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Almost sure rates of mixing for i.i.d. unimodal maps. (English) Zbl 1037.37003
Ann. Sci. Éc. Norm. Supér. (4) 35, No. 1, 77-126 (2002); corrigendum ibid. 36, No. 2, 319-322 (2003).
The authors study ergodic properties of independent identically distributed (i.i.d.) perturbations of “good” (satisfying certain axioms) unimodal interval maps. One of the examples of such good unimodal interval maps is the quadratic family \(x\to a-x^2\) for so-called Collet-Eckman (or Benedicks-Carleson) values of the parameter \(a\). It is known (M. V. Jakobson, M. Benedics and L. Carleson, and others) that a positive measure set of quadratic interval maps admit an absolutely continuous invariant measure and that its correlation functions have an exponential decay. In the works of M. Benedicks and L.-S. Young [Ergodic Theory Dyn. Syst. 12, 13–37 (1992; Zbl 0769.58051)], and V. Baladi and M. Viana [Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, 483–517 (1996; Zbl 0868.58051)], stability of the density and exponential rate of decay of the Markov chain associated to i.i.d. small perturbations were studied.
In the paper, adapting to random systems, (a) partitions associated to hyperbolic times due to J. F. Alves [Ann. Sci. Éc. Norm. Supér., IV. Sér. 33, 1–32 (2000; Zbl 0955.37012)] and (b) a probabilistic coupling method introduced by L.-S. Young [Isr. J. Math. 110, 153–188 (1999; Zbl 0983.37005)] to study rates of mixing, stretched exponential upper bounds for the almost sure rates of mixing are obtained. Apparently, this is the first time that estimates have been obtained for the almost sure rates of mixing in a concrete nonuniformly hyperbolic dynamical setting. Some open questions resulting from the study are posed.

MSC:
37A25 Ergodicity, mixing, rates of mixing
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37H99 Random dynamical systems
37E05 Dynamical systems involving maps of the interval
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References:
[1] Alves J.F. , SRB measures for nonhyperbolic systems with multidimensional expansion , Ann. Scient. Éc. Normale Sup. (4) 33 ( 2000 ) 1 - 32 . Numdam | MR 1743717 | Zbl 0955.37012 · Zbl 0955.37012
[2] Bahnmüller J. , Pesin’s entropy formula of expanding maps , Random Comput. Dynam. 4 ( 1996 ) 99 - 108 . MR 1402414 | Zbl 0882.58037 · Zbl 0882.58037
[3] Baladi V. , Kondah A. , Schmitt B. , Random correlations for small perturbations of expanding maps , Random Comput. Dynam. 4 ( 1996 ) 179 - 204 . MR 1402416 | Zbl 0881.58054 · Zbl 0881.58054
[4] Baladi V. , Viana M. , Strong stochastic stability and rate of mixing for unimodal maps , Ann. Scient. Éc. Normale Sup. (4) 29 ( 1996 ) 483 - 517 . Numdam | MR 1386223 | Zbl 0868.58051 · Zbl 0868.58051
[5] Baladi V. , Young L.-S. , On the spectra of randomly perturbed expanding maps , Comm. Math. Phys. 156 ( 1993 ) 355 - 385 , see also Erratum, Comm. Math. Phys. 166 (1994) 219-220. Article | MR 1233850 | Zbl 0809.60101 · Zbl 0809.60101
[6] Benedicks M. , Carleson L. , On iterations of 1 - ax 2 on ( - 1,1) , Ann. of Math. (2) 122 ( 1985 ) 1 - 25 . Zbl 0597.58016 · Zbl 0597.58016
[7] Benedicks M. , Carleson L. , The dynamics of the Hénon map , Ann. of Math. (2) 133 ( 1991 ) 73 - 169 . MR 1087346 | Zbl 0724.58042 · Zbl 0724.58042
