Stationary measures are one of the key tools in the theory of random dynamical systems, and this naturally motivates the study of their properties. It is known that for an invertible RDS with no finite invariant sets, the corresponding stationary measures cannot have atoms. There are also generalizations of this property; in particular, in a recent joint work with A. Gorodetski and G. Monakov, we have established Hölder regularity of stationary measures for a system of bi-Lipschitz homeomorphisms under very mild assumptions: absence of a common invariant measure and some positive moment condition for the Lipschitz constant. It was also further recently generalized by G. Monakov, obtaining log-Hölder regularity under even milder assumptions for the system.

A next natural direction of generalization would be to attempt to remove the invertibility assumption, considering the maps that are only locally invertible. However, it turns out that such a generalization is impossible! Namely, in a recent joint work with V. Goverse, we construct an example of a smooth random dynamical system on the circle, consisting of locally invertible maps, having no common invariant measure, and for which there is an atomic stationary measure. Moreover, this stationary measure is physical: for a Lebesgue-generic initial point, its trajectory is almost surely distributed with respect to this measure. My talk will be devoted to the presentation of this example.