Stability of the Denjoy-Wolff Theorem

TL;DR

This paper investigates the stability of the Denjoy-Wolff theorem in complex dynamics by analyzing nonautonomous systems of holomorphic self-maps of the unit disc and establishing conditions under which their behavior aligns with the classical theorem.

Contribution

The paper provides a comprehensive analysis of nonautonomous holomorphic dynamical systems and identifies conditions for their behavior to mirror that of autonomous systems governed by the Denjoy-Wolff theorem.

Findings

Conditions for stability of the Denjoy-Wolff theorem under perturbations

Characterization of nonautonomous systems converging to autonomous dynamics

Insights into the dynamical behavior of perturbed holomorphic maps

Abstract

The Denjoy-Wolff theorem is a foundational result in complex dynamics, which describes the dynamical behaviour of the sequence of iterates of a holomorphic self-map $f$ of the unit disc $\mathbb{D}$. Far less well understood are nonautonomous dynamical systems $F_n=f_n\circ f_{n-1} \circ \dots \circ f_1$ and $G_n=g_1\circ g_{2} \circ \dots \circ g_n$, for $n=1,2,\dotsc$, where $f_i$ and $g_j$ are holomorphic self-maps of $\mathbb{D}$. Here we obtain a thorough understanding of such systems $(F_n)$ and $(G_n)$ under the assumptions that $f_n\to f$ and $g_n\to f$. We determine when the dynamics of $(F_n)$ and $(G_n)$ mirror that of $(f^n)$, as specified by the Denjoy-Wolff theorem, thereby providing insight into the stability of the Denjoy-Wolff theorem under perturbations of the map $f$.