Infinitely many stochastically stable attractors

Let f be a diffeomorphism of a compact finite-dimensional boundaryless mani fold M exhibiting infinitely many coexisting attractors. Assume that each a ttractor supports a stochastically stable probability measure and that the union of the basins of attraction of each attractor covers Lebesgue almost all points of M. We prove that the time averages of almost all orbits under random perturbations are given by a finite number of probability measures. Moreover, these probability measures are close to the probability measures supported by the attractors when the perturbations are close to the origin al map f.