Random and deterministic perturbation of a class of skew-product systems
Broomhead, David S.
Nicol, Matthew J.
MetadataShow full item record
This paper is concerned with the stability properties of skew-products T (x, y) = (f(x), g(x, y)) in which (f, X, μ) is an ergodic map of a compact metric space X and g: X x ℝn → ℝn 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), g(x, y)) under deterministic and random perturbation of g. We show that such systems are stable in the sense that for any ε > 0 there is a pairing of orbits of the perturbed and unperturbed system such that paired orbits stay within a distance ε of each other except for a fraction ε 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 'contract on average'.