2024-03-28T17:08:44Zhttps://www.tdx.cat/oai/requestoai:www.tdx.cat:10803/3183722017-08-30T04:30:32Zcom_10803_183col_10803_219
nam a 5i 4500
Billiards
Length spectrum
Exponentially small phenomena
Twist maps
Numerical experiments
High-precision computations
Dual billiards
Singular phenomena in the length spectrum of analytic convex curves
[Barcelona] :
Universitat Politècnica de Catalunya,
2015
Accés lliure
http://hdl.handle.net/10803/318372
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Tamarit Sariol, Anna,
autor
1 recurs en línia (128 pàgines)
Tesi
Doctorat
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I
2015
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I
Tesis i dissertacions electròniques
Ramírez Ros, Rafael,
supervisor acadèmic
Martín de la Torre, Pablo,
supervisor acadèmic
TDX
Consider the billiard map defined inside an analytic closed strictly convex curve Q. Given q>2 and 0<p<q relatively prime integers, there exist at least two (p,q)-periodic trajectories inside Q. The main goal of this thesis is to study the maximal difference of lengths among (p,q)-periodic trajectories on the billiard, D(p,q).
The quantity D(p,q) gives some dynamical and geometrical information. First, it characterizes part of the length spectrum of Q and so it relates to Kac's question, "Can one hear the shape of a drum?''. Second, D(p,q) is an upper bound of Mather's DW(p/q) and so it quantifies the chaotic dynamics of the billiard table.
We first focus on the study of the maximal difference of lengths among (1,q)-periodic orbits. These orbits approach the boundary of the billiard table as q tends to infinity. The study of D(1,q) is twofold.
On the one hand, we obtain an exponentially small upper bound in the period q for D(1,q). The result is obtained on the general
framework of the maximal difference of (p,q)-periodic actions among (p,q)-periodic orbits on analytic exact twist maps.
Precisely, we establish an exponentially small upper bound for differences of (p,q)-periodic actions when the map is analytic on a (m,n)-resonant rotational invariant curve and p/q is ``sufficiently close'' to m/n. The exponent in the upper bound is closely related to the analyticity strip width of a suitable angular variable. The result is obtained in two steps. First, we prove a Neishtadt-like theorem. Second, we apply the MacKay-Meiss-Percival action principle.
This result implies that the lengths of all the (1,q)-periodic billiard trajectories inside analytic strictly convex domains are exponentially close in the period q, which improves the classical result of Marvizi and Melrose about the smooth case. But it also has several other applications in both classical and dual billiards. For instance, we show that the areas of the (1,q)-periodic dual billiard trajectories outside Q are exponentially close in the period q. This result improves Tabachnikov's classical result about the smooth case.
On the other hand, we discuss some exponentially small asymptotic formulas for D(1,q) when the billiard table is a generic axisymmetric
analytic strictly convex curve. In this context, we conjecture that the differences behave asymptotically like an exponentially small factor q^(-3)*exp(-rq) times either a constant or an oscillatory function. Also, 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. This conjecture is strongly supported by numerical experiments. Our computations require a multiple-precision arithmetic and we have used PARI/GP.
The experiments are restricted to some perturbed ellipses and circles, which allow us to compare the numerical results with some analytical Melnikov predictions and also to detect some non-generic behaviors due to the presence of extra symmetries.
The asymptotic formulas we obtain resemble the ones obtained for the splitting of separatrices on many analytic maps, where the behavior of the splitting size is of order h(-m)*exp(-r/h). In such cases, the parameter h>0 is small and continuous so the formulas are exponentially small in 1/h instead. The exponent r has been proved to be (or is strongly numerically supported, depending on the map studied) 2pi times the distance to the real axis of the set of complex singularities of the homoclinic solution of a limit Hamiltonian flow. We propose and study an equivalent limit problem in the billiard setting.
Next, we give some insight on how D(p,q) behaves when (p,q)-periodic orbits do not tend to the boundary of Q but to other regions of the phase space. Namely, we consider the cases of p/q tends to an irrational number or to P/Q. The study of D(p,q) in these cases consists of a phenomenological study based on some numerical results.
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