Schröder Equation in Several Variables and Composition Operator

Let $\phi$ be a holomorphic self-map of the open unit ball $B^n$ of $C^n$ such that $\phi(0)=0$ and that the differential $d\phi_0$ of $\phi$ at 0 is non singular. The study of the Schroder equation in several complex variables $\sigma \circ \phi=d\phi_0 \circ \sigma$ is naturally related to the theory of composition operators on Hardy spaces of holomorphic maps on $B^n$ and to the theory of the 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 Schroder equation