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

In this paper, we study complex analytic aspects of the moduli space ${\text{Bcal}}_d^{fm}$ of degree $d\ge 2$ fixed-point-marked Blaschke products. We define a complex structure on ${\text{Bcal}}_d^{fm}$ 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 ${\mathcal{S}\mathcal{A}}_d^{fm}$, which is a codimension-1 subspace of ${\text{Bcal}}_d^{fm}$.