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A CAT(0)-approach to the marked length spectral rigidity of Sinai billiards

Douglas Finamore, Martin Leguil

TL;DR

The paper develops a CAT(0)-inspired framework to address marked length spectral rigidity for Sinai billiards with finite horizon by embedding the billiard table into a Kourganoff surface family and passing to a CAT(0) limit. It introduces an enriched marked length spectrum ${\mathcal{EL}}_{\mathcal{D}}$ as a limit of spectra from approximating geodesic flows, and proves that equality of enriched spectra for two tables forces an isometry between the tables. The method extends Otal’s rigidity ideas to a non-smooth, CAT(0) setting by leveraging metric gluing, coarse geodesic geometry, and a Liouville current on the boundary at infinity, culminating in angle-preserving conjugacies and an isometry between the domains. This work broadens marked length spectral rigidity beyond smooth Riemannian settings to discontinuous dynamical systems, showing that geometric information is rigidly encoded in spectral data even under discontinuities.

Abstract

We study the spectral rigidity problem for Sinai billiards with finite horizon, specifically asking whether the geometry of the billiard table can be recovered from the lengths of its (marked) periodic trajectories. To address this, we introduce an enriched marked length spectrum EL and prove that two Sinai billiards sharing the same EL must be isometric. Our approach involves approximating the billiard flow using geodesic flows on smooth Riemannian surfaces. In the limit, these flows converge to CAT(0) spaces, which encode both the lengths of periodic orbits and the geometry of the boundary. We adapt Otal's original method -- developed for marked length spectrum rigidity in negatively curved surfaces -- to this new setting. Here, the lack of curvature control is offset by metric comparison estimates. By integrating the analysis of geodesic flows with perturbative techniques for periodic orbits, we establish a rigidity theorem for Sinai billiards with finite horizon. These results extend the classical theory of marked length spectrum rigidity beyond the Riemannian setting, demonstrating that even in discontinuous dynamical systems, geometric information is rigidly encoded in spectral data.

A CAT(0)-approach to the marked length spectral rigidity of Sinai billiards

TL;DR

The paper develops a CAT(0)-inspired framework to address marked length spectral rigidity for Sinai billiards with finite horizon by embedding the billiard table into a Kourganoff surface family and passing to a CAT(0) limit. It introduces an enriched marked length spectrum as a limit of spectra from approximating geodesic flows, and proves that equality of enriched spectra for two tables forces an isometry between the tables. The method extends Otal’s rigidity ideas to a non-smooth, CAT(0) setting by leveraging metric gluing, coarse geodesic geometry, and a Liouville current on the boundary at infinity, culminating in angle-preserving conjugacies and an isometry between the domains. This work broadens marked length spectral rigidity beyond smooth Riemannian settings to discontinuous dynamical systems, showing that geometric information is rigidly encoded in spectral data even under discontinuities.

Abstract

We study the spectral rigidity problem for Sinai billiards with finite horizon, specifically asking whether the geometry of the billiard table can be recovered from the lengths of its (marked) periodic trajectories. To address this, we introduce an enriched marked length spectrum EL and prove that two Sinai billiards sharing the same EL must be isometric. Our approach involves approximating the billiard flow using geodesic flows on smooth Riemannian surfaces. In the limit, these flows converge to CAT(0) spaces, which encode both the lengths of periodic orbits and the geometry of the boundary. We adapt Otal's original method -- developed for marked length spectrum rigidity in negatively curved surfaces -- to this new setting. Here, the lack of curvature control is offset by metric comparison estimates. By integrating the analysis of geodesic flows with perturbative techniques for periodic orbits, we establish a rigidity theorem for Sinai billiards with finite horizon. These results extend the classical theory of marked length spectrum rigidity beyond the Riemannian setting, demonstrating that even in discontinuous dynamical systems, geometric information is rigidly encoded in spectral data.
Paper Structure (21 sections, 37 theorems, 190 equations, 9 figures)

This paper contains 21 sections, 37 theorems, 190 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Liftings and the symbolic space
  4. Periodic orbits.
  5. Geodesic approximations
  6. Kourganoff surfaces
  7. Metric convergence in $\mathcal{K}(\mathcal{D})$
  8. $K_0$ as a metric gluing
  9. Coarse geometry of $(K, d_0)$
  10. The space $\mathcal{G}_0$ and the abstract geodesic flow
  11. The boundary $\partial_{\infty}\tilde{K}_{0}$
  12. The Liouville current on $\partial_{\infty}\tilde{K}_{0}$
  13. Spectral Rigidity
  14. Average angle dissipation
  15. Generic properties of periodic orbits of Sinai billiards
  16. ...and 6 more sections

Key Result

Theorem A

Let $\mathcal{D}_1,\mathcal{D}_2\subset \mathbb{T}^2$ be two Sinai billiards with finite horizon that are diffeomorphic to each other. If the billiards $\mathcal{D}_1,\mathcal{D}_2$ share the same enriched marked length spectrum, i.e., $\mathcal{EL}_{\mathcal{D}_1}=\mathcal{EL}_{\mathcal{D}_2}$, the

Figures (9)

  • Figure 1: A surface $K$ lifting the table $\mathcal{D}$ and a deformation $K_\epsilon$ (Credits: Mickaël Kourganoff, Jos Leys).
  • Figure 2: Lifted table.
  • Figure 3: A smaller obstacle eclipsing a section of the interior of a wave front.
  • Figure 4: Left: A geodesic arc on some surface $K_\epsilon$ and its projection on the billiard table $\mathcal{D}$. Right: for $\epsilon>0$ small and $(t;x,v)\in A_0$, the projection $\{F_\epsilon(s;x,v)\}_{s\in [0,t]}$ closely shadows the billiard orbit segment $\{F_0(s;x,v)\}_{s\in [0,t]}$ (Credits for pictures: Mickaël Kourganoff, Jos Leys).
  • Figure 5: Some loops $\{\sigma_i\}_{i=1}^3$ on the surface $K$, and their projections $\{\gamma_i=p(\sigma_i)\}_{i=1}^3$ on the billiard table $\mathcal{D}$.
  • ...and 4 more figures

Theorems & Definitions (91)

  • Remark 1.2
  • Remark 1.3
  • Theorem A
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • Definition 2.5
  • ...and 81 more