We consider a family of random dynamical systems, each
consisting of two piecewise affine increasing homeomorphisms $f_-, f_+$ of
the unit interval, each with exactly one point of non-differentiability,
iterated randomly according to given probability vector. Since systems of
this type were considered by Alsedà and Misiurewicz, we call them
Alsedà-Misiurewicz systems, or AM-systems. Under certain assumptions, such
a system admits a unique stationary probability measure $\mu$ with no
atoms at the endpoints. In this case, $\mu$ has to be either singular or
absolutely continuous with respect to the Lebesgue measure. We prove
singularity and calculate Hausdorff dimension of the measure $\mu$ and its
support for systems satisfying some resonance conditions. We also prove
that $\mu$ is singular for a certain open set of parameters, verifying a
conjecture by Alsedà and Misiurewicz in this case. This is joint work with
Krzysztof BaraĆski.