Authors: Dragović, Vladimir 
Radnović, Milena
Title: Pseudo-integrable billiards and arithmetic dynamics
Journal: Journal of Modern Dynamics
Volume: 8
Issue: 1
First page: 109
Last page: 132
Issue Date: 1-Jan-2014
Rank: M21
ISSN: 1930-5311
DOI: 10.3934/jmd.2014.8.109
We introduce a new class of billiard systems in the plane, with boundaries formed by finitely many arcs of confocal conics such that they contain some reflex angles. Fundamental dynamical, topological, geometric, and arithmetic properties of such billiards are studied. The novelty, caused by reflex angles on boundary, induces invariant leaves of higher genera and dynamical behavior different from Liouville-Arnold's Theorem. Its analog is derived from the Maier Theorem on measured foliations. The billiard flow generates a measurable foliation defined by a closed 1-form w. Using the closed form, a transformation of the given billiard table to a rectangular cylinder is constructed and a trajectory equivalence between corresponding billiards has been established. A local version of Poncelet Theorem is formulated and necessary algebro-geometric conditions for periodicity are presented. It is proved that the dynamics depends on arithmetic of rotation numbers, but not on geometry of a given confocal pencil of conics.
Keywords: Confocal conics | Interval exchange | Measured foliations | Periodic billiard trajectories | Polygonal billiards | Poncelet theorem
Publisher: American Institute of Mathematical Sciences

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