Holomorphic self-maps of the disk intertwining two linear fractional maps

TL;DR

This paper characterizes holomorphic self-maps of the disk that intertwine two linear fractional maps, providing new insights into their structure, commutation, and iteration roots, with applications to semigroup embeddings.

Contribution

It offers a comprehensive characterization of intertwining maps and roots of linear fractional self-maps of the disk, extending known results and simplifying proofs of recent theorems.

Findings

Characterization of maps intertwining two linear fractional maps

Identification of all roots of such maps in iteration sense

Simplified proof of semigroup embedding theorem

Abstract

We characterize (in almost all cases) the holomorphic self-maps of the unit disk that intertwine two given linear fractional self-maps of the disk. The proofs are based on iteration and a careful analysis of the Denjoy-Wolff points. In particular, we characterize the maps that commute with a given linear fractional map (in the cases that are not already known) and, as an application, determine all "roots" of such maps in the sense of iteration (if any). This yields as a byproduct a short proof of a recent theorem on the embedding of a linear fractional transformation into a semigroup of holomorphic self-maps of the disk.