Stability of the Denjoy–Wolff theorem

Christodoulou, Argyrios and Short, Ian (2021). Stability of the Denjoy–Wolff theorem. Annales Academiæ Scientiarum Fennicæ, 26 pp. 421–431.

DOI: https://doi.org/10.5186/aasfm.2021.4621

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 $f$.

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