ABSTRACT:

A class of integrable billiard tables called Liouvile billiard tables will be considered. A typical example is the billiard inside the ellipsoid. By a billiard table we mean a smooth compact Riemannian manifold with a smoth boundary. The corresponding ``continuous'' dynamical system on it is the ``billiard flow'' which induces a discrete dynamical system on an open subset of the coball bundle of the boundary called billiard ball map. The following problem will be discussed: Let K be a continuous function on the boundary such the mean value of K on any periodic orbit of the billiard ball map is zero. Does it imply K=0? A Radon transform will be introduced and it will be shown that it is one-to-one for a suitable class of continuous functions on a Liouville billiard table.