Nekhoroshev type estimates for billiard ball maps

Gramchev, Todor; Popov, Georgi

Annales de l'Institut Fourier, Tome 45 (1995) no. 3, pp. 859-895.

Cet article est consacré aux estimations de la stabilité effective (du type de Nekhoroshev) du flot du billard pour les domaines bornés strictement convexes de dimension quelconque avec des bords analytiques. Le résultat principal est que toute trajectoire du billard, avec des conditions initiales $\delta -$ près de la variété glissante, reste près de la variété glissante dans un intervalle exponentiellement grand par rapport à $1/\delta$. La démonstration est basée sur une forme normale de l’application du billard dans les classes de Gevrey. Plus généralement, on prouve des estimations de la stabilité effective pour l’application du billard associée à une paire d’hypersurfaces glissantes analytiques dont la variété glissante est compacte.

This paper is devoted to the effective stability estimates (of Nekhoroshev’s type) of the billiard flow for strictly convex bounded domains with analytic boundaries in any dimensions. The main result is that any billiard trajectory with initial data which are $\delta$ - close to the glancing manifold remains close to the glancing manifold in an exponentially large time interval with respect to $1/\delta$. The proof is based on a normal form of the billiard ball map in Gevrey classes. More generally, we prove effective stability estimates for the billiard ball map associated with any pair of analytic glancing hypersurfaces with a compact glancing manifold.

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