On Polygonal Dual Billiard in the Hyperbolic Plane

Given a compact convex plane domain P , one defines the dual billiard transformation F of its exterior as follows. Let x be a point outside of P . There are two support lines to P through x; choose one of them, say, the right one from x’s view-point, and define F (x) to be the reflection of x in the support point. This definition applies if the support point is unique; otherwise F (x) is not defined. The dual billiard map is an outer counterpart of the usual billiard ball map, and it is also known as the “outer billiard”. To the best of our knowledge, the dual billiard system was introduced by Bernhard Neumann [14] ; see also an earlier paper [4] in which the dynamical aspects are not discussed. J. Moser put forward the study of dual billiards in the framework of KAM theory in [12], and since then, the literature on dual billiards has continued to grow — see references at the end of this paper. We are concerned with polygonal dual billiard tables P . The dual billiard map F or its inverse F are not defined on the extensions of the sides of P . The singularity set ∆ of the dual billiard map consists of the points x such that F (x) is not defined for some i ∈ Z; in other words, ∆ is the union of the images and preimages under F of the lines that contain the sides of P . The set ∆ is a countable union of segments and lines; it has zero measure. In the complement of ∆, the map F is a piecewise isometry. We briefly survey what is known about polygonal dual billiards in the Euclidean plane. J. Moser asked in [13] whether the orbits of the polygonal dual billiard map can escape to infinity. A motivation comes from the case when the boundary of the dual billiard table is strictly convex and sufficiently smooth: then every orbit is confined to a compact domain bounded by a KAM invariant curve of the dual billiard map. Moser’s question is still open for general polygons P . However there is a class of polygons, called quasirational, for which every orbit stays bounded.