Length Spectrum of Periodic Rays for Billiard Flow

• Vesselin Petkov ^[1]
  1. [1] University of Bordeaux

     University of Bordeaux

     Arrondissement de Bordeaux, Francia

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 5, 2026
• Idioma: inglés
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• Resumen

  • We study for several compact strictly convex disjoint obstacles the length spectrum L formed by the lengths of all primitive periodic reflecting rays.We prove the existence of sequences { j }, {m j } with j ∈ L, m j ∈ N such that the condition (LB) related to the dynamical zeta function ηD(s) is satisfied. This condition implies the existence of lower bounds for the number of the scattering resonances for Dirichlet Laplacian.

    We construct such sequences under some separation condition for a small subset of L corresponding to lengths of the periodic rays with even reflections. Our separation condition is weaker than the assumption of exponentially separated length spectrum L. Moreover, we show that the periodic orbits in the phase space are exponentially separated.

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