Positive metric entropy for billiards arbitrarily close to an ellipse
For a strictly convex domain $D \subset \mathbb{R}^2$ with smooth boundary of length $|\partial D|$, the billiard map $B$ acts on pairs $(s, \varphi)$, where $s$ is the arclength position of a collision on $\partial D$ and $\varphi \in (0, \pi)$ is the angle of the outgoing direction measured from the positive tangent. It preserves the probability measure $d\mu = \sin\varphi \, ds \, d\varphi / (2|\partial D|)$; write $h_\mu(B)$ for its Kolmogorov–Sinai entropy.
The billiard in an ellipse is integrable: the invariant curves of its billiard map, among them those with the confocal ellipses as caustics, fill out a region with nonempty interior. Berger and Turaev proved that every area-preserving $C^\infty$ diffeomorphism of a surface with an elliptic fixed point can be $C^\infty$-perturbed to one with positive metric entropy; their perturbations are area-preserving maps in general, not billiard maps.
Problem. Is there a noncircular ellipse, with boundary embedding $\gamma_E : S^1 \to \mathbb{R}^2$, and smooth embeddings $\gamma_j$ bounding strictly convex domains $D_j$, such that $\gamma_j \to \gamma_E$ in $C^\infty$ and the billiard maps $B_j$ of the $D_j$ satisfy $h_{\mu_j}(B_j) > 0$ for every $j$? The perturbation must come from changing the billiard table itself.
Source: Berger (2026), Problem 2.17: show that a perturbation of an ellipse defines a billiard with positive metric entropy. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DS — Dynamical Systems
- Secondary
- SG
- Keywords
- AIM Open Applied Problems, convex billiards, metric entropy
References
@incollection{berger2026,
author = {Berger, Pierre},
title = {Wild dynamics on manifolds},
booktitle = {Proceedings of the International Congress of Mathematicians 2026, Volume 5: Invited Lectures (Sections 9--11)},
volume = {5},
year = {2026},
pages = {24--41},
publisher = {Society for Industrial and Applied Mathematics},
doi = {10.1137/25M1810787},
eprint = {2510.12929},
archivePrefix = {arXiv},
}
@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{berger2019,
author = {Berger, Pierre and Turaev, Dmitry},
title = {On {H}erman's positive entropy conjecture},
journal = {Adv. Math.},
volume = {349},
year = {2019},
pages = {1234--1288},
doi = {10.1016/j.aim.2019.04.002},
eprint = {1704.02473},
archivePrefix = {arXiv},
}
@misc{aim-positive-entropy-billiards-near-ellipse,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Positive metric entropy for billiards arbitrarily close to an ellipse}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/143-positive-entropy-billiards-near-ellipse.md},
note = {Entry 143 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-00549, "Positive metric entropy for billiards arbitrarily close to an ellipse", clopen.solutions.