An irrational polygon with nonergodic billiard flow
Let $P \subset \mathbb{R}^2$ be a bounded simple polygon. Its billiard flow moves a point at unit speed along straight lines inside $P$, reflecting at the sides with equal angles of incidence and reflection; the null set of trajectories that hit a vertex is ignored. The flow preserves the normalized Liouville measure $$d\mu = \frac{dx\,dy\,d\theta}{2\pi \operatorname{Area}(P)}$$ on $P \times S^1$. The polygon is rational if all its angles are rational multiples of $\pi$, and irrational otherwise.
The billiard flow of a rational polygon is not ergodic for $\mu$: it decomposes into a one-parameter family of directional flows. Kerckhoff, Masur and Smillie proved that the polygons whose billiard flow is ergodic form a dense $G_\delta$ set among polygons with a given number of sides, and Vorobets proved ergodicity for every polygon whose angles divided by $\pi$ admit a certain super-exponentially fast simultaneous rational approximation. Chaika and Forni proved that the billiard flow is weakly mixing for a dense $G_\delta$ set of polygons.
Problem. Is there a bounded simple polygon $P$ with at least one interior angle an irrational multiple of $\pi$ whose billiard flow is not ergodic, that is, with a measurable flow-invariant set $A \subset P \times S^1$ such that $0 < \mu(A) < 1$?
Source: Gutkin (2012), Problem 8: give an example of an irrational but nonergodic polygon.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DS — Dynamical Systems
- Keywords
- AIM Open Applied Problems, ergodicity, polygonal billiards
References
@article{gutkin2012,
author = {Gutkin, Eugene},
title = {Billiard dynamics: an updated survey with the emphasis on open problems},
journal = {Chaos},
volume = {22},
number = {2},
year = {2012},
pages = {026116},
doi = {10.1063/1.4729307},
eprint = {1301.2547},
archivePrefix = {arXiv},
}
@article{kerckhoff1986,
author = {Kerckhoff, Steven and Masur, Howard and Smillie, John},
title = {Ergodicity of billiard flows and quadratic differentials},
journal = {Ann. of Math. (2)},
volume = {124},
number = {2},
year = {1986},
pages = {293--311},
doi = {10.2307/1971280},
}
@article{vorobets1997,
author = {Vorobets, Ya. B.},
title = {Ergodicity of billiards in polygons},
journal = {Sb. Math.},
volume = {188},
number = {3},
year = {1997},
pages = {389--434},
doi = {10.1070/SM1997v188n03ABEH000211},
}
@article{chaika2026,
author = {Chaika, Jon and Forni, Giovanni},
title = {Weakly mixing polygonal billiards},
journal = {Ann. of Math. (2)},
volume = {203},
number = {3},
year = {2026},
pages = {695--735},
doi = {10.4007/annals.2026.203.3.1},
eprint = {2003.00890},
archivePrefix = {arXiv},
}
@misc{aim-irrational-nonergodic-polygon,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{An irrational polygon with nonergodic billiard flow}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/142-irrational-nonergodic-polygon.md},
note = {Entry 142 at commit a153f46, 2026-10-13},
}
Resolution claims
We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to \(\pi\) is weakly mixing for normalized Liouville measure. Equivalently, the product of the flow with itself is ergodic.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-20 | Open → Partially resolved | The registry's curators | Weak mixing of triangular billiards with an irrational angle v1 recorded from math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00531, "An irrational polygon with nonergodic billiard flow", clopen.solutions.