Counting Periodic Trajectories of Finsler Billiards

TL;DR

This paper establishes lower bounds on the number of periodic billiard trajectories inside convex hypersurfaces in Finsler spaces, extending classical Euclidean results to more general Finsler geometries using Morse and Lusternik-Schnirelmann theories.

Contribution

It provides the first lower bounds for the count of periodic Finsler billiard trajectories, generalizing known Euclidean billiard results to Finsler settings with possibly irreversible metrics.

Findings

Lower bounds of (r-1)(d-2)+1 for prime r

Extension of Euclidean billiard results to Finsler spaces

Stronger bounds in general position cases

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.