Ergodic behaviors of composition operators acting on space of bounded holomorphic functions

TL;DR

This paper characterizes when composition operators on the space of bounded holomorphic functions are mean ergodic, showing they are so if and only if they are uniformly mean ergodic, thus providing a complete understanding of their ergodic behavior.

Contribution

It provides a complete characterization of mean ergodic composition operators on the space of bounded holomorphic functions, linking mean ergodicity to uniform mean ergodicity.

Findings

A composition operator on $H^ abla( abla)$ is mean ergodic iff it is uniformly mean ergodic.

The paper establishes a complete characterization of ergodic behaviors for these operators.

It advances understanding of the dynamics of composition operators on function spaces.

Abstract

We completely characterize the mean ergodic composition operators on $H^\infty(\mathbb{B}_n)$. In particular, we show that a composition operator acting on this space is mean ergodic if and only if it is uniformly mean ergodic.