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Symmetry, Integrability and Geometry: Methods and Applications, 2020, Volume 16, 022, 33 pp.
DOI: https://doi.org/10.3842/SIGMA.2020.022
(Mi sigma1559)
 

This article is cited in 1 scientific paper (total in 1 paper)

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
References:
Abstract: 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.
Keywords: mathematical billiards, Finsler manifolds, magnetic billiards, Morse and Lusternik–Schnirelmann theories, unlabeled cyclic configuration spaces.
Funding agency Grant number
Deutsche Forschungsgemeinschaft
Ministry of Education, Science and Technical Development of Serbia ON 174024
National Science Foundation DMS-1510055
Pavle V. M. Blagojević, Serge Tabachnikov, and Günter M. Ziegler were supported by the DFG via the Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”. Pavle V. M. Blagojević was supported by the grant ON 174024 of Serbian Ministry of Education and Science. Michael Harrison and Serge Tabachnikov were supported by the NSF grant DMS-1510055.
Received: September 11, 2019; in final form March 25, 2020; Published online April 3, 2020
Bibliographic databases:
Document Type: Article
MSC: 37J45; 55R80; 70H12
Language: English
Citation: Pavle V. M. Blagojević, Michael Harrison, S. Tabachnikov, Günter M. Ziegler, “Counting Periodic Trajectories of Finsler Billiards”, SIGMA, 16 (2020), 022, 33 pp.
Citation in format AMSBIB
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\totalpages 33
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  • This publication is cited in the following 1 articles:
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