On the failure of the Denjoy-Wolff Theorem in convex domains

2025/07/14 by Bracci, Filippo, Ökten, Ahmed Yekta · 1 citation
#30C35 #30D05 #30D40 #32F45 #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.10298

Abstract

In this note, we construct examples of bounded smooth convex domains with no non-trivial analytic discs on the boundary which possess a holomorphic self-map without fixed points so that the iterates do not converge to a point (that is, the Denjoy-Wolff theorem does not hold). We also show that, in the case of bounded convex domains with C^1+ε-smooth boundary which have non-trivial analytic discs on the boundary, the cluster set of the orbits of holomorphic self-maps without fixed points can be equal to the principal part of any prime end of any planar bounded simply connected domain.

Citations

• The Denjoy-Wolff Theorem in simply connected domains
• Unbounded visibility domains, the end compactification, and applications
• Abstract boundaries and continuous extension of biholomorphisms
• Horoballs and iteration of holomorphic maps on bounded symmetric domains
• Goldilocks domains, a weak notion of visibility, and applications
• Gromov hyperbolicity and the Kobayashi metric on convex domains of\n finite type
• Canonical Models for holomorphic iteration
• Wolff-Denjoy theorems in non-smooth convex domains
• Non-expanding maps and Busemann functions

Cited by

• Some open questions and conjectures about visibility and iteration in bounded convex domains in \mathbb C^N

Related

• The Denjoy-Wolff Theorem in simply connected domains
• Wolff-Denjoy theorems in non-smooth convex domains
• Pointwise convergence on the boundary in the Denjoy-Wolff Theorem
• A Denjoy-Wolff theorem for bounded symmetric domains
• Denjoy-Wolff points on the bidisk via models