Pseudo-integrable billiards and arithmetic dynamics

TL;DR

This paper introduces a new class of billiard systems with boundaries formed by confocal conic arcs containing reflex angles, revealing unique dynamical and topological properties influenced by arithmetic conditions rather than geometry.

Contribution

It develops a comprehensive study of pseudo-integrable billiards with reflex angles, including invariant leaves of higher genus, a local Poncelet theorem, and connections to interval exchange transformations.

Findings

Invariant leaves of higher genus are induced by reflex angles.

Dynamical behavior depends on arithmetic of rotation numbers, not geometry.

A local Poncelet theorem and conditions for periodicity are established.

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 behaviour different from Liouville-Arnold's theorem. Its analogue is derived from the Maier theorem on measured foliations. A local version of Poncelet theorem is formulated and necessary algebro-geometric conditions for periodicity are presented. The connection with interval exchange transformation is established together with Keane's type conditions for minimality. It is proved that the dynamics depends on arithmetic of rotation numbers, but not on geometry of a given confocal pencil of conics.