Exponential decay of correlations for random Lasota-Yorke maps

We consider random piecewise smooth, piecewise invertible maps mainly on the interval but also in higher dimensions. We assume that, on the average and possibly without any stochastic uniformity: (i) the maps expand distances, (ii) do not have too many pieces, (iii) do not have too large a distortion, and (iv) are strongly mixing. We assume no Markov property. We prove that as in the classical case of the iteration of a fixed piecewise expanding map of the interval, we have exponential decay of random correlations. Our proof builds on the one given by C. Liverani for deterministic, mixing and piecewise expanding interval maps. We demand very little of the stochastic process giving the maps. In particular, if the maps are beta-transformations on [0, 1[(d), i.e., x(n+1) = B(n+1)x(n) mod Z(d) with B-n:R-d --> Rd affine, then our results apply to ail stationary and ergodic processes B-1, B-2,... which expand on the average and satisfy the mixing condition above. We remark that our setting does not imply fast decay of integrated correlations.