Uniform Boundedness of Iterates of Analytic Mappings Implies Linearization: a Simple Proof and Extensions

Abstract

A well-known result in complex dynamics shows that if the iterates of an analytic map are uniformly bounded in a complex domain, then the map is analytically conjugate to a linear map. We present a simple proof of this result in any dimension. We also present several generalizations and relations to other results in the literature.