Infinite genus surfaces and irrational polygonal billiards

TL;DR

This paper proves that the invariant surface of billiards on irrational polygons is topologically equivalent to the Loch Ness monster surface, which has infinite genus and a single end, revealing deep geometric properties of such billiard systems.

Contribution

It establishes a topological classification of invariant surfaces for irrational polygonal billiards, showing they are homeomorphic to the Loch Ness monster surface.

Findings

Invariant surface is homeomorphic to the Loch Ness monster.

The surface has infinite genus and exactly one end.

Provides a topological understanding of irrational polygonal billiards.

Abstract

We prove that the natural invariant surface associated with the billiard game on an irrational polygonal table is homeomorphic to the Loch Ness monster, that is, the only orientable infinite genus topological real surface with exactly one end.