On the dimension of stationary measures for random piecewise affine interval homeomorphisms

Abstract: We study stationary measures for iterated function systems (considered as random dynamical systems) consisting of two piecewise affine interval homeomorphisms, called Alsed`a--Misiurewicz (AM) systems. We prove that for an open set of parameters, the unique non-atomic stationary measure for an AM-system has Hausdorff dimension strictly smaller than $1$. In particular, we obtain singularity of these measures, answering partially a question of Alsed`a and Misiurewicz from 2014.