Schröder Equation in Several Variables and Composition Operator

Abstract

Let $\phi$ be a holomorphic self-map of the open unit ball $\mathbb{B}$ of ${\mathbb{C}}^n$ such that $\phi (0)=0$ and that the differential $d{\phi }_0$ of $\phi$ at $0$ is non singular. The study of the Schröder equation in several complex variables

is naturally related to the theory of composition operators on Hardy spaces of holomorphic maps on $\mathbb{B}$ 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 $\phi$ leads to a simpler approach, and provides new proofs, to results of existence of solutions for the Schröder equation.