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
MathSciNet review:
3884644
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Abstract | References | Similar Articles | Additional Information
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
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
Article copyright:
© Copyright 2018
American Mathematical Society