TITLE:
Existence and non-existence of (convex) caustics

AUTHOR:
Rafael Ramirez-Ros

Departament de Matematica Aplicada I
Universitat Politecnica de Catalunya
Diagonal 647, 08028 Barcelona, Spain

E-MAIL ADDRESSES:
rafael.ramirez@upc.edu

URLs:
www.ma1.upc.edu/~rafael/

ABSTRACT:
We consider the billiard dynamics inside a planar domain ---a billiard table---
whose border is a smooth closed convex curve:
a particle follows straight lines inside the billiard table and
it is reflected at the border following the rule
``the angle of incidence equals the angle of reflection''.

There exist several negative and positive results about convex caustics.
First, we shall describe some qualitative and quantitative non-existence
theorems, which go back to Mather, Gutkin and Katok.
Next, we shall state the classical existence result of Lazutkin,
whose regularity was later improved by R. Douady.
Finally, we shall present a negative result for higher dimensional tables
found by Berger.

KEYWORDS:
caustics, convex billiards