The billiard ball map has been studied since the beginning of the century. Interest in it has recently increased, spurred by new mathematical techniques and the interest in the length spectrum associated to boundary value problems.The goal of this study is to better understand convex planar domains in which the billiard ball map is integrable, that is, where there is a continuous family of caustics for the billiard ball map and this family includes the boundary. The principal result is the theorem of the third chapter which shows that the only smooth convex planar curves satisfying a certain group property are ellipses.This group property and a curvature relating operator leading to both the property and to the proof of the theorem are defined in the second section and in the beginning of the third section.Also important are the motivating example of chapter 4 which shows that the relation between caustics suggested by the theorem of the third chapter is special, and the example of chapter 5 in which the lengths of caustics are completely isolated.
3The first chapter of this work introduces the billiard ball map, invariant circles, and caustics, and explores some basic relations among them. The second chapter gives a characterization of caustics and builds an operator relating caustics which is based on this characterization. This chapter explains geometric relations between caustics in analytic terms and includes the main ingredients of the theorem of the third chapter.The third chapter includes the principal calculation, giving an obstruction to satisfying the group property (also discussed in that chapter), and the fourth and fifth chapters provide relevant examples.
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The Lazutkin parameter for curves which are invariant under the billiard ball map is viewed symplectically in a way which makes it analogous to the sum of the values of a generating function over a closed orbit. This leads to relations among lengths of closed geodesies, lengths of invariant curves for the billiard map, rotation numbers, and the Lazutkin parameter. These relations establish the Birkhoff invariant and the expansion for the lengths of invariant curves in terms of the Lazutkin parameter as symplectic and spectral invariants (for the Dirichlet spectrum) and provide invariants which characterize a family of ellipses among smooth curves with positive curvature. Geodesic flow on a bounded planar region gives rise to several geometric objects among which are closed reflected geodesies and invariant curves -closed curves whose tangents are invariant under reflection at the boundary. On a bounded domain, the map that assigns to each geodesic segment its successor after reflection at the boundary is called the billiard ball map and its dual (in the cotangent bundle for the boundary) is called the boundary map.As shown by Guillemin and Melrose in [9], the lengths of closed geodesies are symplectic invariants associated with a generating function for the boundary map. Moreover, in a convex planar domain Marvizi and Melrose defined the wave invariants (see [12]), obtained via the interpolating hamiltonian of [14] which takes account of the singularity of the boundary map at the boundary. These wave invariants are symplectic and spectral invariants when viewed as functions of the rotation number associated to closed geodesies, but they do not have, as a whole, direct dynamical or geometric interpretations.In this paper a new set of invariants, the caustics' invariants, are introduced. They are the lengths of invariant curves viewed as functions of the Lazutkin parameter of [11]. The Lazutkin parameter has a dynamical interpretation obtained by modifying a generating function for the boundary map.Using these symplectic invariants one can isolate ellipses among planar domains whose curvature is strictly positive. In Sect. 6 it is shown that the caustics' invariants can be obtained from the wave invariants and the lengths of closed geodesies, and
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