Analytic Theory on the Space of Blaschke Products: Simultaneous Uniformization and Pressure Metric

TL;DR

This paper develops a complex analytic framework for the moduli space of fixed-point-marked Blaschke products, establishing a uniformization theorem and analyzing the pressure metric's properties.

Contribution

It introduces a complex structure on the moduli space and proves a simultaneous uniformization theorem for quasi-Blaschke products, advancing the understanding of their geometric and analytic properties.

Findings

Pressure semi-norms are non-degenerate outside the super-attracting locus.

Established a complex structure on the moduli space of Blaschke products.

Proved a simultaneous uniformization theorem for fixed-point-marked quasi-Blaschke products.

Abstract

In this paper, we study complex analytic aspects of the moduli space $\Bcal_d^{fm}$ of degree $d\ge2$ fixed-point-marked Blaschke products. We define a complex structure on $\Bcal_d^{fm}$ and prove the simultaneous uniformization theorem for fixed-point-marked quasi-Blaschke products. As an application, we show that the pressure semi-norms on the space of Blaschke products are non-degenerate outside of the super-attracting locus $\mathcal{SA}^{fm}_d$, which is a codimension-1 subspace of $\Bcal^{fm}_d$.