Baker's Dozen — Periodic hyperbolic outer billiards

Definitions and mathematical statement

Let $P$ be a compact convex geodesic polygon with nonempty interior in the oriented hyperbolic plane $\mathbb H^2$. For an exterior point $x$, choose its right supporting geodesic to $P$. Except when the support contains a side, it touches a unique vertex $v$. Define $T_P(x)$ by the hyperbolic half-turn about $v$: the points $x,v,T_P(x)$ lie on a geodesic and $v$ is their midpoint. The other orientation gives the inverse convention.

Each half-turn extends to the ideal boundary $\partial_\infty\mathbb H^2$. On that boundary, adjacent sector formulas agree at their common endpoint: both half-turns interchange the ideal endpoints of the intervening side. They therefore define a continuous, piecewise projective circle map $\widehat T_P$ on the entire ideal boundary.

The conjecture is that, for every such $P$, there are an integer $m\ge1$ and either an ordinary exterior point $x$ with $T_P^m(x)=x$, whose orbit avoids the singular supporting-side rays, or an ideal point $\xi$ with $\widehat T_P^m(\xi)=\xi$. Every ideal point is permitted, including sector endpoints. Thus a periodic orbit at infinity is permitted. No uniform bound on $m$ is requested, and a limiting sequence of almost periodic trajectories alone does not supply the required orbit.

Short English statement

Must every convex polygon in the hyperbolic plane have a periodic outer billiard trajectory, allowing the trajectory to lie on the ideal boundary?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 600004 (AMR-005-0004), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Tabachnikov - A Baker's Dozen of Problems (2015), Conjecture 2. Original source indexed by AMR-005-0004. https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/.