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
Full-text PDF Free Access
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.
- [1] Emil Artin, Ein mechanisches system mit quasiergodischen bahnen, Abh. Math. Sem. Univ. Hamburg 3 (1924), no. 1, 170–175 (German). MR 3069425, https://doi.org/10.1007/BF02954622
- [2] Jean Bourgain and Alex Kontorovich, Beyond expansion II: low-lying fundamental geodesics, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 5, 1331–1359. MR 3635355, https://doi.org/10.4171/JEMS/694
- [3] Simon Castle, Norbert Peyerimhoff, and Karl Friedrich Siburg, Billiards in ideal hyperbolic polygons, Discrete Contin. Dyn. Syst. 29 (2011), no. 3, 893–908. MR 2773157, https://doi.org/10.3934/dcds.2011.29.893
- [4]
F. Douma, English translation of E. Artin's article [1] Ein mechanisches System mit quasiergodischen Bahnen at http://www.maths.dur.ac.uk/
dma0np/
- [5] John Hamal Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 2, Matrix Editions, Ithaca, NY, 2016. Surface homeomorphisms and rational functions. MR 3675959
- [6] Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors. MR 1215481
- [7] Steven P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235–265. MR 690845, https://doi.org/10.2307/2007076
- [8] Caroline Series, The modular surface and continued fractions, J. London Math. Soc. (2) 31 (1985), no. 1, 69–80. MR 810563, https://doi.org/10.1112/jlms/s2-31.1.69
- [9] Scott Wolpert, Noncompleteness of the Weil-Petersson metric for Teichmüller space, Pacific J. Math. 61 (1975), no. 2, 573–577. MR 422692
- [10] Scott Wolpert, On the Weil-Petersson geometry of the moduli space of curves, Amer. J. Math. 107 (1985), no. 4, 969–997. MR 796909, https://doi.org/10.2307/2374363
- [11] Scott A. Wolpert, Geodesic length functions and the Nielsen problem, J. Differential Geom. 25 (1987), no. 2, 275–296. MR 880186
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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
