Open Problems on Billiards and Geometric Optics

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A flow has time parameter $t\in\mathbb R$; a cascade is an invertible discrete-time system with $t\in\mathbb Z$. They satisfy $g^{t+s}=g^t\circ g^s$ and $g^0=\mathrm{id}$. The trapped set is $T_g=\{x:\{g^t x:t\ge0\}\text{ is bounded and }\{g^t x:t\le0\}\text{ is unbounded}\}$. Hausdorff dimension is the metric dimension in the Euclidean plane. The source asks for $C^m$ regularity in $x$, or in $(x,t)$ for flows.

Statement recorded in the supplied source.

Let $g^t:\mathbb{R}^2\to\mathbb{R}^2$ be a Lebesgue-measure-preserving flow or cascade. A point is trapped if its positive semiorbit is bounded and its negative semiorbit is unbounded; let $T_g$ be the trapped set. What is the maximum Hausdorff dimension of $T_g$? More specifically, for $C^m$ systems determine the maximal dimension $d(m)$, between $1$ and $2$. [S2]

Short English statement

Find the largest Hausdorff dimension of points that stay bounded forward in time but escape backward in area-preserving planar flows or discrete systems, and determine its dependence on differentiability.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5200010 (AMR-051-0010). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 M. Bialy, C. Fierobe, A. Glutsyuk, M. Levi, A. Plakhov and S. Tabachnikov, Open problems on billiards and geometric optics (2021), arXiv:2110.10750v2, Alexander Plakhov, Problem 1, p. 8. https://arxiv.org/pdf/2110.10750v2.