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Periodic ellipsoidal billiard trajectories and extremal polynomials. (English) Zbl 1432.37056
Authors’ abstract: A comprehensive study of periodic trajectories of billiards within ellipsoids in \(d\)-dimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between periodic billiard trajectories and extremal polynomials on the systems of \(d\) intervals on the real line. By leveraging deep, but yet not widely known results of the Krein-Levin-Nudelman theory of generalized Chebyshev polynomials [M. G. Kreĭn et al., in: Functional analysis, optimization, and mathematical economics. A collection of papers dedicated to the memory of L. V. Kantorovich. New York: Oxford University Press. 56–114 (1990; Zbl 0989.41505)], fundamental properties of billiard dynamics are proven for any \(d\), viz., that the sequences of winding numbers are monotonic. By emploing the potential theory we prove the injectivity of the frequency map. As a byproduct, for \(d=2\) a new proof of the monotonicity of the rotation number is obtained. The case study of trajectories of small periods \(T, d\leq T \leq 2d\) is given. In particular, it is proven that all 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 \(d=3\).”
The proven properties of the billiard dynamics answer positively all the conjectures from the paper by R. Ramírez-Ros [Nonlinearity 27, No. 5, 1003–1028 (2014; Zbl 1317.37063)].

MSC:
37C83 Dynamical systems with singularities (billiards, etc.)
37C27 Periodic orbits of vector fields and flows
33C47 Other special orthogonal polynomials and functions
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