Abstract
<p>This paper is concerned with the stability properties of skew-products <i>T</i> (,i>x</i>,<i>y</i>) = (<i>f</i>(<i>x</i>), <i>g</i>(<i>x</i>,<i>y</i>)) in which (<i>f</i>,<i>X</i>,<i>mu</i>) is an ergodic map of a compact metric space <i>X</i> and <i>g</i>: <i>X</i> x <b>R</b><i><sup>n</i></sup> → <b>R</b><i><sup>n</i></sup> is continuous. We assume that the skew-product has a negative maximal Lyapunov exponent in the fibre. </p> <p>We study the orbit stability and stability of mixing of <i>T</i> (,i>x</i>,<i>y</i>) = (<i>f</i>(<i>x</i>), <i>g</i>(<i>x</i>,<i>y</i>)) under deterministic and random perturbation of <i>g</i>. We show that such systems are stable in the sense that for any <i>epsilon</i> > 0 there is a pairing of orbits of the perturbed and unperturbed system such that paired orbits stay within a distance <i>epsilon</i> of each other except for a fraction <i>epsilon</i> of the time. </p> <p>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'.</p>