Let Omega⊂R^2 be a bounded domain with real-analytic boundary containing the segment joining the two centres, and let B_h denote the billiard map on the fixed-energy phase space M_h, for h≥ 0. If partial Omega is not an ellipse confocal with the two centres, then every real-analytic function F colon M_h→R satisfying F circ B_h=F is constant.

Two-centre Birkhoff–Poritsky conjecture

The conjecture asserts analytic rigidity for planar two-centre billiards: in the non-confocal case, every analytic billiard invariant should be constant in the relevant non-negative-energy regime. No proposer or original date is identified in the retrieved sources.

September 2026 analytic-rigidity claim

A preprint claims that, for every non-confocal $C^1$ table and non-negative energy, one can construct trajectories with prescribed sufficiently large winding itineraries, full-shift-type invariant sets, periodic trajectories, and arbitrarily high entropy. It also claims that every analytic billiard invariant is constant for real-analytic boundaries, which would settle the stated conjecture in this regime. The claim is unverified.

Current status (as of September 2026): The planar analytic-rigidity claim for non-confocal tables at non-negative energy is reported as proved but remains unverified; broader smooth or higher-dimensional analogues remain open.

Two-centre Birkhoff–Poritsky conjecture

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