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Reconstruction of an ellipse from two β-values

Determine whether an elliptic Birkhoff billiard can be reconstructed within the class of ellipses from two values of the Mather β-function.

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Background

The β-function encodes action spectra of twist maps and has been used to deduce rigidity results for billiards, including recovery of ellipses from β on an interval.

The authors ask whether two pointwise β-values suffice to recover an ellipse within the family of ellipses.

References

In this section we formulate natural open questions related to the results discussed in previous sections. (2) Can one reconstruct an elliptic Birkhoff billiard in the class of ellipses by two values of $\beta$-function?

— Integrable Billiards and Related Topics (2510.03790 - Bialy et al., 4 Oct 2025) in Section 9 (Open questions), Subsection Mather β-function, item (2)