The iterates of composition operators on Banach spaces of holomorphic functions

In this paper we study uniform convergence, strong convergence, weak convergence, and ergodicity of the iterates of composition operators C, 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;' do not converge in the weak operator topology and ap is mean ergodic if ep is an elliptic automorphism, (ii) C,:,2 converge uniformly if the Denjoy-Wolff point of co is in ID, (iii) Cy, is not mean ergodic if the Denjoy-Wolff point of ep lies on the boundary (C) 2020 Elsevier Inc. All rights reserved.