Schröder Equation in Several Variables and Composition Operator

Abstract

Let $\varphi$ be a holomorphic self-map of the open unit ball $\B$ of $\C^n$ such that $\varphi(0)=0$ and that the differential $d \varphi_{0}$ of $\varphi$ at $0$ is non singular. The study of the Schr\"oder equation in several complex variables

is naturally related to the theory of composition operators on Hardy spaces of holomorphic maps on $\B$ and to the theory of discrete, complex dynamical systems. An extensive use of the infinite matrix which represents the composition operator associated to the map $\varphi$ leads to a simpler approach, and provides new proofs, to results of existence of solutions for the Schr\"oder equation.