2019/10/22 by Otto Vaughn Osterman, Osterman, Otto Vaughn
Mathematics · Physics and Astronomy · #37D40 #37D50 #37J25 #70B05 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1910.10290
openalex publication_date 2019/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Dynamical billiards, or the behavior of a particle traveling in a planar\nregion D undergoing elastic collisions with the boundary, has been\nextensively studied and is used to model many physical phenomena such as a\nBoltzmann gas. Of particular interest are the dispersing billiards, where D\nconsists of a union of finitely many open convex regions. These billiard flows\nare known to be ergodic and to possess the K-property. However, Turaev and\nRom-Kedar (1998) proved that for dispersing systems permitting singular\nperiodic orbits, there exists a family of smooth Hamiltonian flows with regions\nof stability near such orbits, converging to the billiard flow. They conjecture\nthat systems possessing such singular periodic orbits are dense in the space of\nall dispersing billiard systems and remark that if this conjecture is true then\nevery dispersing billiard system is arbitrarily close to a non-ergodic smooth\nHamiltonian flow with regions of stability. In this paper, we consider billiard\ntables consisting of the complement to a union of open unit disks with disjoint\nclosures. We present a partial solution to this conjecture by showing that if\nthe system possesses a near-singular periodic orbit satisfying certain\nconditions, then it can be perturbed to a system that permits a singular\nperiodic orbit. We comment on the assumptions of our theorem that must be\nremoved to prove the conjecture of Turaev and Rom-Kedar for these systems.\n