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

TL;DR

This paper demonstrates that the classical Denjoy-Wolff theorem fails in certain smooth convex domains by constructing specific examples where iterates of holomorphic self-maps do not converge, revealing limitations of the theorem.

Contribution

It provides explicit examples of convex domains where the Denjoy-Wolff theorem does not hold, especially in smooth convex domains with no boundary analytic discs.

Findings

Existence of convex domains with no boundary analytic discs where the theorem fails

Construction of holomorphic self-maps with non-converging iterates

Cluster sets of orbits can match any prime end in certain domains

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 $C^{1+\varepsilon}$-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.