Math Conjectures

Ivrii periodic-billiard conjecture

OPENMajorConjectureProposed 1980 · Full conjecture

Canonical statement

Let Ω⊂Rn\Omega\subset\mathbb R^n be a bounded domain with C∞C^\infty boundary, and let B∗∂Ω={(x,ξ)∈T∗∂Ω:∣ξ∣<1}B^*\partial\Omega=\{(x,\xi)\in T^*\partial\Omega:|\xi|<1\}, equipped with its canonical Liouville measure. The set of points of B∗∂ΩB^*\partial\Omega that are periodic under the billiard map has measure zero.
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Let \(\Omega\subset\mathbb R^n\) be a bounded domain with \(C^\infty\) boundary, and let \(B^*\partial\Omega=\{(x,\xi)\in T^*\partial\Omega:|\xi|<1\}\), equipped with its canonical Liouville measure. The set of points of \(B^*\partial\Omega\) that are periodic under the billiard map has measure zero.
DYN-006:[Ivrii1980SecondTermSpectral][Gutkin2003BilliardSurvey][Glutsyuk2017FourReflective][Weinreich2025AlgebraicBilliards]
Dynamical systemsGeometry and topologyAnalysis and functional analysis

Notes

Ivrii's conjecture asserts that in a bounded domain Ω⊂Rn\Omega\subset\mathbb R^n with smooth boundary, the points of billiard phase space that are periodic under the billiard map form a set of Liouville measure zero. Ivrii arrived at the question in 1980 through spectral asymptotics: this measure-zero condition is exactly the hypothesis needed for the second term in Weyl's law for the Laplacian on Ω\Omega [Ivrii1980SecondTermSpectral].

Partial results are organized by period. The planar cases of periods 33 and 44 are known, as are analytic and generic classes of boundaries and various special geometries [Gutkin2003BilliardSurvey]. Complex-algebraic methods have been effective for quadrilateral orbits in planar algebraic billiards [Glutsyuk2017FourReflective] and, more recently, for billiards in algebraic curves such as the Fermat hyperbola [Weinreich2025AlgebraicBilliards].

Each period so far requires its own argument, and no method is uniform over all periods for an arbitrary smooth boundary. The conjecture — and with it the unconditional two-term spectral asymptotics that motivated it — remains open.

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.