LAUSR

Abstract In this paper we study uniform convergence, strong convergence, weak convergence, and ergodicity of the iterates of composition operators C φ 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 C φ n do not converge in the weak operator topology and C φ is mean ergodic if φ is an elliptic automorphism, (ii) C φ n converge uniformly if the Denjoy-Wolff point of φ is in D , (iii) C φ is not mean ergodic if the Denjoy-Wolff point of φ lies on the boundary ∂ D .