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Real Bers embedding on the line: Fisher-Rao linearization, Schwarzian curvature, and scattering coordinates

Hy Lam

TL;DR

The paper constructs a real Bers embedding for the decay-controlled diffeomorphism group Diff^{-\,\

Abstract

We develop a real-analytic counterpart of the Bers embedding for the Fréchet Lie group $\Diff^{-\infty}(\R)$ of decay-controlled diffeomorphisms of the line, and establish its connection to $L^p$ Fisher-Rao geometry on densities. For $p\in[1,\infty)$, the $p$-root map $\varphi\mapsto p(\varphi'^{1/p}-1)$ isometrically linearizes the homogeneous $\dot W^{1,p}$ Finsler metric on $\Diff^{-\infty}(\R)$, yielding explicit geodesics and a canonical flat connection whose Eulerian geodesic equation is the generalized Hunter-Saxton equation; for $p=\infty$, logarithmic coordinates $\varphi\mapsto\log\varphi'$ provide a global isometry and the Schwarzian derivative emerges as the projective curvature. We construct a real Bers map $β^{-\infty}\colon\Diff^{-\infty}(\R)/\Aff(\R)\to W^{\infty,1}(\R)$ via this Schwarzian, prove it is a Fréchet-smooth injective immersion whose linearization admits a tame right inverse given by an explicit Volterra operator, and characterize its image through Sturm-Liouville spectral theory. We introduce an $L^p$-Schwarzian family $S_p$ that interpolates between affine and projective cocycles, establish full asymptotic expansions as $p\to\infty$ in Fréchet and Orlicz-Sobolev scales, and extend the Bers embedding to Orlicz diffeomorphism groups. Through the Jacobian correspondence, these structures transfer to a manifold of densities asymptotic to Lebesgue measure, where the nonlinear Eulerian transport reduces to a pointwise Riccati law and the Schwarzian becomes the score curvature governing Fisher information. The compact-manifold $L^p$ Fisher-Rao linearization of Bauer, Bruveris, Harms, and Michor is recalled as a guiding framework.

Real Bers embedding on the line: Fisher-Rao linearization, Schwarzian curvature, and scattering coordinates

TL;DR

The paper constructs a real Bers embedding for the decay-controlled diffeomorphism group Diff^{-\,\

Abstract

We develop a real-analytic counterpart of the Bers embedding for the Fréchet Lie group of decay-controlled diffeomorphisms of the line, and establish its connection to Fisher-Rao geometry on densities. For , the -root map isometrically linearizes the homogeneous Finsler metric on , yielding explicit geodesics and a canonical flat connection whose Eulerian geodesic equation is the generalized Hunter-Saxton equation; for , logarithmic coordinates provide a global isometry and the Schwarzian derivative emerges as the projective curvature. We construct a real Bers map via this Schwarzian, prove it is a Fréchet-smooth injective immersion whose linearization admits a tame right inverse given by an explicit Volterra operator, and characterize its image through Sturm-Liouville spectral theory. We introduce an -Schwarzian family that interpolates between affine and projective cocycles, establish full asymptotic expansions as in Fréchet and Orlicz-Sobolev scales, and extend the Bers embedding to Orlicz diffeomorphism groups. Through the Jacobian correspondence, these structures transfer to a manifold of densities asymptotic to Lebesgue measure, where the nonlinear Eulerian transport reduces to a pointwise Riccati law and the Schwarzian becomes the score curvature governing Fisher information. The compact-manifold Fisher-Rao linearization of Bauer, Bruveris, Harms, and Michor is recalled as a guiding framework.
Paper Structure (92 sections, 141 theorems, 783 equations)

This paper contains 92 sections, 141 theorems, 783 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Relation to prior work
  4. Organization
  5. Relation to Part II
  6. Isometric linearization and geodesic structure of the diffeomorphism group
  7. The group $\mathop{\mathrm{Diff}}\nolimits^{-\infty}(\mathbb{R})$ and the $\dot W^{1,p}$ metric
  8. Fiberwise trivialization and proto-bundle geometry
  9. Isometric embedding via $p$-root coordinates
  10. The multiplicative cocycle
  11. Distance and geodesics
  12. The canonical flat connection and a projective deformation
  13. Parallel transport
  14. Geodesic action functional and Hamiltonian formulation
  15. Asymptotics as $p\to\infty$
  16. ...and 77 more sections

Key Result

Theorem 1.1

For $p\in[1,\infty)$, the map is an isometric embedding from $(\mathop{\mathrm{Diff}}\nolimits^{-\infty}(\mathbb{R}), F_{\dot W^{1,p}})$ into $(W^{\infty,1}(\mathbb{R}), \left\|\cdot\right\|_{L^p})$. Its image is the open set $\{f\in W^{\infty,1}(\mathbb{R}) \mid f > -p\}$. For $p=\infty$, the map $\Phi_\infty(\varphi)=\log(\var

Theorems & Definitions (346)

  • Theorem 1.1: Isometric linearization, Theorem \ref{['thm_Diff_isometry']}
  • Theorem 1.2: $L^p$-Schwarzian, Propositions \ref{['prop_Lp_schwarzian']} and \ref{['prop_Sp_p_to_infty']}
  • Theorem 1.3: Real Bers embedding, Theorem \ref{['thm_real_bers_frechet']}
  • Theorem 1.4: Scattering Bers map, Theorem \ref{['thm_scattering_injective']}
  • Definition 2.1
  • Proposition 2.2: BBM2012
  • Proposition 2.3: Tangent bundle trivialization
  • proof
  • Definition 2.4
  • Definition 2.5: Jacobian density manifold
  • ...and 336 more