We consider a generalized type of unique factorization of the positive integers with restrictions on the exponents and view them as a family of arithmetic convolutions and divisibility relations, similar to the convolutions defined by Narkewicz [On a class of arithmetical convolutions, Colloq. Math. 10 (1963) 81–94]. We introduce special types of multiplicativity corresponding to these convolutions, and discuss algebraic properties of the associated arithmetic convolutions and analogs of the Möbius functions. We also prove asymptotics for analogs of the totient function, totient summatory function, and divisor summatory function.

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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.