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Conformal Geometry and Dynamics

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Minimizing length of billiard trajectories in hyperbolic polygons


Authors: John R. Parker, Norbert Peyerimhoff and Karl Friedrich Siburg
Journal: Conform. Geom. Dyn. 22 (2018), 315-332
MSC (2010): Primary 37D40; Secondary 32G15, 53A35, 37F30
DOI: https://doi.org/10.1090/ecgd/328
Published electronically: December 7, 2018
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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.


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Additional Information

John R. Parker
Affiliation: Department of Mathematical Sciences, Durham University, Science Laboratories, South Road, Durham, DH1 3LE, United Kingdom
Email: j.r.parker@durham.ac.uk

Norbert Peyerimhoff
Affiliation: Department of Mathematical Sciences, Durham University, Science Laboratories, South Road, Durham, DH1 3LE, United Kingdom
Email: norbert.peyerimhoff@durham.ac.uk

Karl Friedrich Siburg
Affiliation: Fakultät für Mathematik, Technische Universität Dortmund, Lehrstuhl LS IX, Vogelpothsweg 87, 44 227 Dortmund, Germany
Email: karlfriedrich.siburg@uni-dortmund.de

DOI: https://doi.org/10.1090/ecgd/328
Received by editor(s): October 23, 2016
Published electronically: December 7, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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