arXiv subjects

Otto Vaughn Osterman

Publications and source records attributed to Otto Vaughn Osterman.

3 recordsLinked to original sources

arXiv2026

On the Factor Complexity Associated with a Family of Multidimensional Continued Fraction Algorithms

We study the complexity of $S$-adic sequences corresponding to a family of 216 multidimensional continued fractions maps, called Triangle Partition maps (TRIP maps), with an emphasis on those with low upper bounds on complexity. Our main result is to prove that the complexity of $S$-adic sequences corresponding to the triangle map (called the $(e,e,e)$-TRIP map in this paper) has upper bound at most $3n$. Our second main result is to prove an upper bound of $2n+1$ on complexity for another TRIP map. We discuss a dynamical phenomenon, which we call ``hidden $\R^2$ behavior,'' that occurs in this map and its relationship to complexity. Combining this with previously known results and a list of counter-examples, we provide a complete list of the TRIP maps which have upper bounds on complexity of at most $3n$, except for one remaining case for which we conjecture such an upper bound to hold.

math.DS↗

arXiv2022

On Length Spectrum Rigidity of Dispersing Billiard Systems

We consider the class of dispersing billiard systems in the plane formed by removing three convex analytic scatterers satisfying the non-eclipse condition. The collision map in this system is conjugated to a subshift, providing a natural labeling of periodic points. We study the problem of marked length spectrum rigidity for this class of systems. We show that two such systems have the same marked length spectrum if and only if their collision maps are analytically conjugate to each other near a homoclinic orbit and that two scatterers and the marked length spectrum together uniquely determine the third scatterer.

math.DS↗

arXiv2019

On the Density of Dispersing Billiard Systems with Singular Periodic Orbits

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.

math.DS↗