Abstract
We consider billiard trajectories in ideal hyperbolic polygons and present a conjecture about the minimality of the average length of cyclically related billiard trajectories in regular hyperbolic polygons. We prove this conjecture in particular cases, using geometric and algebraic methods from hyperbolic geometry.
Highlights
In this article we study billiards in polygons of the hyperbolic plane
The polygons Π ⊂ D under consideration are ideal, which means that all vertices of Π lie in the boundary at infinity ∂D = {z ∈ C | |z| = 1}
We identify all shifts in the set of all periodic billiard sequences Sper(Π)
Summary
In this article we study billiards in polygons of the hyperbolic plane. Our main model of the hyperbolic plane is the Poincare unit disk D = {z ∈ C | |z| < 1}. An−1 a closed billiard trajectory which starts and ends at the side a0 and hits the sides of the polygon Π in the order a1, . The (finite) hyperbolic length of this closed piecewise geodesic curve is denoted by L(Π, a) In this geometric interpretation, the shift a1, . An−1, but to change the counter-clockwise enumeration of the sides of Π, i.e., to choose a different side with the label 1 This leads to different closed billiard trajectories in Π which have, the same combinatorial structure. Note that “shifts” and “being cyclically related” are completely different concepts: for example, in a pentagon, 1524 and its shifts (e.g., 5241) represent the same periodic billiard sequence, whereas 1524 and its cyclically related sequences (e.g., 2135) are different elements in Sper(Π). For results about dual (or outer) billiards in the hyperbolic plane see, e.g., the survey [TaDo-05] and the recent article [Ta-07]
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