We consider a family of iterated function systems, each
consisting of two (symmetric)
piecewise affine increasing homeomorphisms of the unit interval, each
with exactly one point of non-differentiability.
We call them Alsedà-Misiurewicz systems (as they were introduced and
studied by Alsedà and Misiurewicz).
Under certain assumptions, such a system admits a unique stationary
probability measure with no atoms at the endpoints.
It has to be either singular or absolutely continuous with respect to
the Lebesgue measure.
Alsedà and Misiurewicz conjectured that typically such measures should
be singular.
We prove that singularity holds for a certain open set of parameters, as
well as systems satisfying some resonance conditions.
In the latter case, we calculate or bound Hausdorff dimension of the
stationary measure and its support.

This is a joint work with Krzysztof Barański.