Local rigidity shows that domains near ellipses must be ellipses if their billiard map is integrable, suggesting strong geometric constraints.
We show a local rigidity result for the integrability of symplectic billiards. We prove that any domain which is close to an ellipse, and for which the symplectic billiard map is rationally integrable must be an ellipse as well. This is in spirit of the result of [2] for Birkhoff billiards.
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Daniel Tsodikovich (2025) studied this question.
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