Pseudo-integrable billiards and arithmetic dynamics

Vladimir Dragović; Milena Radnović

doi:10.3934/jmd.2014.8.109

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2014, Volume 8, Issue 1: 109-132. Doi: 10.3934/jmd.2014.8.109

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Pseudo-integrable billiards and arithmetic dynamics

Vladimir Dragović^1, and
Milena Radnović^2,

1.
Department of Mathematical Sciences, University of Texas at Dallas, FO 35, 800West Campbell Road, TX 75080 USA, Mathematical Institute SANU, Kneza Mihaila 36, Belgrade
2.
Mathematical Institute SANU, Kneza Mihaila 36, Belgrade

Received: August 31, 2013

Published: June 2014

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Abstract

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.

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Mathematics Subject Classification: Primary: 37D50, 37J35, 37A05; Secondary: 28D05.

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