Math Conjectures

Birkhoff billiard conjecture

OPENLandmarkConjectureProposed 1927 · Standard version

Canonical statement

Let Ω⊂R2\Omega\subset\mathbb R^2 be bounded with C∞C^\infty, strictly convex boundary. Its billiard phase space is the open annulus A={(x,v):x∈∂Ω, ∣v∣=1, v points inward and is not tangent to ∂Ω}\mathcal A=\{(x,v):x\in\partial\Omega,\ |v|=1,\ v\text{ points inward and is not tangent to }\partial\Omega\}, with the billiard map sending one reflected state to the next. If A\mathcal A is foliated by invariant circles homotopic to its boundary, then ∂Ω\partial\Omega is an ellipse.
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Let \(\Omega\subset\mathbb R^2\) be bounded with \(C^\infty\), strictly convex boundary. Its billiard phase space is the open annulus \(\mathcal A=\{(x,v):x\in\partial\Omega,\ |v|=1,\ v\text{ points inward and is not tangent to }\partial\Omega\}\), with the billiard map sending one reflected state to the next. If \(\mathcal A\) is foliated by invariant circles homotopic to its boundary, then \(\partial\Omega\) is an ellipse.
DYN-005:[Birkhoff1927DynamicalSystems][Poritsky1950BilliardProperty][KaloshinSorrentino2018LocalBirkhoff][KaloshinZhang2024NearlyCentralBirkhoff]
Dynamical systemsGeometry and topology

Notes

Billiard motion in a bounded strictly convex planar domain induces a map on an annulus of boundary positions and directions, and in an ellipse this phase annulus is foliated by invariant curves: elliptic billiards are integrable. The Birkhoff conjecture asserts the converse — if the open phase annulus of a smooth strictly convex billiard is foliated by invariant essential circles, the boundary must be an ellipse. It stems from Birkhoff's 1927 treatment of billiards as a model dynamical system [Birkhoff1927DynamicalSystems], with an early explicit study by Poritsky [Poritsky1950BilliardProperty].

The strongest results to date are perturbative or conditional. Kaloshin and Sorrentino proved a local version of the conjecture for convex billiards near ellipses [KaloshinSorrentino2018LocalBirkhoff], and subsequent work treats nearly centrally symmetric domains [KaloshinZhang2024NearlyCentralBirkhoff]; further cases are known under symmetry or rational-integrability hypotheses.

For an arbitrary smooth strictly convex table, however, global integrability has not been classified, and the conjecture remains open in its full strength.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.