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Эта публикация цитируется в 1 научной статье (всего в 1 статье)
Counting Periodic Trajectories of Finsler Billiards
Pavle V. M. Blagojevićab, Michael Harrisonc, S. Tabachnikovd, Günter M. Zieglera a Institut für Mathematik, FU Berlin, Arnimallee 2, 14195 Berlin, Germany
b Mathematical Institut SASA, Knez Mihailova 36, 11000 Beograd, Serbia
c Department of Mathematical Science, Carnegie Mellon University,
Pittsburgh, PA 15213, USA
d Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
Аннотация:
We provide lower bounds on the number of periodic Finsler billiard trajectories inside a quadratically convex smooth closed hypersurface $M$ in a $d$-dimensional Finsler space with possibly irreversible Finsler metric. An example of such a system is a billiard in a sufficiently weak magnetic field. The $r$-periodic Finsler billiard trajectories correspond to $r$-gons inscribed in $M$ and having extremal Finsler length. The cyclic group $\mathbb{Z}_r$ acts on these extremal polygons, and one counts the $\mathbb{Z}_r$-orbits. Using Morse and Lusternik–Schnirelmann theories, we prove that if $r\ge 3$ is prime, then the number of $r$-periodic Finsler billiard trajectories is not less than $(r-1)(d-2)+1$. We also give stronger lower bounds when $M$ is in general position. The problem of estimating the number of periodic billiard trajectories from below goes back to Birkhoff. Our work extends to the Finsler setting the results previously obtained for Euclidean billiards by Babenko, Farber, Tabachnikov, and Karasev.
Ключевые слова:
mathematical billiards, Finsler manifolds, magnetic billiards, Morse and Lusternik–Schnirelmann theories, unlabeled cyclic configuration spaces.
Поступила: 11 сентября 2019 г.; в окончательном варианте 25 марта 2020 г.; опубликована 3 апреля 2020 г.
Образец цитирования:
Pavle V. M. Blagojević, Michael Harrison, S. Tabachnikov, Günter M. Ziegler, “Counting Periodic Trajectories of Finsler Billiards”, SIGMA, 16 (2020), 022, 33 pp.
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/sigma1559 https://www.mathnet.ru/rus/sigma/v16/p22
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