007).  doi: 10.1017/CBO9780511618628.  Google Scholar

[8]

M. Boshernitzan, Quantitative recurrence results,, Invent. Math., 113 (1993), 617.  doi: 10.1007/BF01244320.  Google Scholar

[9]

P. Diaconis and D. Freedman, Iterated random functions,, SIAM Rev., 41 (1999), 45.  doi: 10.1137/S0036144598338446.  Google Scholar

[10]

K. Falconer, "Techniques in Fractal Geometry,", John Wiley & Sons Ltd., (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.   Google Scholar

[12]

Y. Kifer, "Ergodic Theory of Random Transformations,", Progress in Probability and Statistics, (1986).   Google Scholar

[13]

Y. Kifer and P.-D. Liu, Random dynamics,, Handbook of Dynamical Systems, 1B (2006), 379.  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.  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.  doi: 10.2977/prims/1195182976.  Google Scholar

[16]

M. Pollicott and M. Yuri, "Dynamical Systems and Ergodic Theory,", London Mathematical Society Student Texts, (1998).   Google Scholar

[17]

J. Rousseau and B. Saussol, Poincaré recurrence for observations,, Trans. Amer. Math. Soc., 362 (2010), 5845.  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.  doi: 10.3934/dcds.2006.15.259.  Google Scholar

[19]

M. Viana, "Stochastic Dynamics of Deterministic Systems,", Brazilian Math. Colloquium, (1997).   Google Scholar

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