Minimising length of closed billiard trajectories on hyperbolic polygons

TL;DR

This paper investigates the minimal average length of certain closed billiard trajectories in hyperbolic polygons, proving that symmetry and regularity minimize these lengths using Teichmüller theory.

Contribution

It establishes the minimal average length configurations for closed billiard trajectories in even-sided right-angled polygons and Lambert quadrilaterals with acute angles.

Findings

Regular even-sided right-angled polygons minimize average length.

Lambert quadrilaterals with reflective symmetry minimize average length.

Teichmüller theory techniques are used to prove these minimizations.

Abstract

In a hyperbolic polygon any finite collection of closed billiard trajectories can be assigned an average length function. In this paper, we consider the average length of the collection of cyclically related closed billiard trajectories in even-sided right-angled polygons and the collection of reflectively related closed billiard trajectories in Lambert quadrilaterals with acute angle $\pi/k$. We show that in the former case the average length is minimised by the regular evensided right-angled polygon, and in the latter case it is minimised by the Lambert quadrilateral with a reflective symmetry about its long axis. We use techniques from Teichmueller theory to prove the main theorems.