Stability of the Denjoy-Wolff theorem - Christodoulou, Argyrios; Short, Ian (Annales Fennici Mathematici, 2021)
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 \(D\). Far less well understood are nonautonomous dynamical systems \(F_n=f_n° f_n-1 ° … ° f_1\) and \(G_n=g_1° g_2 ° … ° g_n\), for \(n=1,2,…\), where \(f_i\) and \(g_j\) are holomorphic self-maps of \(D\). Here we obtain a thorough understanding of such systems \((F_n)\) and \((G_n)\) under the assumptions that \(f_n→ f\) and \(g_n→ f\). 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 \(f\).