Random and deterministic perturbation of a class of skew-product systems

This paper is concerned with the stability properties of skew-products T (X, y) = (f(x), g(x, y)) in which (f, X,(mu)) is an ergodic map of a compact metric space X and g: X * R^n - R^2 is continuous. We assume that the skew-product has a negative maximal Lyapunov exponent in the fibre. We study the orbit stability and stability of mixing of T (x, y) = (f(x)y g(x, y)) under deterministic and random perturbation of g. We show that such systems are stable in the sense that for any (epsilon) > 0 there is a pairing of orbits of the perturbed and unperturbed system such that paired orbits stay within a distance 8 of each other except for a fraction (epsilon) of the time. Furthermore, we show that the invariant measure for the perturbed system is continuous (in the Hutchinson metric) as a function of the size of the perturbation to g (Lipschitz topology) and the noise distribution. Our results have applications to the stability of Iterated Function Systems which ʹcontract on averageʹ.