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Finiteness in polygonal billiards on hyperbolic plane. (English) Zbl 1490.37039

Summary: J. Hadamard [Journ. de Math. (5) 4, 27–73 (1898; JFM 29.0522.01)] 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 Typ’ 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.

MSC:

37C83 Dynamical systems with singularities (billiards, etc.)
37B10 Symbolic dynamics
37B51 Multidimensional shifts of finite type
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)

Citations:

JFM 29.0522.01

References:

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