TY - CHAP

T1 - Describing Blaschke Products by Their Critical Points

AU - Ivrii, Oleg

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

PY - 2021

Y1 - 2021

N2 - In this talk, I will discuss a question which originates in complex analysis but is really a problem in non-linear elliptic PDE. A finite Blaschke product is a proper holomorphic self-map of the unit disk, just like a polynomial is a proper holomorphic self-map of the complex plane. A celebrated theorem of Heins says that up to post-composition with a Möbius transformation, a finite Blaschke product is uniquely determined by its critical points. Dyakonov suggested that it may be interesting to extend this result to infinite degree, however, one must be careful since infinite Blaschke products may have identical critical sets. I will first explain how to parametrize inner functions F of finite entropy (i.e. which have derivative in the Nevanlinna class) in terms of InnF′. The answer involves measures on the unit circle that do not charge Beurling–Carleson sets. Afterwards, I will discuss how one might parametrize arbitrary inner functions using 1-generated invariant subspaces of the weighted Bergman space A12. The proofs rely on the Liouville correspondence which provides a bridge between complex analysis and non-linear elliptic PDE.

AB - In this talk, I will discuss a question which originates in complex analysis but is really a problem in non-linear elliptic PDE. A finite Blaschke product is a proper holomorphic self-map of the unit disk, just like a polynomial is a proper holomorphic self-map of the complex plane. A celebrated theorem of Heins says that up to post-composition with a Möbius transformation, a finite Blaschke product is uniquely determined by its critical points. Dyakonov suggested that it may be interesting to extend this result to infinite degree, however, one must be careful since infinite Blaschke products may have identical critical sets. I will first explain how to parametrize inner functions F of finite entropy (i.e. which have derivative in the Nevanlinna class) in terms of InnF′. The answer involves measures on the unit circle that do not charge Beurling–Carleson sets. Afterwards, I will discuss how one might parametrize arbitrary inner functions using 1-generated invariant subspaces of the weighted Bergman space A12. The proofs rely on the Liouville correspondence which provides a bridge between complex analysis and non-linear elliptic PDE.

UR - http://www.scopus.com/inward/record.url?scp=85119696656&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-74417-5_14

DO - 10.1007/978-3-030-74417-5_14

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.chapter???

AN - SCOPUS:85119696656

T3 - Trends in Mathematics

SP - 89

EP - 98

BT - Trends in Mathematics

PB - Springer Science and Business Media Deutschland GmbH

ER -