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Centralizers of non-elliptic univalent self-maps and the embeddability problem in the unit disc

Manuel D. Contreras, Santiago Díaz-Madrigal, Pavel Gumenyuk

Abstract

The embeddability problem is a very old and hard problem in discrete holomorphic iteration which deals with determining general conditions on a given univalent self-map $\varphi$ of the unit disc $\mathbb D$ in order to be contained in a continuous one-parameter semigroup. In this paper, we tackle this embedding problem by establishing different dichotomy results about the centralizer of $\varphi$ (i.e. the set of all univalent self-maps commuting with $\varphi$) which depend strongly on the dynamical character of $\varphi$. Our approach is, in part, based on a new technique to obtain simultaneous linearizations of two non-elliptic univalent self-maps of the unit disc, which might be interesting on their own. We also introduce and study several closed additive subsemigroups of the complex plane that collect the main features of the centralizer of $\varphi$ and which play a prominent position in those dichotomy results.

Centralizers of non-elliptic univalent self-maps and the embeddability problem in the unit disc

Abstract

The embeddability problem is a very old and hard problem in discrete holomorphic iteration which deals with determining general conditions on a given univalent self-map of the unit disc in order to be contained in a continuous one-parameter semigroup. In this paper, we tackle this embedding problem by establishing different dichotomy results about the centralizer of (i.e. the set of all univalent self-maps commuting with ) which depend strongly on the dynamical character of . Our approach is, in part, based on a new technique to obtain simultaneous linearizations of two non-elliptic univalent self-maps of the unit disc, which might be interesting on their own. We also introduce and study several closed additive subsemigroups of the complex plane that collect the main features of the centralizer of and which play a prominent position in those dichotomy results.
Paper Structure (14 sections, 26 theorems, 101 equations)

This paper contains 14 sections, 26 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Holomorphic self-maps of the unit disc
  5. Commuting holomorphic self-maps
  6. Holomorphic models for univalent self-maps
  7. One-parameter semigroups in the unit disc and embeddability problem
  8. First results on the centralizer
  9. Simultaneous solutions to Abel's equation
  10. Parabolic self-maps of zero hyperbolic step
  11. Hyperbolic self-maps
  12. Parabolic self-maps of positive hyperbolic step
  13. Examples
  14. Appendix

Key Result

Theorem 1.1

Let $\varphi\in\mathcal{U}(\mathbb{D})$ be a non-elliptic self-map.

Theorems & Definitions (83)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5: Canonicalmodel
  • Remark 2.6
  • Theorem 2.7: Cowen, see also Canonicalmodel
  • Remark 2.8
  • Definition 2.9
  • ...and 73 more