Symmetry and regularity of the maps

Billiards is a dynamical system describing the motion of a particle inside a smooth closed curve moving inside the enclosed region in straight lines and bouncing off the walls according the equal-angle-law of reflection (see [21] for a survey). Parameterizing the point of impact by a curve-parameter and the angle of impact with , the Poincaré map inherits all the information of the billiard flow. The map is a map on the annulus preserving the Lebesgue area. The compact space X (also called the phase space) has the invariant boundary and . The billiard map T has been studied first by Birkhoff and is a celebrated example of a monotone twist map, a class of conservative dynamical systems introduced by Poincaré. Every billiard map has the involutive symmetry making T reversible: .

We are studying the family of billiard maps defined by the one-parameter family of curves . Since the tables are symmetric by reflections at the x and y axes, this symmetry is also present in the phase space X. Together with the involutive symmetry S, the whole phase space X has a dihedral symmetry. We can factor this symmetry out by identifying points if and . In Fig. 2, we see numerical computations of orbits for some values of p.

We investigate first the continuity and smoothness of the maps as well as of the map from to a topological space of maps. Note that the billiard map in a convex domain is in general not continuous as the example of a general triangle shows. The billiard map has at least the degree of smoothness of the normal map at the curve . By the symmetry, we can restrict to the quadrant , where the curve is given by and where the normal is . The map is if (x(s),y(s)) is away from the coordinate axes . Denote with C(X,X) the space of homeomorphisms of X with the topology given by the metric . Define for any compact set and

and the topology on by if and only if for all and all compact .

Proof. A billiard map in a convex curve is in general not a homeomorphism. In our case, the boundary has four symmetric points, where the curvature may be infinite. By imposing at these points the symmetric normal direction, we find for a continuous continuation of the normal direction field on the whole curve . In the limiting cases p=1 and , the table is a square, and the billiard map can also be extended to the singular set in the phase space by defining in the four vertices of the square the normal direction pointing to the center of the table. For p=2n, the curve is algebraic and so real analytic which implies that the map is real analytic. Given a curve (t,f(t)), the curvature is given by

This means for

This is continuous as a function of t for p;SPMgt;2, leading to a continuously differentiable map for p;SPMgt;2. The curvature is unbounded in places for p;SPMlt;2, so that for p;SPMlt;2, the map is no longer continuously differentiable. However, for , the curvature depends continuously on p which shows that is continuous.
End of the proof.

Remark. The map is not continuous at p=2, because for p=2, the curvature at s=0 is 1 and for p;SPMgt;2, the curvature at s=0 is vanishing. One can therefore not apply KAM theory to conclude that for p;SPMgt;2, near enough to 2, the map has invariant curves. Indeed, as we will see below, there are no invariant curves for p;SPMgt;2.

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