  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.
