On the Density of Dispersing Billiard Systems with Singular Periodic Orbits

TL;DR

This paper investigates the density of dispersing billiard systems with singular periodic orbits, providing partial results towards a conjecture that such systems are dense in the space of all dispersing billiards, with implications for ergodicity.

Contribution

It offers a partial proof that systems with near-singular periodic orbits can be perturbed to have actual singular periodic orbits in certain billiard tables.

Findings

If a system has a near-singular periodic orbit satisfying specific conditions, it can be perturbed to admit a singular periodic orbit.

The paper discusses the assumptions needed to fully prove the conjecture of Turaev and Rom-Kedar.

Provides insights into the structure of dispersing billiards with singular orbits.

Abstract

Dynamical billiards, or the behavior of a particle traveling in a planar region $D$ undergoing elastic collisions with the boundary, has been extensively studied and is used to model many physical phenomena such as a Boltzmann gas. Of particular interest are the dispersing billiards, where $D$ consists of a union of finitely many open convex regions. These billiard flows are known to be ergodic and to possess the $K$-property. However, Turaev and Rom-Kedar (1998) proved that for dispersing systems permitting singular periodic orbits, there exists a family of smooth Hamiltonian flows with regions of stability near such orbits, converging to the billiard flow. They conjecture that systems possessing such singular periodic orbits are dense in the space of all dispersing billiard systems and remark that if this conjecture is true then every dispersing billiard system is arbitrarily close to a non-ergodic smooth Hamiltonian flow with regions of stability. In this paper, we consider billiard tables consisting of the complement to a union of open unit disks with disjoint closures. We present a partial solution to this conjecture by showing that if the system possesses a near-singular periodic orbit satisfying certain conditions, then it can be perturbed to a system that permits a singular periodic orbit. We comment on the assumptions of our theorem that must be removed to prove the conjecture of Turaev and Rom-Kedar for these systems.