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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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