Finite Blaschke Products and Decomposition

TL;DR

This paper investigates the conditions under which finite Blaschke products can be decomposed into compositions of lower-degree Blaschke products, focusing on the role of automorphisms of the unit disk.

Contribution

It characterizes when a finite Blaschke product can be expressed as a composition of simpler Blaschke products, linking this to automorphisms of the unit disk.

Findings

Provides criteria for Blaschke product decomposition

Relates decomposition to automorphisms of the unit disk

Enhances understanding of Blaschke product structure

Abstract

Let $B(z)$ be a finite Blaschke product of degree $n$. We consider the problem when a finite Blaschke product can be written as a composition of two nontrivial Blaschke products of lower degree related to the condition $% B\circ M=B$ where $M$ is a M\"{o}bius transformation from the unit disk onto itself.