Minimizing length of billiard trajectories in hyperbolic polygons

Parker, John R; Peyerimhoff, Norbert; Siburg, Karl Friedrich

Abstract

Closed billiard trajectories in a polygon in the hyperbolic plane can be coded by the order in which they hit the sides of the polygon. In this paper, we consider the average length of cyclically related closed billiard trajectories in ideal hyperbolic polygons and prove the conjecture that this average length is minimized for regular hyperbolic polygons. The proof uses a strict convexity property of the geodesic length function in Teichmüller space with respect to the Weil-Petersson metric, a fundamental result established by Wolpert.

Citation

Parker, J. R., Peyerimhoff, N., & Siburg, K. F. (2018). Minimizing length of billiard trajectories in hyperbolic polygons. Conformal Geometry and Dynamics, 22, 315-332. https://doi.org/10.1090/ecgd/328

• Journal Article Type: Article
• Acceptance Date: Oct 29, 2018
• Online Publication Date: Dec 7, 2018
• Publication Date: Nov 1, 2018
• Deposit Date: Oct 30, 2018
• Publicly Available Date: Oct 31, 2018
• Journal: Conformal Geometry and Dynamics
• Print ISSN: 1088-4173
• Publisher: American Mathematical Society
• Peer Reviewed: Peer Reviewed
• Volume: 22
• Pages: 315-332
• DOI: https://doi.org/10.1090/ecgd/328
• Public URL: https://durham-repository.worktribe.com/output/1315083