Schroeder equation in several variables and composition operators

Let φ be a holomorphic self-map of the open unit ball Bn of Cn such that φ(0) = 0 and the differential dφ0 of φ at 0 is non-singular. The study of the Schröder equation in several complex variables σ O φ = dφ0 O σ is naturally related to the theory of composition operators on Hardy spaces of holomorphic maps on Bn 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 φ leads to a simpler approach, and provides new proofs, to results on existence of solutions for the Schröder equation.