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 $\ddot{I}†$ 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 $\ddot{I}†$ (i.e. the set of all univalent self-maps commuting with $\ddot{I}†$) which depend strongly on the dynamical character of $\ddot{I}†$. 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 $\ddot{I}†$ and which play a prominent position in those dichotomy results.

Ver3: The most important changes concern our viewpoint on the result about the simultaneous linearization of commuting self-maps, Theorem 4.1. Accordingly, a part of Introduction and Section 4 have been re-organized. Ver2: an error in the list of references and a misprint in the Introduction are corrected; references [7,23] are added