Kirby Problem 2.42

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 translation surface is a surface with Euclidean coordinate charts whose transition maps are translations. Its straight-line flow in direction $\theta\in[0,2\pi)$ is locally translation by $t(\cos\theta,\sin\theta)$, away from singularities, and preserves Euclidean area. The flow is ergodic for area measure if every measurable invariant set has either zero area or full area.

The standard Chamanara surface is obtained from a unit square by repeatedly pairing, by translations, the right half of the remaining top edge with the left half of the remaining bottom edge, and the upper half of the remaining right edge with the lower half of the remaining left edge; corners, cut points and their identifications are omitted. This fixes parameter $\alpha=1/2$. [S2]

For the Bruin--Lukina surfaces, define the dyadic interval exchange $a$ to send each interval $I_j=(1-2^{1-j},1-2^{-j})$ by translation to $(2^{-j},2^{1-j})$, for $j\geq1$. Glue the top of the unit square to the bottom by $(x,1)\sim(a(x),0)$, removing cut endpoints and corners. Identifying $(1,y)\sim(0,y)$ gives their basic surface $L$. More generally take $q\geq2$, a permutation $\pi\in S_q$ of $\{0,\ldots,q-1\}$, and an integer $p\geq q$. Set $r=p\bmod q$, $s(i)=q-1-i$, $t(i)=i+r\bmod q$, and $\pi'=s\circ t^{-1}\circ\pi\circ s^{-1}$. Divide both vertical sides into $p$ open intervals of length $1/p$, numbered from bottom to top. Match left interval $j$ to right interval $\pi'(j)$ for $0\leq j<q$, and match left interval $j$ to right interval $j$ for $q\leq j<p$, always by translations; remove all singular endpoints. Together with the same horizontal gluing this defines $L_{\pi,p}$. [S3]

Statement recorded in the supplied source.

Is the straight-line flow on the standard Chamanara surface ergodic with respect to area for Lebesgue-almost every $\theta\in[0,2\pi)$? Does the same almost-every-direction assertion hold for the basic Bruin--Lukina surface $L$ and for their family $L_{\pi,p}$ with the parameters above? [S2]

Short English statement

Do almost all directional geodesic flows preserve no nontrivial measurable subsets on the Chamanara surface and on the related finite-area infinite-genus surfaces of Bruin and Lukina?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2790 (KP-2.42). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman (eds.), K3: A New Problem List in Low-Dimensional Topology, AMS Mathematical Surveys and Monographs 295 (2026), author preliminary edition. Problem2.42 and all remarks, pages120--121. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 Henk Bruin and Olga Lukina, Rotated odometers, arXiv:2101.00868v3 (2023); Journal of the London Mathematical Society107 (2023),1983--2024. Section2.1 equation(3), Section2.3 equation(4), Figures3--4. https://arxiv.org/pdf/2101.00868v3.