Schröder equation in several variables and composition operators. (Q997581)

Summary: Let \(\varphi\) be a holomorphic self-map of the open unit ball \(\mathbb{B}^n\) of \(\mathbb{C}^n\) such that \(\varphi (0)=0\) and that the differential \(d \varphi_0\) of \(\varphi\) at \(0\) is non-singular. The study of the Schröder equation in several complex variables \[ \sigma \circ \varphi=d \varphi_{0} \circ \sigma \] is naturally related to the theory of composition operators on Hardy spaces of holomorphic maps on \(\mathbb{B}^n\) 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 Schröder equation.