Decomposable Blaschke products of degree 2ⁿ

Aksoy, Asuman; Arici, Francesca; Celorrio, M.; Gorkin, Pamela

doi:10.1090/tran/8937

Decomposable Blaschke products of degree $2^n$
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by Asuman Güven Aksoy, Francesca Arici, M. Eugenia Celorrio and Pamela Gorkin;

Trans. Amer. Math. Soc. 376 (2023), 6341-6369

DOI: https://doi.org/10.1090/tran/8937

Published electronically: June 13, 2023

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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, Poncelet’s 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 $2$.

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