Finite Blaschke Products as Compositions of Other Finite Blaschke Products

TL;DR

This paper characterizes when a finite Blaschke product can be decomposed into a composition of two non-trivial finite Blaschke products, linking the problem to group theory and normal subgroups.

Contribution

It introduces a method to determine such factorizations using a group constructed from the Blaschke product and its local inverses, connecting to Ritt's classical work.

Findings

A group can be computed from a finite Blaschke product and its local inverses.

Compositional factorizations correspond to normal subgroups of this group.

The paper provides a criterion for non-trivial decompositions of finite Blaschke products.

Abstract

These notes answer the question "When can a finite Blaschke product $B$ be written as a composition of two finite Blaschke products $B_1$ and $B_2$, that is, $B=B_1\circ B_2$, in a non-trivial way, that is, where the order of each is greater than 1." It is shown that a group can be computed from $B$ and its local inverses, and that compositional factorizations correspond to normal subgroups of this group. This manuscript was written in 1974 but not published because it was pointed out to the author that this was primarily a reconstruction of work of Ritt from 1922 and 1923, who reported on work on polynomials. It is being made public now because of recent interest in this subject by several mathematicians interested in different aspects of the problem and interested in applying these ideas to complex analysis and operator theory.