Singular phenomena in the length spectrum of analytic convex curves

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.Considereu l'aplicació billard definida dins d'una corba tancada, analítica i estrictament convexa Q. Per q>2 i 00 és petit i contínu i les fórmules són exponencialment petites en 1/h. S'ha demostrat (o està recolzat fortament per experiments numèriques) que l'exponent r és 2Pi vegades la distància a l'eix real del conjunt de singularitats complexes de la solució homoclínica del flux d'un Hamiltonià límit. Proposem i estudiem un equivalent a problema límit per l'aplicació billar. A continuació, comentem com es comporta D(p,q) per òrbites (p,q)-periòdiques que tendeixen a regions de l'espai de fases diferents de la frontera de Q. En concret, considerem els casos de p/q tendint a un nombre irracional o a P/Q. L'estudi de D(p,q) en aquests casos es basa en un estudi numèric dels fenòmens