Atomic physical measures for non-invertible random dynamical systems
arXiv:2607.11421
The authors construct a random dynamical system on the circle, using locally invertible maps, that has an atomic stationary measure which is physical—typical random orbits converge to it—demonstrating that Hölder regularity of stationary measures does not extend to such non‑invertible systems.
Abstract
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 . Moreover, this measure is physical: for Lebesgue-almost every initial point , the Cesà ro averages of its random trajectory almost surely converge to . 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.
33 pages, 8 figures