Conjugacy and counterexample in random iteration.
This article is concerned with the structure of the Fatou set for random iteration of polynomials. ``Random iteration of polynomials'' is the composition of a sequence of polynomials which are not necessarily equal to each other, but which usually satisfy some uniform bounds. The Fatou and Julia sets for random iteration are defined similarly as for ordinary iteration. A fundamental theorem of Sullivan states that for every iterated rational map, every Fatou component is eventually periodic. As a consequence, every local limit function on the Fatou set must be either constant or univalent, and the diameters of the iterates of any Fatou component must be eventually periodic. In this article, counterexamples to both statements are given for random iteration of polynomials: there is a Fatou component on which there are both constant and nonconstant limit functions, and the diameters of the iterates of some Fatou component have positive lim sup but zero lim inf.
scientific article; zbMATH DE number 2101100