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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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