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A billiard table close to an ellipse is deformationally spectrally rigid among dihedrally symmetric domains

Corentin Fierobe, Vadim Kaloshin, Alfonso Sorrentino

TL;DR

The paper proves deformational spectral rigidity for billiard domains: near an ellipse, any dihedrally symmetric deformation preserving the length spectrum must be a rigid motion (translation and rotation). The authors encode isospectral deformations via deformation maps $n_{\\tau}$ and leverage Mather’s $\\beta$-function together with KAM theory to derive an isospectral operator whose invertibility forces $n_{\\tau}=0$. A key innovation is combining two data types—periodic orbits with rotation number $1/q$ and invariant Diophantine curves—to establish orthogonality and then invertibility even in nearby elliptical geometries. This provides a rigorous local rigidity result in the billiard setting, contributing to the broader understanding of inverse spectral problems and integrable billiard dynamics akin to Birkhoff-type conjectures.

Abstract

In this paper, we show that any dihedrally symmetric deformation with constant length spectrum of a domain close to an ellipse is obtained by translating and rotating the initial domain. A domain is said to be dihedrally symmetric if it is axis-symmetric and centrally symmetric. In this result, the topology set on domains' boundaries is a finitely smooth Whitney topology depending on the ellipse we are considering.

A billiard table close to an ellipse is deformationally spectrally rigid among dihedrally symmetric domains

TL;DR

The paper proves deformational spectral rigidity for billiard domains: near an ellipse, any dihedrally symmetric deformation preserving the length spectrum must be a rigid motion (translation and rotation). The authors encode isospectral deformations via deformation maps and leverage Mather’s -function together with KAM theory to derive an isospectral operator whose invertibility forces . A key innovation is combining two data types—periodic orbits with rotation number and invariant Diophantine curves—to establish orthogonality and then invertibility even in nearby elliptical geometries. This provides a rigorous local rigidity result in the billiard setting, contributing to the broader understanding of inverse spectral problems and integrable billiard dynamics akin to Birkhoff-type conjectures.

Abstract

In this paper, we show that any dihedrally symmetric deformation with constant length spectrum of a domain close to an ellipse is obtained by translating and rotating the initial domain. A domain is said to be dihedrally symmetric if it is axis-symmetric and centrally symmetric. In this result, the topology set on domains' boundaries is a finitely smooth Whitney topology depending on the ellipse we are considering.

Paper Structure

This paper contains 20 sections, 25 theorems, 127 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Global rigidity of billiards.
  3. Local rigidity of billiards.
  4. Outline of the proof
  5. Plan of the paper
  6. Acknowledgments
  7. Isospectral deformations with dihedral symmetry
  8. Mather's $\beta$-function
  9. Deformation map
  10. $\tau$-variation of Mather's $\beta$-function
  11. Isospectral orthogonality
  12. Periodic orbits of rotation number $1/q$
  13. Isospectral operator
  14. Proof of Theorem \ref{['theorem:main']}
  15. The family $\sin^{2j}$, $j\geq 0$
  16. ...and 5 more sections

Key Result

Theorem 2

Let $\mathscr E$ be an ellipse. There exists an integer $r = r(\mathscr E)>0$ and $\varepsilon=\varepsilon(\mathscr E)>0$ such that any strongly convex domain $\Omega$ with $\mathscr C^{r}$-smooth boundary having dihedral symmetry and $\varepsilon$-$\mathscr C^r$-close to $\mathscr E$ is rigid under

Figures (2)

  • Figure 1: Two successive billiard impact points in a strictly convex billiard domain $\Omega$. Here $f(s,\varphi)=(s',\varphi')$ and $L(s,s')$ measures the distance between the two impact points.
  • Figure 2: An axially symmetric billiard orbit of rotation number $\omega = 1/8$.

Theorems & Definitions (54)

  • Definition 1
  • Theorem 2
  • Remark 3
  • Definition 4
  • Definition 5
  • Theorem 6: Gole
  • proof : Sketch of proof
  • Proposition 7
  • proof
  • Definition 8
  • ...and 44 more