Ivrii's measure-zero conjecture for billiards
Let $d \ge 2$ and let $\Omega \subset \mathbb{R}^d$ be a bounded strictly convex domain with $C^\infty$ boundary. Unit-speed rays in $\Omega$ reflect specularly at $\partial\Omega$ (angle of incidence equals angle of reflection). The billiard map acts on the inward-pointing, non-grazing unit vectors based at points of $\partial\Omega$ and preserves the Liouville measure there; a point is $k$-periodic if it is fixed by the $k$-th iterate. Ivrii proved that the Dirichlet and Neumann eigenvalue counting functions of a bounded domain with smooth boundary have a two-term asymptotic expansion, with second term proportional to the area of the boundary, whenever the periodic billiard trajectories form a set of measure zero. The $3$-periodic points have measure zero: Rychlik proved this for planar convex domains with smooth boundary and Vorobets in every dimension. Glutsyuk and Kudryashov proved that the $4$-periodic points of any planar billiard with piecewise $C^4$ boundary have measure zero.
Problem. Does the union over all integers $k \ge 2$ of the sets of $k$-periodic points of the billiard map have Liouville measure zero, for every $d \ge 2$ and every such $\Omega$?
Source: Ivrii, in connection with his two-term Weyl law (1980), for every billiard with smooth boundary; see Glutsyuk and Kudryashov (2012). Stated by AIM — Open Applied Problems for strictly convex domains.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DS — Dynamical Systems
- Secondary
- SP
- Keywords
- AIM Open Applied Problems, billiards, periodic orbits, Weyl's law
References
@article{ivrii1980,
author = {Ivrii, V. Ya.},
title = {Second term of the spectral asymptotic expansion of the {L}aplace--{B}eltrami operator on manifolds with boundary},
journal = {Funct. Anal. Appl.},
volume = {14},
number = {2},
year = {1980},
pages = {98--106},
doi = {10.1007/BF01086550},
}
@article{glutsyuk2012,
author = {Glutsyuk, Alexey and Kudryashov, Yury},
title = {No planar billiard possesses an open set of quadrilateral trajectories},
journal = {J. Mod. Dyn.},
volume = {6},
number = {3},
year = {2012},
pages = {287--326},
doi = {10.3934/jmd.2012.6.287},
}
@article{rychlik1989,
author = {Rychlik, Marek Ryszard},
title = {Periodic points of the billiard ball map in a convex domain},
journal = {J. Differential Geom.},
volume = {30},
number = {1},
year = {1989},
pages = {191--205},
doi = {10.4310/jdg/1214443290},
}
@article{vorobets1994,
author = {Vorobets, Ya. B.},
title = {On the measure of the set of periodic points of the billiard},
journal = {Math. Notes},
volume = {55},
number = {5},
year = {1994},
pages = {455--460},
doi = {10.1007/BF02110371},
}
@misc{aim-ivrii-periodic-billiards,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Ivrii's measure-zero conjecture for billiards}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/002-ivrii-periodic-billiards.md},
note = {Entry 002 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00290, "Ivrii's measure-zero conjecture for billiards", clopen.solutions.