The iterates of composition operators on Banach spaces of holomorphic functions

Abstract

In this paper we study uniform convergence, strong convergence, weak convergence, and ergodicity of the iterates of composition operators Cφ on various Banach spaces of holomorphic functions on the unit disk, such as Bergman spaces, Dirichlet spaces, weighted Banach spaces with sup-norm, and weighted Bloch spaces. For many spaces, the following results are proved:(i)the iterates Cφn do not converge in the weak operator topology and Cφ is mean ergodic if φ is an elliptic automorphism,(ii)Cφn converge uniformly if the Denjoy-Wolff point of φ is in D,(iii)Cφ is not mean ergodic if the Denjoy-Wolff point of φ lies on the boundary ∂D.