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Texas A&M University

Geometry Seminar

Date: February 14, 2020

Time: 4:00PM - 5:00PM

Location: BLOC 628

Speaker: Vladimir Dragovic, UT Dallas


Title: Periodic ellipsoidal billiards and Chebyshev polynomials on several intervals

Abstract: A comprehensive study of periodic trajectories of the billiards within ellipsoids in the d-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between the periodic billiard trajectories and the extremal polynomials of the Chebyshev type on the systems of d intervals on the real line. As a byproduct, for d = 2 a new proof of the monotonicity of the rotation number is obtained and the result is generalized for any d. The case study of trajectories of small periods T, d < T < 2d is given. In particular, it is proven that all d-periodic trajectories are contained in a coordinate-hyperplane and that for a given ellipsoid, there is a unique set of caustics which generates d + 1-periodic trajectories. A complete catalog of billiard trajectories with small periods is provided for d = 3. This  talk is based on: V. Dragovic, M. Radnovic, Periodic ellipsoidal billiard trajectories and extremal poly- nomials, arXiv 1804.02515, Comm. Math. Physics. 019-03552-y, 2019, Vol. 372, p. 183-211. V. Dragovic, M. Radnovic, Caustics of Poncelet polygons and classical extremal polynomials, arXiv 1812.02907, Regular and Chaotic Dynamics, (2019), Vol. 24, No. 1, p. 1-35. A. Adabrah, V. Dragovic, M. Radnovic, Periodic billiards within conics in the Min- kowski plane and Akhiezer polynomials, arXiv; 1906.0491, Regular and Chaotic Dy- namics, No. 5, Vol. 24, 2019, p. 464-501.