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

Abstract

This paper concerns the stability properties of skew-products T (x y) = ( f (x) g (x y)) in which ( f X ) is an ergodic map of a compact metric space X and g : X R n ! R n is continuous. We assume that the skew-product has a negative maximal Lyapunov exponent in the bre.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 s h o w t h a t s u c h systems are stable in the sense that for any > 0 there is a pairing of orbits of the perturbed and unperturbed system such that paired orbits stay within a distance of each other except for a fraction of the time.We also show that such systems preserve ergodicity and higher order mixing properties under deterministic perturbation of g and perturbation of g by i.i.d. additive n o i s e . Furthermore we s h o w 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".

Contraction in mean and transfer operators

“…This should be compared with the results of Broomhead et al (1999) and Hadjiloucas et al (2000). Of course, Proposition 2 holds in the more general setting that we allow the weights p i to be functions, provided they are in the space B …1 † .…”

Physical measures for certain class of non-uniformly hyperbolic endomorphisms on the solid torus

“…Now, we consider the global attractor Λ of Ψ which is obtained from the previous remark. By the construction of Ψ, condition (5) of the smooth arc H given by (15) and the covering property (12) in Theorem 2.5, there exists a topological ball B ⊂ X so that…”

“…Now, we prove that the maximal fiber Lyapunov exponent of G Ψ is also negative. In fact, as a consequence of the modified Multiplicative Ergodic Theorem (see [15,Proosition 2.2]), there exists ε > 0 (which is obtained by semi-conjugacy π) so that for ν-almost every t ∈ S there exists a constant C(t) such that…”

“…< C(t)e (λ+ε)n , for all n > 0. Now we apply the approach used in the proof of [15,Thm. 3.1] to ensure that for fixed x ∈ X and a.e.…”

“…In the non-uniform case, when the fiber map is contracting on average [5,24,25] (for instance when the skew product map possesses a negative maximal Lyapunov exponent in the fibre), less is known about the stability of the dynamics under additive noise. In [15], the authors have proved that this kind of system preserves ergodicity and higher order mixing properties under deterministic perturbations of the fiber map and perturbation by i.i.d. additive noise.…”