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

Filippo Bracci, Ahmed Yekta Ökten

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.

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

Paper Structure

This paper contains 4 sections, 11 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Convex wedges with no Denjoy-Wolff property
  4. Target set on convex domains with non-trivial analytic discs on the boundary

Key Result

Theorem 1.2

There exists a bounded convex domain $\Omega\subset \mathbb C^2$, such that $\partial \Omega$ is $C^\infty$-smooth, $\partial \Omega$ does not contain non-trivial analytic discs and $\Omega$ does not have the Denjoy-Wolff property.

Theorems & Definitions (25)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • ...and 15 more