Attractors and Time Averages for Random Maps

Abstract

Considering random noise in finite dimensional parameterized families of diffeomorphisms of a compact finite dimensional boundaryless manifold M, we show the existence of time averages for almost every orbit of each point of M, imposing mild conditions on the families. Moreover these averages are given by a finite number of physical absolutely continuous stationary probability measures. We use this result to deduce that situations with infinitely many sinks and Henon-like attractors are not stable under random perturbations, e.g., Newhouse's and Colli's phenomena in the generic unfolding of a quadratic homoclinic tangency by a one-parameter family of diffeomorphisms.