Special Colloquium
On the failure of the Denjoy-Wolff Theorem
Speaker: Filippo Bracci (Università degli Studi di Firenze)Date: Friday, 09 October 2026 - 12:00Place: Aula Naranja, ICMAT
Abstract:
The classical Denjoy-Wolff theorem states that if f is a holomorphic self-map of the unit disc without fixed points, then the iterates of f converge uniformly on compacta to a unique point on the boundary. This theorem has been generalised in various contexts, including metric spaces that are Gromov hyperbolic (provided the iterates of the map are compactly divergent). Consequently, it holds in any “Gromov model domain” in C^n such as bounded convex domains that are Gromov hyperbolic with respect to the Kobayashi distance.
One feature of Gromov hyperbolicity is the concept of “visibility,” which roughly means that the geodesics “bend inside.” Bharali and Maitra proved that visibility (for the Euclidean boundary of a bounded domain) is sufficient to obtain the Denjoy-Wolff theorem. However, in a recent paper with A. M. Benini, we demonstrated that visibility in general is not necessary for the Denjoy-Wolff theorem to hold. This result follows from a characterisation we obtain of planar simply connected domains for which the Denjoy-Wolff theorem holds.
Despite this, I had a conjecture that for bounded convex domains, visibility and the Denjoy-Wolff theorem are equivalent. The challenge in proving or disproving this conjecture lies in constructing holomorphic self-maps for which the Denjoy-Wolff theorem does not hold—the only easy case being domains that are products. To address this, in a recent paper with Y. Ökten, we constructed examples of non-visible smooth bounded convex domains in C^n without non-trivial analytic discs on their boundaries that fail to satisfy the Denjoy-Wolff theorem.
Furthermore, our construction allows us to show that if D is a smooth bounded convex domain with a non-trivial analytic disc on its boundary, and if K is a compact set in the complex plane that can be realised as the principal part of a prime end of a simply connected domain, then there exists a holomorphic self-map of D such that the omega-limit of its iterates is (up to homotheties) K.
The aim of this talk is to explain these previous results and raise some open questions.
The talk is based on the papers:
- A. M. Benini, F. Bracci, The Denjoy-Wolff Theorem in simply connected domains, Trans. Amer. Math. Soc., 2026
- F. Bracci, A. Y. Ökten, On the failure of the Denjoy-Wolff theorem in convex domains, Proc. Amer. Math. Soc., 2026
