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.