Counting Periodic Trajectories of Finsler Billiards
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...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2020 |
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| Language: | English |
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Інститут математики НАН України
2020
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210588 |
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| Cite this: | Counting Periodic Trajectories of Finsler Billiards. Pavle V.M. Blagojević, Michael Harrison, Serge Tabachnikov and Günter M. Ziegler. SIGMA 16 (2020), 022, 33 pages |
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Blagojević, Pavle V.M. Harrison, Michael Tabachnikov, Serge Ziegler, Günter M. 2025-12-12T10:31:40Z 2020 Counting Periodic Trajectories of Finsler Billiards. Pavle V.M. Blagojević, Michael Harrison, Serge Tabachnikov and Günter M. Ziegler. SIGMA 16 (2020), 022, 33 pages 1815-0659 2020 Mathematics Subject Classification: 37J45; 55R80; 70H12 arXiv:1712.07930 https://nasplib.isofts.kiev.ua/handle/123456789/210588 https://doi.org/10.3842/SIGMA.2020.022 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 ℤᵣ acts on these extremal polygons, and one counts the ℤᵣ-orbits. Using Morse and Lusternik-Schnirelmann theories, we prove that if 𝑟 ≥ 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. We are grateful to Sergei Ivanov for useful discussions on Finsler geometry, and we are grateful to the following sources of funding. Pavle V. M. Blagojevic, Serge Tabachnikov, and Gunter M. Ziegler were supported by the DFG via the Collaborative Research Center TRR 109 Discretization in Geometry and Dynamics. Pavle V. M. Blagojevic was supported by the grant ON 174024 of the Serbian Ministry of Education and Science. Michael Harrison and Serge Tabachnikov were supported by the NSF grant DMS-1510055. We are also grateful to the referees for their suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Counting Periodic Trajectories of Finsler Billiards Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Counting Periodic Trajectories of Finsler Billiards |
| spellingShingle |
Counting Periodic Trajectories of Finsler Billiards Blagojević, Pavle V.M. Harrison, Michael Tabachnikov, Serge Ziegler, Günter M. |
| title_short |
Counting Periodic Trajectories of Finsler Billiards |
| title_full |
Counting Periodic Trajectories of Finsler Billiards |
| title_fullStr |
Counting Periodic Trajectories of Finsler Billiards |
| title_full_unstemmed |
Counting Periodic Trajectories of Finsler Billiards |
| title_sort |
counting periodic trajectories of finsler billiards |
| author |
Blagojević, Pavle V.M. Harrison, Michael Tabachnikov, Serge Ziegler, Günter M. |
| author_facet |
Blagojević, Pavle V.M. Harrison, Michael Tabachnikov, Serge Ziegler, Günter M. |
| publishDate |
2020 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
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 ℤᵣ acts on these extremal polygons, and one counts the ℤᵣ-orbits. Using Morse and Lusternik-Schnirelmann theories, we prove that if 𝑟 ≥ 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.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/210588 |
| citation_txt |
Counting Periodic Trajectories of Finsler Billiards. Pavle V.M. Blagojević, Michael Harrison, Serge Tabachnikov and Günter M. Ziegler. SIGMA 16 (2020), 022, 33 pages |
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| first_indexed |
2025-12-17T12:03:33Z |
| last_indexed |
2025-12-17T12:03:33Z |
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