Decomposable Blaschke products of degree \(2^n\)

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 \(2\).