Conjugacy and Counterexample in Random Iteration

We consider counterexamples in the field of random iteration to two well-known theorems of classical complex dynamics Sullivan’s non-wandering theorem and the classification of periodic Fatou components. Random iteration which was first introduced by Fornaess and Sibony (1991) is a generalization of standard complex dynamics where instead of considering iterates of a fixed rational function, one allows the mappings to vary at each stage of the iterative process. In this setting one can produce oscillatory behaviour of a type forbidden in classical rational iteration. The technique of the proof requires us to extend the classical notion of conjugacy between dynamical systems to random iteration and we prove some basic results concerning conjugacy in this setting.