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On the lengths of the periodic trajectories of a billiard. (Sur les longueurs des trajectoires périodiques d’un billard.) (French) Zbl 0599.58039

Géométrie symplectique et de contact: autour du théorème de Poincaré-Birkhoff, Journ. lyonnaises Soc. math. France 1983, Sémin. sud-rhodanien Géom. III, 122-139 (1984).
[For the entire collection see Zbl 0528.00006.]
It is known that the singular support of the zeta-function of the Dirichlet Laplacian of a bounded domain \(\Omega\) is contained in the set \(L\) of lengths of periodic billiard trajectories in \(\Omega\) and that for generic \(\Omega\) the two sets coincide. In view of the famous problem “Can one hear the shape of a drum?” posed by Mark Kac it becomes important to study the length spectra of billiards. The author studies the asymptotics of length of periodic trajectories in \(\Omega\) which are close to stable periodic trajectory. Then he applies the results to billiards with the symmetries of an ellipse for which the small axis of symmetry is stable. He obtains the following rigidity result. If \(\Omega_ t\) is a real analytic family of billiards satisfying the conditions above and having the same length spectrum, then \(\Omega_ t\) is isometric to \(\Omega_ 0\).
Also published in Zbl 1126.37307.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
Full Text: EuDML