ABSTRACT. -It has been known since the pioneering work of Jakobson and subsequent work by Benedicks and Carleson and others that a positive measure set of quadratic maps admit an absolutely continuous invariant measure.

Abstract. We study the decay of correlations for towers. Using Birkhoff's projective metrics, we obtain a rate of mixing of the form:where α(n) goes to zero in a way related to the asymptotic mass of upper floors, f is some Lipschitz norm and g 1 is some L 1 norm. The fact that the dependence on g is given by an L 1 norm is useful to study asymptotic laws of successive entrance times.

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We estimate density and regression functions for weak dependant datas. Using an exponential inequality obtained in [DeP] and in [Mau2], we control the deviation between the estimator and the function itself. These results are applied to a large class of dynamical systems and lead to estimations of invariant densities and on the mapping itself.2000 Mathematics Subject Classification. 37A50, 60E15, 37D20.