Journal of Modern Dynamics (JMD)

Pseudo-integrable billiards and arithmetic dynamics

Pages: 109 - 132, Issue 1, March 2014      doi:10.3934/jmd.2014.8.109

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Vladimir Dragović - Department of Mathematical Sciences, University of Texas at Dallas, FO 35, 800West Campbell Road, TX 75080 USA, Mathematical Institute SANU, Kneza Mihaila 36, Belgrade, Serbia (email)
Milena Radnović - Mathematical Institute SANU, Kneza Mihaila 36, Belgrade, Serbia (email)

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.

Keywords:  Confocal conics, Poncelet theorem, periodic billiard trajectories, interval exchange, polygonal billiards, measured foliations.
Mathematics Subject Classification:  Primary: 37D50, 37J35, 37A05; Secondary: 28D05.

Received: September 2013;      Available Online: July 2014.