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Title:	Exponentially small asymptotic formulas for the length spectrum in some billiard tables
Author:	Martín, P.; Ramírez Ros, Rafael; Tamarit Sariol, A.
Other authors:	Universitat Politècnica de Catalunya. Departament de Matemàtiques; Universitat Politècnica de Catalunya. SD - Sistemes Dinàmics de la UPC
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 ¿ +8. 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.
Abstract:	Peer Reviewed
Subject(s):	-Àrees temàtiques de la UPC::Matemàtiques i estadística -Hamiltonian systems -Billiards -exponentially small phenomena -length spectrum -Melnikov method -numeric experiments -Sistemes hamiltonians -Classificació AMS::37 Dynamical systems and ergodic theory::37E Low-dimensional dynamical systems -Classificació AMS::35 Partial differential equations::35J Partial differential equations of elliptic type -Classificació AMS::65 Numerical analysis::65P Numerical problems in dynamical systems -Classificació AMS::52 Convex and discrete geometry::52A General convexity
Rights:	http://creativecommons.org/licenses/by-nc-nd/3.0/es/
Document type:	Article - Submitted version Article
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