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

Yan Mary He, Homin Lee, Insung Park

TL;DR

The paper develops a complex-analytic framework for the moduli space $\\mathcal{B}_d^{fm}$ of degree-$d$ fixed-point-marked Blaschke products, introducing a holomorphic structure and proving a Bers-type simultaneous uniformization for fixed-point-marked quasi-Blaschke products. It establishes that the Weil–Petersson semi-norm $||\cdot||_{WP}$ is non-degenerate off the super-attracting locus $\\mathcal{SA}^{fm}_d$, and consequently induces a path metric $d_{WP}$ on $\\mathcal{B}_d^{fm}$; the WP-norm is closely tied to the pressure semi-norm via $||\cdot||_{WP}=\tfrac{1}{2}||\cdot||_{\\mathcal{P}}$. A key construction is a holomorphic simultaneous uniformization map $\\mathcal{U}$ linking pairs of Blaschke products to quasi-Blaschke products, realized through a mating framework and Beltrami-surgery arguments, with a diagonal identification recovering the Blaschke locus. These results extend McMullen’s WP-norm framework to quasi-Blaschke dynamics, reinforcing Sullivan’s dictionary and providing a robust holomorphic/metrics toolkit for studying degenerations and deformations in the space of Blaschke-type dynamical systems.

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 $\Bcal^{fm}_d$.

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

TL;DR

The paper develops a complex-analytic framework for the moduli space of degree- fixed-point-marked Blaschke products, introducing a holomorphic structure and proving a Bers-type simultaneous uniformization for fixed-point-marked quasi-Blaschke products. It establishes that the Weil–Petersson semi-norm is non-degenerate off the super-attracting locus , and consequently induces a path metric on ; the WP-norm is closely tied to the pressure semi-norm via . A key construction is a holomorphic simultaneous uniformization map linking pairs of Blaschke products to quasi-Blaschke products, realized through a mating framework and Beltrami-surgery arguments, with a diagonal identification recovering the Blaschke locus. These results extend McMullen’s WP-norm framework to quasi-Blaschke dynamics, reinforcing Sullivan’s dictionary and providing a robust holomorphic/metrics toolkit for studying degenerations and deformations in the space of Blaschke-type dynamical systems.

Abstract

In this paper, we study complex analytic aspects of the moduli space of degree fixed-point-marked Blaschke products. We define a complex structure on 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 , which is a codimension-1 subspace of .

Paper Structure

This paper contains 17 sections, 17 theorems, 47 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Pressure semi-norms and degenerating vectors
  3. Simultaneous uniformization of quasi-Blaschke products
  4. Summary of the proofs
  5. Preliminaries
  6. The space of fixed-point-marked quasi-Blaschke products
  7. Matings of Blaschke products
  8. McMullen's Weil--Petersson semi-norm
  9. Simultaneous uniformization of quasi-Blaschke products
  10. Simultaneous uniformization with central components of polynomials
  11. Simultaneous uniformization of quasi-Blaschke products with Blaschke products (Proof of Theorem \ref{['main theorem 1']}).
  12. Proofs of Theorems \ref{['main theorem 2']}, \ref{['main theorem 3']} and \ref{['theorem:Path metric']}
  13. Auxiliary lemmas
  14. Weil--Petersson semi-norm and almost complex structure
  15. Positivity of lengths of $C^1$-paths and Theorem \ref{['theorem:Path metric']}
  16. ...and 2 more sections

Key Result

Theorem 1

For $d\ge 2$ and $\lambda\in\Delta$, we define Consider $T\mathcal{A}_d^{fm}:=\bigcup_{\lambda\in \Delta} T\mathcal{A}^{fm}_d(\lambda)$ as a sub-bundle of $T\mathcal{B}_d^{fm}$. Then the following hold.

Figures (2)

  • Figure 1: ${\rm Mate}\colon \mathcal{B}_d^{fm}\times \mathcal{B}_d^{fm}\to \mathcal{QB}_d^{fm}$
  • Figure 2: Construction of the map $\Theta$.

Theorems & Definitions (40)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 2.1: Quasi-Blaschke products
  • Definition 2.2: Space of quasi-Blaschke products
  • Definition 2.3: Space of Blaschke products
  • Definition 2.4: Standard representatives
  • Lemma 2.5
  • proof
  • ...and 30 more