Mean Ergodic Composition Operators on Banach spaces of holomorphic functions

TL;DR

This paper characterizes when composition operators induced by holomorphic functions on the unit disc are mean ergodic or uniformly mean ergodic on Banach spaces of holomorphic functions, based on the asymptotic behavior of the symbol's iterates.

Contribution

It provides new conditions linking the ergodic properties of composition operators to the asymptotic behavior of their symbols' iterates on Banach spaces of holomorphic functions.

Findings

Conditions for mean ergodicity of composition operators

Conditions for uniform mean ergodicity

Asymptotic behavior of symbols' iterates as key factor

Abstract

Given a symbol $\varphi,$ i.e., a holomorphic endomorphism of the unit disc, we consider the composition operator $C_{\varphi}(f)=f\circ\varphi$ defined on the Banach spaces of holomorphic functions $A(\mathbb{D})$ and $H^{\infty}(\mathbb{D})$. We obtain different conditions on the symbol $\varphi$ which characterize when the composition operator is mean ergodic and uniformly mean ergodic in the corresponding spaces. These conditions are related to the asymptotic behaviour of the iterates of the symbol. As an appendix, we deal with some particular case in the setting of weighted Banach spaces of holomorphic functions.