Magic billiards: the case of elliptic boundaries

V. Dragović^ab, M. Radnović<sup>cbd

a</sup> Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX, USA
^b Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Republic of Serbia
^c School of Mathematics and Statistics, University of Sydney, Sydney, Australia
^d University of New South Wales, Sydney, Australia

Abstract: We introduce a novel concept of magic billiards, which can be viewed as an umbrella unifying several well-known generalisations of mathematical billiards. We analyse the properties of magic billiards in the case of elliptic boundaries. We provide explicit conditions for periodicity in algebro-geometric, analytic and polynomial forms. A topological description of these billiards is given using Fomenko graphs.
Bibliography: 45 titles.

Keywords: elliptical billiards, topological billiards, Fomenko graphs, periodic trajectories, divisors on elliptic curves, polynomial Pell equations.

MSC: 37C83, 37J35, 37J39

Received: 27.08.2024 and 18.02.2025

DOI: 10.4213/sm10181

 English version: Sbornik: Mathematics, 2025, 216:5, 654–673

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