A note on the boundary dynamics of holomorphic iterated function systems

TL;DR

This paper studies how the boundary behavior of holomorphic iterated function systems on the unit disc can be extended from interior dynamics, generalizing classical theorems using new perturbation techniques.

Contribution

It introduces a sufficient condition for extending interior dynamics of holomorphic iterated function systems to the boundary, generalizing the Denjoy--Wolff Theorem.

Findings

Provides a boundary extension criterion for holomorphic iterated function systems

Generalizes the Denjoy--Wolff Theorem to systems of functions

Uses new perturbation techniques with elliptic Möbius transformations

Abstract

We consider the boundary dynamics of iterated function systems of holomorphic self-maps of the unit disc. Our main result provides a sufficient condition which guarantees that the dynamical behaviour of a left iterated function system in the interior of the unit disc can be extended to the boundary. This generalises an extension of the classical Denjoy--Wolff Theorem, due to Bourdon, Matache and Shapiro, to the setting of iterated function systems. To do so we modify estimates for the Hardy norm of composition operators and combine them with a new technique of perturbing a left iterated function system by elliptic M\"obius transformations.