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.
