Stability of the Denjoy-Wolff theorem

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 \(\mathbf{D}\). 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 \(n=1,2,\ldots\), where \(f_i\) and \(g_j\) are holomorphic self-maps of \(\mathbf{D}\). 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 \(f\).