Minimising length of closed billiard trajectories on hyperbolic polygons
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 π/k. We show that in the former case the average length is minimized by the regular even-sided right-angled polygon, and in the latter case it is minimized by the Lambert quadrilateral with a reflective symmetry about its long axis. We use techniques from Teichmüller theory to prove the main theorems.
Fields of Study & Subjects
Mathematical Dynamics and Fractals
Mathematics and Applications
Geometric and Algebraic Topology
Publication details
- Journal / source
- Dynamical Systems
- Volume
- 41
- Issue
- 3
- Pages
- 365–380
- Publication date
- 6 March 2026
- Language
- EN