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Elliptic non-degeneracy and spectral rigidity of classical billiards

Georgi Popov, Peter Topalov

Abstract

In this paper we show that the billiard ball map of the Liouville billiard tables of classical type on the ellipsoid is non-degenerate at the elliptic fixed point. As a corollary we obtain a spectral rigidity result.

Elliptic non-degeneracy and spectral rigidity of classical billiards

Abstract

In this paper we show that the billiard ball map of the Liouville billiard tables of classical type on the ellipsoid is non-degenerate at the elliptic fixed point. As a corollary we obtain a spectral rigidity result.
Paper Structure (9 sections, 6 theorems, 91 equations)

This paper contains 9 sections, 6 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Liouville billiard tables
  3. Liouville billiard tables on the ellipsoid
  4. Applications to the spectral rigidity
  5. Proof of Theorem \ref{['th:hessian_elliptic_point']}
  6. Non-degeneracy of the Liouville billiard tables on the ellipsoid
  7. Spectral rigidity of billiard tables
  8. Spectral rigidity of billiard tables on the ellipsoid
  9. Preceding results

Key Result

Theorem 1

For any Liouville billiard table of classical type the Hamiltonian $L$ is well defined and $C^\infty$-smooth in a neighborhood of zero in ${\mathbb R}_+$ and satisfies and

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Corollary 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Definition 1
  • ...and 2 more