Finiteness in polygonal billiards on hyperbolic plane

Anima Nagar, Pradeep Singh

Abstract: J. Hadamard studied the geometric properties of geodesic flows on surfaces of negative curvature, thus initiating ``Symbolic Dynamics". In this article, we follow the same geometric approach to study the geodesic trajectories of billiards in ``rational polygons'' on the hyperbolic plane. We particularly show that the billiard dynamics resulting thus are just `Subshifts of Finite Type' or their dense subsets. We further show that `Subshifts of Finite Type' play a central role in subshift dynamics and while discussing the topological structure of the space of all subshifts, we demonstrate that they approximate any shift dynamics.