Pseudo-integrable billiards and arithmetic dynamics

Abstract

We introduce a new class of billiard systems in the plane, withboundaries formed by finitely many arcs of confocal conics such thatthey 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 invariantleaves of higher genera and dynamical behavior different fromLiouville--Arnold's Theorem. Its analog is derived from the MaierTheorem on measured foliations. The billiard flow generates ameasurable foliation defined by a closed 1-form $w$. Using theclosed form, a transformation of the given billiard table to arectangular cylinder is constructed and a trajectory equivalencebetween corresponding billiards has been established. A local versionof Poncelet Theorem is formulated and necessary algebro-geometricconditions for periodicity are presented. It is proved that thedynamics depends on arithmetic of rotation numbers, but not ongeometry of a given confocal pencil of conics.