The iterated monodromy group of a rational function of degree d ≥ 2 acts naturally on the complete d-ary rooted tree of preimages of a generic point. In this talk, I will introduce the fixed-point process of iterated monodromy groups motivated by its application to study the proportion of periodic points of rational functions. A fixed-point process will be eventually constant if it’s a martingale. I will give a construction of families of rational functions that have a non-martingale fixed-point process with fixed-point proportion 0.