Noisy attractors of Markov maps

It is shown that Markov maps when subjected to weakly continuous random perturbations have an attractive invariant measure that incorporates the dispersive effects of perturbations as well as the ordering effects of the mapping. Under any translational invariant perturbation a Markov map always has an attractive invariant measure. Interesting features of the noisy invariant measure are that it shows details much finer than the length scale settled by the noise amplitude and that the self-similar property of the unperturbed invariant measure is lost. At small noise amplitudes a degraded self-similarity is retained.