On the length and area spectrum of analytic convex domains

Area-preserving twist maps have at least two different $(p,q)$-periodic orbits and every $(p,q)$-periodic orbit has its $(p,q)$-periodic action for suitable couples $(p,q)$. We establish an exponentially small upper bound for the differences of $(p,q)$-periodic actions when the map is analytic on a $(m,n)$-resonant rotational invariant curve (resonant RIC) and $p/q$ is "sufficiently close" to $m/n$. The exponent in this 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, in which the $n$-th power of the twist map is written as an integrable twist map plus an exponentially small remainder on the distance to the RIC. Second, we apply the MacKay-Meiss-Percival action principle. We apply our exponentially small upper bound to several billiard problems. The resonant RIC is a boundary of the phase space in almost all of them. For instance, we show that the lengths (respectively, areas) of all the $(1,q)$-periodic billiard (respectively, dual billiard) trajectories inside (respectively, outside) analytic strictly convex domains are exponentially close in the period $q$. This improves some classical results of Marvizi, Melrose, Colin de Verdi\`ere, Tabachnikov, and others about the smooth case

Exponentially small asymptotic formulas for the length spectrum in some billiard tables

Let $q \ge 3$ be a period. There are at least two $(1,q)$-periodic trajectories inside any smooth strictly convex billiard table, and all of them have the same length when the table is an ellipse or a circle. We quantify the chaotic dynamics of axisymmetric billiard tables close to their borders by studying the asymptotic behavior of the differences of the lengths of their axisymmetric $(1,q)$-periodic trajectories as $q \to +\infty$. 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 $q^{-3} e^{-r q}$ 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 restricted to some perturbed ellipses and circles, which allows us to compare the numerical results with some analytical predictions obtained by Melnikov methods and also to 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.Comment: 21 pages, 37 figure

On the length and area spectrum of analytic convex domains

Area-preserving twist maps have at least two different (p, q)-periodic orbits and every (p, q)-periodic orbit has its (p, q)-periodic action for suitable couples (p, q). We establish an exponentially small upper bound for the differences of (p, q)-periodic actions when the map is analytic on a (m, n)-resonant rotational invariant curve (resonant RIC) and p/q is 'sufficiently close' to m/n. The exponent in this 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, in which the n-th power of the twist map is written as an integrable twist map plus an exponentially small remainder on the distance to the RIC. Second, we apply the MacKay-Meiss-Percival action principle. We apply our exponentially small upper bound to several billiard problems. The resonant RIC is a boundary of the phase space in almost all of them. For instance, we show that the lengths (respectively, areas) of all the (1, q)-periodic billiard (respectively, dual billiard) trajectories inside (respectively, outside) analytic strictly convex domains are exponentially close in the period q. This improves some classical results of Marvizi, Melrose, Colin de Verdiere, Tabachnikov, and others about the smooth case.Peer ReviewedPostprint (author's final draft

On Birkhoff's conjecture about convex billiards

Birkhoff conjectured that the elliptic billiard was the only integrable convex billiard. Here we prove a local version of this conjecture: any non-trivial symmetric entire perturbation of an elliptic billiard is non-integrable

Singular separatrix splitting and the Poincare-Melnikov method for area preserving maps

The splitting of separatrices of area preserving maps close to the identity is one of the most paradigmatic examples of an exponentially small or singular phenomenon. The intrinsic small parameter is the characteristic exponent h > 0 of the saddle fixed point. A standard technique to measure the splitting of separatrices is the so-called Poincaré-Melnikov method, which has several specic features in the case of analytic planar maps. The aim of this talk is to compare the predictions for the splitting of separatrices provided by the Poincaré-Melnikov method, with the analytic and numerical results in a simple example where computations in multiple-precision arithmetic are performed

