Baker's Dozen — Completely periodic hyperbolic outer billiards

Definitions and mathematical statement

For a compact convex geodesic polygon $P\subset\mathbb H^2$, orient the plane and define the outer billiard map $T_P$ as follows. From a point $x$ outside $P$, take the right supporting geodesic and its contact vertex $v$, and apply the half-turn centered at $v$. The map is undefined when that support is not a unique vertex. Write $\Omega_P$ for the set of exterior points whose forward and backward iterates avoid all such singularities.

Call $P$ completely periodic when \[ \text{for every }x\in\Omega_P\text{ there is an integer }m(x)\ge1 \text{ with }T_P^{m(x)}(x)=x. \] The requested task is a necessary and sufficient geometric characterization of the polygons with this property, up to hyperbolic isometry. In particular, the characterization should describe restrictions on side lengths, angles, and their arrangement, rather than merely restating periodicity of the map.

The period may depend on the starting point; a common finite order for the entire map is not part of the definition. Singular orbits are excluded, so the classification does not depend on assigning arbitrary images at side extensions. The question concerns all ordinary exterior orbits, not just the dynamics induced on the circle at infinity.

Short English statement

Classify the hyperbolic polygons for which every nonsingular exterior outer billiard trajectory eventually returns to its starting phase point.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 600005 (AMR-005-0005), 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), Section 4 second unnumbered problem. Original source indexed by AMR-005-0005. https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/.