Nekhoroshev type estimates for billiard ball maps

Todor Gramchev; Georgi Popov

Annales de l'institut Fourier (1995)

  • Volume: 45, Issue: 3, page 859-895
  • ISSN: 0373-0956

Abstract

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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 δ - close to the glancing manifold remains close to the glancing manifold in an exponentially large time interval with respect to 1 / δ . 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.

How to cite

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Gramchev, Todor, and Popov, Georgi. "Nekhoroshev type estimates for billiard ball maps." Annales de l'institut Fourier 45.3 (1995): 859-895. <http://eudml.org/doc/75141>.

@article{Gramchev1995,
abstract = {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.},
author = {Gramchev, Todor, Popov, Georgi},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {3},
pages = {859-895},
publisher = {Association des Annales de l'Institut Fourier},
title = {Nekhoroshev type estimates for billiard ball maps},
url = {http://eudml.org/doc/75141},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Gramchev, Todor
AU - Popov, Georgi
TI - Nekhoroshev type estimates for billiard ball maps
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 3
SP - 859
EP - 895
AB - 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.
LA - eng
UR - http://eudml.org/doc/75141
ER -

References

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