Poincaré-Melnikov-Arnold method for twist maps

The Poincar\'e--Melnikov--Arnold method is the standard tool for detecting splitting of invariant manifolds for systems of ordinary differential equations close to ``integrable'' ones with associated separatrices. This method gives rise to an integral (continuous sum) known as the Melnikov function (or Melnikov integral). If this function is not identically zero, the separatrices split. Moreover, the non-degenerate zeros of this function are associated to transversal intersections of the perturbed invariant (stable and unstable) manifolds. There exists a similar theory for planar maps, and in this case the Melnikov function is not a continuous sum anymore, but an infinite and (a priori) analytically uncomputable (discrete) sum. In a previous work, we have given a method to compute explicitly this kind of sums in terms of elliptic functions, under hypotheses of meromorphicity over the functions in the sum. This method allows us to obtain a strong non-integrability criterion and to apply it to perturbations of elliptic billiards and integrable standard-like maps like the McMillan map. Explicit estimates of the splitting angles are also given. Our aim is extend this method to the study of the splitting of doubly asymptotic manifolds (separatrices) associated to hyperbolic fixed points of twist maps in arbitrary dimensions. We work with maps generated globally by a generating function. Using the variational principle satisfied by these maps, we associate the non-degenerated critical points of a scalar function (here called Melnikov potential) to the transversal intersections of the perturbed asymptotic manifolds. We want to stress the difference of this point of view with the usual one in the literature, that is based in the study of non-degenerated zeros of a vectorial function. The symplectic structure and the variational principle play a fundamental role in our construction. As a first example where this theory can be applied, we study standard-like perturbations of a $2d$-dimensional twist map given by~R. McLachlan, for $d\ge 2$. This map is a multidimensional generalization of the McMillan map. We prove, among other results, that any entire perturbation destroys the separatrix of the McLachlan map

Singular separatrix splitting and Melnikov method: An experimental study

We consider families of analytic area-preserving maps depending on two pa- rameters: the perturbation strength E and the characteristic exponent h of the origin. For E=0, these maps are integrable with a separatrix to the origin, whereas they asymptote to flows with homoclinic connections as h->0+. For fixed E!=0 and small h, we show that these connections break up. The area of the lobes of the resultant turnstile is given asymptotically by E exp(-Pi^2/h)Oª(h), where Oª(h) is an even Gevrey-1 function such that Oª(0)!=0 and the radius of convergence of its Borel transform is 2Pi^2. As E->0 the function Oª tends to an entire function Oº. This function Oº agrees with the one provided by the Melnikov theory, which cannot be applied directly, due to the exponentially small size of the lobe area with respect to h. These results are supported by detailed numerical computations; we use an expensive multiple-precision arithmetic and expand the local invariant curves up to very high order

Melnikov potential for exact symplectic maps

The splitting of separatrices of hyperbolic fixed points for exact symplectic maps of $n$ degrees of freedom is considered. The non-degenerate critical points of a real-valued function (called the Melnikov potential) are associated to transverse homoclinic orbits and an asymptotic expression for the symplectic area between homoclinic orbits is given. Moreover, if the unperturbed invariant manifolds are completely doubled, it is shown that there exist, in general, at least $4$ primary homoclinic orbits ($4n$ in antisymmetric maps). Both lower bounds are optimal. Two examples are presented: a $2n$-dimensional central standard-like map and the Hamiltonian map associated to a magnetized spherical pendulum. Several topics are studied about these examples: existence of splitting, explicit computations of Melnikov potentials, transverse homoclinic orbits, exponentially small splitting, etc

Homoclinic orbits of twist maps and billiards

The splitting of separatrices for hyperbolic fixed points of twist maps with $d$ degrees of freedom is studied through a real-valued function, called the Melnikov potential. Its non-degenerate critical points are associated to transverse homoclinic orbits and an asymptotic expression for the symplectic area between homoclinic orbits is given. Moreover, Morse theory can be applied to give lower bounds on the number of transverse homoclinic orbits. This theory is applied first to elliptic billiards, where non-integrability holds for any non-trivial entire symmetric perturbation. Next, symmetrically perturbed prolate billiards with $d>1$ degrees of freedom are considered. Several topics are studied about these billiards: existence of splitting, explicit computations of Melnikov potentials, existence of $8$ or $8d$ transverse homoclinic orbits, exponentially small splitting, et