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Pressure metrics in geometry and dynamics

Yan Mary He, Homin Lee, Insung Park

TL;DR

The article surveys pressure metrics across deformation spaces defined via thermodynamic formalism, showing that pressure forms often recover familiar geometric metrics (notably the Weil–Petersson metric on Teichmüller spaces) and extend to broader settings such as quasi-Fuchsian spaces, Anosov representations, and hyperbolic components in the moduli space of rational maps. It introduces a unified framework of pressure norms and forms by pulling back symbolically defined pressure data to geometric spaces, yielding both nondegeneracy results and explicit degeneracy phenomena. In particular, degeneracy of the pressure semi-norm on quasi-Blaschke spaces is characterized, with degeneracy occurring precisely along pure bending directions on the Blaschke boundary; the main degeneracy theorem clarifies when the semi-norm vanishes and when it remains nondegenerate. Overall, the work connects thermodynamic formalism with complex-analytic and geometric deformation theories, illuminating how dynamical and conformal structures govern metric properties in diverse moduli spaces.

Abstract

In this article, we first provide a survey of pressure metrics on various deformation spaces in geometry, topology, and dynamics. Then we discuss pressure semi-norms and their degeneracy loci in the space of quasi-Blaschke products

Pressure metrics in geometry and dynamics

TL;DR

The article surveys pressure metrics across deformation spaces defined via thermodynamic formalism, showing that pressure forms often recover familiar geometric metrics (notably the Weil–Petersson metric on Teichmüller spaces) and extend to broader settings such as quasi-Fuchsian spaces, Anosov representations, and hyperbolic components in the moduli space of rational maps. It introduces a unified framework of pressure norms and forms by pulling back symbolically defined pressure data to geometric spaces, yielding both nondegeneracy results and explicit degeneracy phenomena. In particular, degeneracy of the pressure semi-norm on quasi-Blaschke spaces is characterized, with degeneracy occurring precisely along pure bending directions on the Blaschke boundary; the main degeneracy theorem clarifies when the semi-norm vanishes and when it remains nondegenerate. Overall, the work connects thermodynamic formalism with complex-analytic and geometric deformation theories, illuminating how dynamical and conformal structures govern metric properties in diverse moduli spaces.

Abstract

In this article, we first provide a survey of pressure metrics on various deformation spaces in geometry, topology, and dynamics. Then we discuss pressure semi-norms and their degeneracy loci in the space of quasi-Blaschke products
Paper Structure (16 sections, 16 theorems, 93 equations)

This paper contains 16 sections, 16 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Pressure metrics in geometry and dynamics
  3. Thermodynamic formalism
  4. Pressure metrics on Teichmüller spaces and the space of Blaschke products
  5. Pressure metrics on Teichmüller spaces
  6. Pressure forms and semi-norms on the spaces of Blaschke products
  7. Pressure forms on quasi-Fuchsian spaces
  8. Pressure metrics on deformation spaces of Anosov representations
  9. Pressure forms on hyperbolic components in the moduli space of rational maps
  10. Symmetric bilinear forms $\langle\cdot,\cdot\rangle_G$ and semi-norms $||\cdot||_G$ on hyperbolic components in ${\rm Rat}_d$
  11. Semi-norms $||\cdot||_G$ on hyperbolic components in ${\rm rat}_d$
  12. Conformal equivalence
  13. Non-degeneracy conditions for the pressure semi-norm
  14. Degeneracy loci of the pressure semi-norms in quasi-Blaschke products spaces
  15. The space of fixedpoint-marked quasi-Blaschke products
  16. ...and 1 more sections

Key Result

Proposition 2.1

Fix a smooth path $\phi_t$ in $C^{\alpha}(\Sigma_A^+)$. Denote the equilibrium measure of $\phi_0$ by $m = m(\phi_0)$. Then we have and, if the first derivative of $\mathcal{P}(\phi_t)$at $t=0$ is zero, then

Theorems & Definitions (24)

  • Proposition 2.1: McMullen08
  • Theorem 2.2: McMullen08
  • Theorem 2.3: McMullen08
  • Theorem 2.4: McMullen08
  • Theorem 2.5: BridgemanTaylorBridgeman_WPMetricQF
  • Theorem 2.6
  • Proposition 2.7: BCLS
  • Theorem 2.8: HeNie_MetricHypComp
  • Proposition 2.9: HeNie_MetricHypComp
  • Corollary 2.10: HeNie_MetricHypComp
  • ...and 14 more