TY - JOUR

T1 - Counting periodic trajectories of finsler billiards

AU - Blagojević, Pavle V.M.

AU - Harrison, Michael

AU - Tabachnikov, Serge

AU - Ziegler, Günter M.

N1 - Funding Information:
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. Blagojević, Serge Tabachnikov, and Günter M. Ziegler were supported by the DFG via the Collaborative Research Center TRR 109 “Dis-cretization in Geometry and Dynamics”. Pavle V. M. Blagojevićwas 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. We are also grateful to the referees for their suggestions.
Publisher Copyright:
© 2020, Institute of Mathematics. All rights reserved.

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

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U2 - 10.3842/SIGMA.2020.022

DO - 10.3842/SIGMA.2020.022

M3 - Article

AN - SCOPUS:85085891461

VL - 16

JO - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

JF - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SN - 1815-0659

M1 - 022

ER -