We consider C 2 families t → ft of C 4 nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure µt of ft, as a function of t on the set of Collet-Eckmann (CE) parameters:Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set of CE parameters ∆, and, for each t 0 ∈ ∆, a set ∆ 0 ⊂ ∆ of polynomially recurrent parameters containing t 0 as a Lebesgue density point, and constants C ≥ 1, Γ > 4, so that, for every 1/2-Hölder function A,In addition, for all t ∈ ∆ 0 , the renormalisation period Pt of ft satisfies Pt ≤ Pt 0 , and there are uniform bounds on the rates of mixing of f Pt t for all t with Pt = Pt 0 . If ft(x) = tx(1 − x), the set ∆ contains almost all CE parameters.Lower bounds: Assuming existence of a transversal mixing Misiurewicz-Thurston parameter t 0 , we find a set of CE parameters ∆ ′ M T accumulating at t 0 , a constant C ≥ 1, and a C ∞ function A 0 , so that C|t − t 0 | 1/2 ≥ A 0 dµt − A 0 dµt 0 ≥ C −1 |t − t 0 | 1/2 , ∀t ∈ ∆ ′

Let I ⊂ R be an interval and Ta : [0, 1] → [0, 1], a ∈ I, a one-parameter family of piecewise expanding maps such that for each a ∈ I the map Ta admits a unique absolutely continuous invariant probability measure µa. We establish sufficient conditions on such a one-parameter family such that a given point x ∈ [0, 1] is typical for µa for a full Lebesgue measure set of parameters a, i.e.,for Lebesgue almost every a ∈ I. In particular, we consider C 1,1 (L)-versions of βtransformations, piecewise expanding unimodal maps, and Markov structure preserving one-parameter families. For families of piecewise expanding unimodal maps we show that the turning point is almost surely typical whenever the family is transversal.

We show that for almost every map in a transversal one-parameter family of piecewise expanding unimodal maps the Birkhoff sum of suitable observables along the forward orbit of the turning point satisfies the law of iterated logarithm. This result will follow from an almost sure invariance principle for the Birkhoff sum, as a function on the parameter space. Furthermore, we obtain a similar result for general one-parameter families of piecewise expanding maps on the interval.

We study an extension of the Viana map where the base dynamics is a discontinuous expanding map, and prove the existence of two positive Lyapunov exponents.

We study a class of skew-products of quadratic maps-also called Viana mapswhere the base dynamics is given by a high enough iteration of a Misiurewicz-Thurston quadratic map. We show that these systems admit two positive Lyapunov exponents.