Fitting Invariant Curves on Billiard Tables and the Birkhoff-Herman Theorem
A two-dimensional billiard table is geometrically integrable when the phase space is foliated by continuous invariant curves. When an integrable planar domain has a C 4 boundary with strictly positive curvature, a neighborhood of the boundary is foliated by invariant circles. This family of invariant circles can lose convexity only after developing a singularity and if it developes a singularity, the boundary contains a segment of an ellipse. An important role in this result is played by the Birkhoff-Herman thoerem which shows that differentiability of enveloped curves cannot be lost without a change in homotopy type.
Keywordsbilliard map integrable invariant curves Birkhoff-Herman theorem
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- A. M. Abdrakhmanov, Integrable Billiards, Vestnik Moskov Univ. Ser. I, Mat. Mekh. (1990), no. 6, 28–33.Google Scholar
- V. Bangent, Mather sets for twist maps and geodesics on tori, Dynamics Reported 1 (1988), 1–45.Google Scholar
- S. V. Bbolotin, Integrable Birkhoff Billiards, Vestnik Moskov Univ. Ser. I, Mat. Mekh. (1990), no. 2, 45–49.Google Scholar
- M. R. Herman, Sur les Courbes Invariant par les Difféomorphism.es de I’Anneau,. Astérisque 103-104 (1983).Google Scholar
- J. Moser, Various aspects of integrable Hamiltonian systems, Progr. Math. 8 (1980), 223–289.Google Scholar