Minimizing length of billiard trajectories in hyperbolic polygons
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- by John R. Parker, Norbert Peyerimhoff and Karl Friedrich Siburg
- Conform. Geom. Dyn. 22 (2018), 315-332
- 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.References
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Bibliographic Information
- John R. Parker
- Affiliation: Department of Mathematical Sciences, Durham University, Science Laboratories, South Road, Durham, DH1 3LE, United Kingdom
- MR Author ID: 319072
- ORCID: 0000-0003-0513-3980
- 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
- MR Author ID: 290247
- 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
- MR Author ID: 332558
- Email: karlfriedrich.siburg@uni-dortmund.de
- Received by editor(s): October 23, 2016
- Published electronically: December 7, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Conform. Geom. Dyn. 22 (2018), 315-332
- MSC (2010): Primary 37D40; Secondary 32G15, 53A35, 37F30
- DOI: https://doi.org/10.1090/ecgd/328
- MathSciNet review: 3884644