  We consider the free motion of a point particle inside a circular billiard
with periodically moving boundary, with the assumption that the collisions of
the particle with the boundary are elastic so that the energy of the particle
is not preserved. It is known that if the motion of the boundary is regular
enough then the energy is bounded due to the existence of invariant curves. We
show that it is nevertheless possible that the motion of the particle is
chaotic, also under regularity assumptions for the moving boundary. More
precisely, we show that there exists a class of functions describing the motion
of the boundary for which the billiard map admits invariant probability
measures with positive metric entropy. The proof relies on variational
techniques based on Aubry-Mather theory.
