Minimizing length of billiard trajectories in hyperbolic polygons

Parker, John; Peyerimhoff, Norbert; Siburg, Karl

doi:10.1090/ecgd/328

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.

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