Exponentially small asymptotic formulas for the length spectrum in some billiard tables

TL;DR

This paper investigates the asymptotic behavior of length differences of periodic trajectories in convex billiard tables, revealing exponentially small formulas and conjecturing their form based on numerical experiments near ellipses and circles.

Contribution

It introduces a conjecture on the exponential asymptotics of length differences in axisymmetric billiards, supported by numerical experiments and analytical predictions.

Findings

Length differences decay exponentially with q

Asymptotic behavior involves an exponential factor and oscillations

Numerical results align with Melnikov method predictions

Abstract

Let $q \ge 3$ be a period. There are at least two $(1,q)$-periodic trajectories inside any smooth strictly convex billiard table, and all of them have the same length when the table is an ellipse or a circle. We quantify the chaotic dynamics of axisymmetric billiard tables close to their borders by studying the asymptotic behavior of the differences of the lengths of their axisymmetric $(1,q)$-periodic trajectories as $q \to +\infty$. Based on numerical experiments, we conjecture that, if the billiard table is a generic axisymmetric analytic strictly convex curve, then these differences behave asymptotically like an exponentially small factor $q^{-3} e^{-r q}$ times either a constant or an oscillating function, and the exponent $r$ is half of the radius of convergence of the Borel transform of the well-known asymptotic series for the lengths of the $(1,q)$-periodic trajectories. Our experiments are restricted to some perturbed ellipses and circles, which allows us to compare the numerical results with some analytical predictions obtained by Melnikov methods and also to detect some non-generic behaviors due to the presence of extra symmetries. Our computations require a multiple-precision arithmetic and have been programmed in PARI/GP.