The Wolff-Denjoy iteration property in complex Banach spaces (Q2821996)

Let \(\mathbb{D}\) be the open unit disc in \(\mathbb{C}\) and \(f: \mathbb{D}\to\mathbb{D}\) a holomorphic map without fixed points. The classical Wolff-Denjoy theorem [\textit{J. Wolff}, C. R. 182, 42--43 and 255--257 (1926; JFM 52.0309.02)] states that then there exists a unique point \(\xi\in\partial\mathbb{D}\) such that the sequence \((f^n)_n\) of iterates of \(f\) converges, uniformly on compact subsets of \(\mathbb{D}\), to the constant function \(f(z)\equiv\xi\).NEWLINENEWLINE In this paper the authors study the following related problem: given a bounded convex domain in a complex Banach space, suppose that the sequence \((f^n(\widetilde x))_n\) of iterates of a holomorphic map \(f: D\to D\) for one point \(\widetilde x\) converges to a point \(\xi\in\partial D\). Under what hypotheses this implies the convergence of all sequences of iterates to \(\xi\)? The main tools are several geometric variants of the Wolff-Denjoy property and the Kadec-Klee property.