Non-Existence of Invariant Curves for Some Billiard Systems

A mathematical billiard consists of a planar region as a table and a
ball moving in the table and reflecting on the boundary. By
considering the moving as a discrete dynamical system, the billiard
map is an area-preserving twist map. In this thesis, we show the
result of non-existence of invariant curves for two different billiard
maps. The first result presents the sufficient condition of the
non-existence of invariant curves. In the second one, we prove the
non-existence of invariant curves near the boundary for Halpern’s
billiard, which has a convergent billiard orbit.