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 - 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

SN - 1815-0659

VL - 16

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

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

M1 - 022

ER -