Decomposable Blaschke products of degree $2^n$

TL;DR

This paper investigates the decomposability of degree $2^n$ Blaschke products into quadratic factors, exploring their geometric, algebraic, and group-theoretic properties, including connections to classical theorems and the monodromy group.

Contribution

It establishes conditions for decomposability of degree $2^n$ Blaschke products and characterizes their monodromy groups as wreath products of cyclic groups, linking geometry and algebra.

Findings

Blaschke products of degree $2^n$ with elliptical curves have at most $n$ critical values.

Decomposable Blaschke products have monodromy groups as wreath products of cyclic groups of order 2.

Elliptical range theorem relates to the factorization of Blaschke products.

Abstract

We study the decomposability of a finite Blaschke product $B$ of degree $2^n$ into $n$ degree-$2$ Blaschke products, examining the connections between Blaschke products, the elliptical range theorem, Poncelet theorem, and the monodromy group. We show that if the numerical range of the compression of the shift operator, $W(S_B)$, with $B$ a Blaschke product of degree $n$, is an ellipse then $B$ can be written as a composition of lower-degree Blaschke products that correspond to a factorization of the integer $n$. We also show that a Blaschke product of degree $2^n$ with an elliptical Blaschke curve has at most $n$ distinct critical values, and we use this to examine the monodromy group associated with a regularized Blaschke product $B$. We prove that if $B$ can be decomposed into $n$ degree-$2$ Blaschke products, then the monodromy group associated with $B$ is the wreath product of $n$ cyclic groups of order $2$. Lastly, we study the group of invariants of a Blaschke product $B$ of order $2^n$ when $B$ is a composition of $n$ Blaschke products of order $2$.