Exponentially Small Asymptotic Formulas for the Length Spectrum in Some Billiard Tables

ABSTRACT

Let q ≥ 3 be a period. There are at least two (1, q)-periodic trajectories inside any smooth strictly convex billiard table. We quantify the chaotic dynamics of axisymmetric billiard tables close to their boundaries by studying the asymptotic behavior of the differences of the lengths of their axisymmetric (1, q)-periodic trajectories as q → +∞. 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⁻³e^−rq 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 focused on some perturbed ellipses and circles, so we can compare the numerical results with some analytical predictions obtained by Melnikov methods. We also 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.

KEYWORDS:

• billiards
• exponentially small phenomena
• length spectrum
• Melnikov method
• numeric experiments

2000 AMS Subject Classification:

• 37E40, 37J50, 65P10, 52A10

Acknowledgments

We thank T. M. Seara and C. Simó for very useful remarks and comments. We also appreciate the assistance of A. Granados and P. Roldán in the use of the UPC Applied Math cluster for our experiments. We acknowledge the use of the UPC Applied Math cluster system for research computing.

Funding

The authors were supported in part by CUR-DIUE Grant 2014SGR504 (Catalonia) and MINECO-FEDER Grant MTM2012-31714 (Spain).