Baker's Dozen — Periodic billiard trajectories

Definitions and mathematical statement

Let $K\subset\mathbb R^2$ be a compact strictly convex region with smooth boundary $\gamma$. An inner billiard trajectory consists of straight segments in $K$, with the tangential component of velocity preserved and the normal component reversed at each reflection. Its least period is the smallest positive number of reflections returning to the same boundary point and outgoing direction. A one-parameter family means a nonconstant smooth family of periodic phase points, rather than repeated traversal of one orbit.

For two distinct integers $p,q\ge2$, ask whether a nonelliptic $\gamma$ can carry both a one-parameter family of trajectories of least period $p$ and one of least period $q$. Determine the possible pairs and the restrictions they impose on the table. This is a simultaneous periodic-family question; it does not assume that the whole billiard phase space is foliated by invariant curves.

Ask also the outer billiard version. Fix an orientation and, for $x$ outside $K$, choose the supporting tangent of the prescribed side, with contact point $t$; set $T(x)=2t-x$. Replace the inner billiard map above by $T$, and require nonconstant smooth families with two distinct least periods. An ellipse means an affine image of a Euclidean circle.

Short English statement

Can a smooth convex billiard table other than an ellipse support continuous families of periodic motions of two different least periods? Ask the same question for reflection in exterior tangent points.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 600002 (AMR-005-0002), 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), Problem 1. Original source indexed by AMR-005-0002. https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/14-01/.