Problem 3 — (Renormalization of billiards in polygons).

Definitions and mathematical statement

For a planar polygon $P$, its billiard dynamics moves a unit-speed point along straight segments and reflects at an edge by equal incoming and outgoing angles. If $n$ is a unit normal at an edge, the velocity changes by \[ v\longmapsto v-2(v\cdot n)n. \] Trajectories hitting vertices are singular and require exclusion or a specified continuation convention. A billiard system includes its polygon and the corresponding flow or collision map; directional restrictions may also be part of the data.

Problem 3 (renormalization of billiards in polygons). Is there a natural dynamical system on a space of polygonal billiard systems that yields a useful renormalization procedure? The desired dynamics acts on the space of billiard systems, not merely by moving a particle in a single fixed polygon. It should relate billiard dynamics at different scales or through an appropriate induction/rescaling scheme in a way useful for studying the original dynamics.

The excerpt does not define the parameter space, topology, equivalence under similarities, polygon class, directional data, or the precise relation between a renormalized billiard and the initial system. It also does not specify a mathematical criterion for "natural" or "useful." In particular, it does not restrict to rational-angle polygons, convex polygons, or polygons unfolding to compact translation surfaces. No such restriction, or extension of a known rational-polygon framework to arbitrary polygons, is silently assumed. A precise existence theorem requires these choices and a stated dynamical conclusion to be supplied.

Short English statement

Construct a useful renormalization dynamics on the space of polygonal billiard systems. The source leaves the polygon class, parameter space, rescaling relation, and success criterion open.

Sources

• Local UnsolvedMath registry, permanent ID 11000210, source code catalogue entry; no proposer or proposal year supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics, Problem 3, PDF page 256; https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The registry records PDF-font-structure extraction and access on 2026-07-29; its inventory distinguishes 329 labeled headings by source page.
• No separate authoritative reference or candidate renormalization construction is supplied. The research summary says its literature status cannot be verified.