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

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 axisy...

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Vydáno v:	Experimental mathematics Ročník 25; číslo 4; s. 416 - 440
Hlavní autoři:	Martín, P., Ramírez-Ros, R., Tamarit-Sariol, A.
Médium:	Journal Article Publikace
Jazyk:	angličtina
Vydáno:	Taylor & Francis 01.10.2016
Témata:	35 Partial differential equations 35J Partial differential equations of elliptic type 37 Dynamical systems and ergodic theory 37E Low-dimensional dynamical systems 37E40, 37J50, 65P10, 52A10 52 Convex and discrete geometry 52A General convexity 65 Numerical analysis 65P Numerical problems in dynamical systems Billiards Classificació AMS exponentially small phenomena Hamiltonian systems length spectrum Matemàtiques i estadística Melnikov method numeric experiments Sistemes hamiltonians Àrees temàtiques de la UPC
ISSN:	1058-6458, 1944-950X
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Shrnutí:	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 −3 e −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.
ISSN:	1058-6458 1944-950X
DOI:	10.1080/10586458.2015.1076361