Decomposable Blaschke products of degree 2ⁿ

Abstract

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