Homotopical rigidity of polygonal billiards

TL;DR

This paper investigates the homotopical equivalence of billiard flows in polygonal shapes, establishing conditions under which such equivalence implies geometric similarity or affine similarity, especially for rational polygons.

Contribution

It introduces and analyzes homotopical equivalence of billiard flows, connecting it to geometric similarity for rational polygons, and compares it with existing equivalence notions.

Findings

Homotopically equivalent billiard flows imply geometric similarity for rational polygons.

For polygons with all sides vertical and horizontal, homotopical equivalence implies affine similarity.

Homotopical equivalence relates closely to other known equivalence relations in billiard dynamics.

Abstract

Consider two $k$-gons $P$ and $Q$. We say that the billiard flows in $P$ and $Q$ are homotopically equivalent if the set of conjugacy classes in the fundamental group of $P$ which contain a periodic billiard orbit agrees with the analogous set for $Q$. We study this equivalence relationship and compare it to the equivalence relations, order equivalence and code equivalence, introduced in \cite{BT1,BT2}. In particular we show if $P$ is a rational polygon, and $Q$ is homotopically equivalent to $P$, then $P$ and $Q$ are similar, or affinely similar if all sides of $P$ are vertical and horizontal.