Atomic physical measures for non-invertible random dynamical systems

We construct an example of a random dynamical system on the circle, formed by maps that are only locally invertible, which possesses an atomic stationary measure $\nu$. Moreover, this measure is physical: for Lebesgue-almost every initial point $x_0$, the Cesàro averages of its random trajectory almost surely converge to $\nu$. This shows that the Hölder regularity of stationary measures, known for (non-measure-preserving) random dynamical systems formed by diffeomorphisms, cannot be generalized to this class of systems. We also provide some related examples, including ones where a stationary measure charges a proper submanifold, despite the absence of a closed common invariant submanifold.