2019/07/22 by Christodoulou, Argyrios, Short, Ian · 1 citation
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1907.09366
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 \mathbbD. Far less well understood are nonautonomous dynamical systems Fn=fn∘ fn-1 ∘ … ∘ f1 and Gn=g1∘ g2 ∘ … ∘ gn, for n=1,2,\dotsc, where fi and gj are holomorphic self-maps of \mathbbD. Here we obtain a thorough understanding of such systems (Fn) and (Gn) under the assumptions that fn→ f and gn→ f. We determine when the dynamics of (Fn) and (Gn) mirror that of (fn), as specified by the Denjoy-Wolff theorem, thereby providing insight into the stability of the Denjoy-Wolff theorem under perturbations of the map f.