Fitting Invariant Curves on Billiard Tables and the Birkhoff-Herman Theorem

  • Edoh Y. Amiran
Conference paper

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.


billiard map integrable invariant curves Birkhoff-Herman theorem 


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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Edoh Y. Amiran
    • 1
  1. 1.Western Washington UniversityUSA

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