Gregory Galperin

600 Lincoln Avenue, Charleston IL, 61920-3099
Depatment of Mathematics, Eastern Illinois University


Albeverio S., Galperin G. A., Nizhnik I. L., Nizhnik L. P.
A constructive description of generalized billiards is given, the billiards being inside an infinite strip with a periodic law of reflection off the strip's bottom and top boundaries. Each of the boundaries is equipped with the same periodic lattice, where the number of lattice's nodes between any two successive reflection points may be prescribed arbitrarily. For such billiards, a full description of the structure of the set of billiard trajectories is provided, the existence of spatial chaos is found, and the exact value of the spatial entropy in the class of monotonic billiard trajectories is found.
Keywords: billiards, dynamical systems, spatial chaos, entropy
Citation: Albeverio S., Galperin G. A., Nizhnik I. L., Nizhnik L. P.,  Generalized billiards inside an infinite strip with periodic laws of reflection along the strip's boundaries , Regular and Chaotic Dynamics, 2005, vol. 10, no. 3, pp. 285-306
DOI: 10.1070/RD2005v010n03ABEH000316
Galperin G. A.
Counting collisions in a simple dynamical system with two billiard balls can be used to estimate $\pi$ to any accuracy.
Citation: Galperin G. A.,  Playing pool with $\pi$ (the number $\pi$ from a billiard point of view), Regular and Chaotic Dynamics, 2003, vol. 8, no. 4, pp. 375-394
Galperin G. A.
A relationship between the three metrics — Billiard, Euclidean, and Lobachevskian (Hyperbolic) — is established in the article. This relationship is applied to a billiard problem on generalized diagonals of a Euclidean multidimensional convex polyhedron.
Citation: Galperin G. A.,  Relationship between Euclidean, Lobachevskian (hyperbolic), and billiard metrics and its application to a billiard problem in $R^d$, Regular and Chaotic Dynamics, 2003, vol. 8, no. 4, pp. 441-448
Chernov N., Galperin G. A.
Search light in billiard tables
2003, vol. 8, no. 2, pp.  225-241
We investigate whether a search light, $S$, illuminating a tiny angle ("cone") with vertex $A$ inside a bounded region $Q \in \mathbb{R}^2$ with the mirror boundary $\partial Q$, will eventually illuminate the entire region $Q$. It is assumed that light rays hitting the corners of $Q$ terminate. We prove that: $(\mathbf{I})$ if $Q =$ a circle or an ellipse, then either the entire $Q$ or an annulus between two concentric circles/confocal ellipses (one of which is $\partial Q$) or the region between two confocal hyperbolas will be illuminated; $(\mathbf{II})$ if $Q =$ a square, or $(\mathbf{III})$ if $Q =$ a dispersing (Sinai) or semidespirsing billiards, then the entire region $Q$ is will be illuminated.
Citation: Chernov N., Galperin G. A.,  Search light in billiard tables, Regular and Chaotic Dynamics, 2003, vol. 8, no. 2, pp. 225-241
Galperin G. A., Zvonkine D.
Periodic Billiard Trajectories in Right Triangle
2003, vol. 8, no. 1, pp.  29-44
This paper is dedicated to periodic billiard trajectories in right triangles and right-angled tetrahedra. We construct a specific type of periodic trajectories and show that the trajectories of this type fill the right triangle entirely. Then we establish the instability of all known types of periodic trajectories in right triangles. Finally, some of these results are generalized to the $n$-dimensional case and are given a mechanical interpretation.
Citation: Galperin G. A., Zvonkine D.,  Periodic Billiard Trajectories in Right Triangle, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 29-44
Galperin G. A.
Convex Polyhedra without Simple Closed Geodesics
2003, vol. 8, no. 1, pp.  45-58
It is proved that, in contrast to the smooth convex surfaces homeomorphic to the two-dimensional sphere, the most of convex polyhedra in three dimensional space are free of closed geodesics without self-intersections.
Citation: Galperin G. A.,  Convex Polyhedra without Simple Closed Geodesics, Regular and Chaotic Dynamics, 2003, vol. 8, no. 1, pp. 45-58

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