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

SIGMA 16 (2020), 022, 33 pages      arXiv:1712.07930      https://doi.org/10.3842/SIGMA.2020.022
Contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry Fuchs

Counting Periodic Trajectories of Finsler Billiards

Pavle V.M. Blagojević ab, Michael Harrison c, Serge Tabachnikov d and Günter M. Ziegler a
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

Received September 11, 2019, in final form March 25, 2020; Published online April 03, 2020

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.

Key words: mathematical billiards; Finsler manifolds; magnetic billiards; Morse and Lusternik-Schnirelmann theories; unlabeled cyclic configuration spaces.

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