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.
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Y. Amiran, E. (2004). Fitting Invariant Curves on Billiard Tables and the Birkhoff-Herman Theorem. In: Delgado, J., Lacomba, E.A., Llibre, J., Pérez-Chavela, E. (eds) New Advances in Celestial Mechanics and Hamiltonian Systems. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-9058-7_2
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