Counting Periodic Trajectories of Finsler Billiards

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2020
• Main Authors: Blagojević, Pavle V.M., Harrison, Michael, Tabachnikov, Serge, Ziegler, Günter M.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2020
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/210588
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• 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

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.