[8] Benedicks M. , Young L.-S. , Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps , Erg. Theory Dyn. Syst. 12 ( 1992 ) 13 - 37 . MR 1162396 | Zbl 0769.58051 · Zbl 0769.58051
[9] Blokh A.M. , Lyubich M.Yu. , Measurable dynamics of S-unimodal maps of the interval , Ann. Scient. Éc. Normale Sup. 24 ( 1991 ) 545 - 573 . Numdam | MR 1132757 | Zbl 0790.58024 · Zbl 0790.58024
[10] Bogenschütz T. , Stochastic stability of equilibrium states , Random Comput. Dynam. 4 ( 1996 ) 85 - 98 . MR 1402413 | Zbl 0881.58048 · Zbl 0881.58048
[11] Bougerol P. , Lacroix J. , Products of Random Matrices with Applications to Schrödinger Operators , Birkhäuser , Boston , 1985 . MR 886674 | Zbl 0572.60001 · Zbl 0572.60001
[12] Bruin H. , Luzzatto S. , van Strien S. , Decay of correlations in one-dimensional dynamics , Preprint, 1999. arXiv · Zbl 1039.37021
[13] Buzzi J. , SRB measures for random Lasota-Yorke maps , Trans. Amer. Math. Soc. 352 ( 2000 ) 3289 - 3304 . MR 1707698 | Zbl 0956.37021 · Zbl 0956.37021
[14] Buzzi J. , Exponential decay of correlations for random Lasota-Yorke maps , Comm. Math. Phys. 208 ( 1999 ) 25 - 54 . MR 1729876 | Zbl 0974.37038 · Zbl 0974.37038
[15] Collet P. , Ergodic properties of some unimodal mappings of the interval , Preprint 11 , Institute Mittag Leffler, 1984 (unpublished). · Zbl 0553.58019
[16] Collet P. , A remark about uniform de-correlation prefactors , Preprint, 1999.
[17] Hennion H. , Limit theorems for products of positive random matrices , Ann. Probab. 25 ( 1997 ) 1545 - 1587 . Article | MR 1487428 | Zbl 0903.60027 · Zbl 0903.60027
[18] Katok A. , Kifer Y. , Random perturbations of transformations of an interval , J. Anal. Math. 47 ( 1986 ) 193 - 237 . MR 874051 | Zbl 0616.60064 · Zbl 0616.60064
[19] Khanin K. , Kifer Y. , Thermodynamic formalism for random transformations and statistical mechanics , in: Sinai’s Moscow Seminar on Dynamical Systems , Math. Soc. Trans. Ser. 2 , 171 , American Mathematical Society , Providence, RI , 1996 , pp. 107 - 140 . MR 1359097 | Zbl 0841.58017 · Zbl 0841.58017
[20] Kifer Y. , Limit theorems for random transformations and processes in random environments , Trans. Amer. Math. Soc. 350 ( 1998 ) 1481 - 1518 . MR 1451607 | Zbl 0995.37033 · Zbl 0995.37033
[21] Ledrappier F. , Young L.-S. , Entropy formula for random transformations , Probab. Theory Related Fields 80 ( 1988 ) 217 - 240 . MR 968818 | Zbl 0638.60054 · Zbl 0638.60054
[22] Pollicott M. , Yuri M. , Dynamical Systems and Ergodic Theory , London Mathematical Society Student Texts , 40 , Cambridge University Press , Cambridge , 1998 . MR 1627681 | Zbl 0897.28009 · Zbl 0897.28009
[23] Seneta E. , Nonnegative Matrices and Markov Chains , Springer Series in Statistics , Springer-Verlag , New York, Berlin , 1981 . MR 719544 | Zbl 0471.60001 · Zbl 0471.60001
[24] Viana M. , Multidimensional nonhyperbolic attractors , Inst. Hautes Etudes Sci. Publ. Math. 85 ( 1997 ) 63 - 96 . Numdam | MR 1471866 | Zbl 1037.37016 · Zbl 1037.37016
[25] Young L.-S. , Statistical properties of systems with some hyperbolicity including certain billiards , Ann. of Math. (2) 147 ( 1998 ) 585 - 650 . MR 1637655 | Zbl 0945.37009 · Zbl 0945.37009
[26] Young L.-S. , Recurrence times and rates of mixing , Israel J. Math. 110 ( 1999 ) 153 - 188 . MR 1750438 | Zbl 0983.37005 · Zbl 0983.37005
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