Counting Periodic Trajectories of Finsler Billiards

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.

References in corpus (3)

• On the category of Euclidean configuration spaces and associated fibrations
• Remarks on magnetic flows and magnetic billiards, Finsler metrics and a magnetic analog of Hilbert's fourth problem
• On Totally integrable magnetic billiards on constant curvature surface

Cited by in corpus (2)

• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems
• Billiards in generic convex bodies have positive topological entropy