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Symmetry and regularity of the maps

Billiards is a dynamical system describing the motion of a particle inside a smooth closed curve tex2html_wrap_inline405 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 tex2html_wrap_inline407 and the angle of impact with tex2html_wrap_inline409 , the Poincaré map tex2html_wrap_inline411 inherits all the information of the billiard flow. The map tex2html_wrap_inline413 is a map on the annulus tex2html_wrap_inline415 preserving the Lebesgue area. The compact space X (also called the phase space) has the invariant boundary tex2html_wrap_inline419 and tex2html_wrap_inline421 . 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 tex2html_wrap_inline425 making T reversible: tex2html_wrap_inline429 .

We are studying the family of billiard maps tex2html_wrap_inline289 defined by the one-parameter family of curves tex2html_wrap_inline433 . Since the tables are symmetric by reflections at the x and y axes, this tex2html_wrap_inline439 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 tex2html_wrap_inline447 if tex2html_wrap_inline449 and tex2html_wrap_inline451 . In Fig. 2, we see numerical computations of orbits for some values of p.

We investigate first the continuity and smoothness of the maps tex2html_wrap_inline289 as well as of the map tex2html_wrap_inline457 from tex2html_wrap_inline459 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 tex2html_wrap_inline289 has at least the degree of smoothness of the normal map tex2html_wrap_inline463 at the curve tex2html_wrap_inline465 . By the tex2html_wrap_inline439 symmetry, we can restrict to the quadrant tex2html_wrap_inline469 , where the curve is given by tex2html_wrap_inline471 and where the normal is tex2html_wrap_inline473 . The map tex2html_wrap_inline289 is tex2html_wrap_inline477 if (x(s),y(s)) is away from the coordinate axes tex2html_wrap_inline481 . Denote with C(X,X) the space of homeomorphisms of X with the topology given by the metric tex2html_wrap_inline487 . Define for any compact set tex2html_wrap_inline489 and tex2html_wrap_inline491

displaymath493

and the topology on tex2html_wrap_inline495 by tex2html_wrap_inline497 if and only if tex2html_wrap_inline499 for all tex2html_wrap_inline501 and all compact tex2html_wrap_inline489 .

lemma85

Proof. A billiard map in a convex curve tex2html_wrap_inline405 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 tex2html_wrap_inline525 a continuous continuation of the normal direction field on the whole curve tex2html_wrap_inline291 . In the limiting cases p=1 and tex2html_wrap_inline327 , 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 tex2html_wrap_inline289 is real analytic. Given a curve (t,f(t)), the curvature is given by

displaymath539

This means for tex2html_wrap_inline541

displaymath543

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 tex2html_wrap_inline553 , the curvature depends continuously on p which shows that tex2html_wrap_inline557 is continuous.
End of the proof.

Remark. The map tex2html_wrap_inline559 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 tex2html_wrap_inline289 has invariant curves. Indeed, as we will see below, there are no invariant curves for p;SPMgt;2.

Next: Numerical computation of orbits Up: Billiards in the unit Previous: Introduction

Oliver Knill, Jul 10 1998