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Minimizing length of billiard trajectories in hyperbolic polygons

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

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Authors

Karl Friedrich Siburg



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

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