Dual billiards in the hyperbolic plane*

Abstract

We study an area preserving map of the exterior of a smooth convex curve in the hyperbolic plane,defined by a natural geometrical construction and called the dual billiard map. We consider twoproblems: stability and the area spectrum. The dual billiard map is called stable if all its orbitsare bounded. We show that both stable and unstable behaviours may occur. Ifthe map at infinity has a hyperbolic periodic orbit, then the dual billiard map has orbits escaping toinfinity. On the other extreme, if the map at infinity is smoothly conjugated to a Diophantineirrational rotation of the circle, then the dual billiard map is stable. The area spectrum is the set of extremal areas of n-gons, circumscribed about the dual billiardcurve; this is to the dual billiard what the length spectrum is to the usual, inner, one. We show thatthe area spectrum has an asymptotic expansion in even negative powers of n as n→∞. Thefirst coefficient of this expansion is the area of the dual billiard curve, and the next is, up to aconstant, the cubed integral of the cube root of its curvature. We describe the curves that arerelative extrema of these two functionals and show that they are the trajectories ofthe pseudospherical pendulum with various gravity directions.