Infinitely many stochastically stable attractors (Q2722671)

Suppose that~\(f\) is a diffeomorphism of a compact smooth Riemann manifold, which has (a possibly infinite number of) local attractors, such that the union of their basins of attraction has full Lebesgue measure. On each of these local attractors~\(f\) is assumed to have a dense orbit, and each of the attractors is assumed to support a stochastically stable invariant probability measure. It is shown that then for a suitable random perturbation of~\(f\), which is a product of random i. i. d.\ diffeomorphisms from a suitable subset of a small neighbourhood of~\(f\), there exists a finite number of `physical' invariant probability measures (the number of which grows as the intensity of the perturbation decreases), which determine the time averages of the system. Furthermore, as the intensity of the perturbation tends to zero, these probability measures converge to the convex combination of the invariant measures on the attractors of~\(f\) with the weights being the values of the Lebesgue measure of the corresponding basins of attraction.