Alsed\`a-Misiurewicz systems with place-dependent probabilities

Abstract

We consider systems of two specific piecewise linear homeomorphisms of the unit interval, so called the Alsed\`a-Misiurewicz systems, and investigate the basic properties of Markov chains which arise when these two transformations are applied randomly with probabilities depending on the point of the interval. Though this iterated function system is not contracting in average and known methods do not apply, stability and the strong law of large numbers are proven.