The frequency map for billiards inside ellipsoids

The billiard motion inside an ellipsoid Q \subset \Rset^{n+1} is completely integrable. Its phase space is a symplectic manifold of dimension $2n$, which is mostly foliated with Liouville tori of dimension $n$. The motion on each Liouville torus becomes just a parallel translation with some frequency $\omega$ that varies with the torus. Besides, any billiard trajectory inside $Q$ is tangent to $n$ caustics $Q_{\lambda_1},...,Q_{\lambda_n}$, so the caustic parameters $\lambda=(\lambda_1,...,\lambda_n)$ are integrals of the billiard map. The frequency map $\lambda \mapsto \omega$ is a key tool to understand the structure of periodic billiard trajectories. In principle, it is well-defined only for nonsingular values of the caustic parameters. We present four conjectures, fully supported by numerical experiments. The last one gives rise to some lower bounds on the periods. These bounds only depend on the type of the caustics. We describe the geometric meaning, domain, and range of $\omega$. The map $\omega$ can be continuously extended to singular values of the caustic parameters, although it becomes "exponentially sharp" at some of them. Finally, we study triaxial ellipsoids of \Rset^3. We compute numerically the bifurcation curves in the parameter space on which the Liouville tori with a fixed frequency disappear. We determine which ellipsoids have more periodic trajectories. We check that the previous lower bounds on the periods are optimal, by displaying periodic trajectories with periods four, five, and six whose caustics have the right types. We also give some new insights for ellipses of \Rset^2.Comment: 50 pages, 13 figure

The frequency map for billiards inside ellipsoids

The billiard motion inside an ellipsoid Q Rn+1 is completely integrable. Its phase space is a symplectic manifold of dimension 2n, which is mostly foliated with Liouville tori of dimension n. The motion on each Liouville torus becomes just a parallel translation with some frequency ! that varies with the torus. Besides, any billiard trajectory inside Q is tangent to n caustics Q 1 ; : : : ;Q n, so the caustic parameters = ( 1; : : : ; n) are integrals of the billiard map. The frequency map 7! ! is a key tool to understand the structure of periodic billiard trajectories. In principle, it is well-defined only for nonsingular values of the caustic parameters. We present four conjectures, fully supported by numerical experiments. The last one gives rise to some lower bounds on the periods. These bounds only depend on the type of the caustics. We describe the geometric meaning, domain, and range of !. The map ! can be continuously extended to singular values of the caustic parameters, although it becomes “exponentially sharp” at some of them. Finally, we study triaxial ellipsoids of R3. We compute numerically the bifurcation curves in the parameter space on which the Liouville tori with a fixed frequency disappear. We determine which ellipsoids have more periodic trajectories. We check that the previous lower bounds on the periods are optimal, by displaying periodic trajectories with periods four, five, and six whose caustics have the right types. We also give some new insights for ellipses of R2.Preprin

3D billiards: visualization of regular structures and trapping of chaotic trajectories

The dynamics in three-dimensional billiards leads, using a Poincar\'e section, to a four-dimensional map which is challenging to visualize. By means of the recently introduced 3D phase-space slices an intuitive representation of the organization of the mixed phase space with regular and chaotic dynamics is obtained. Of particular interest for applications are constraints to classical transport between different regions of phase space which manifest in the statistics of Poincar\'e recurrence times. For a 3D paraboloid billiard we observe a slow power-law decay caused by long-trapped trajectories which we analyze in phase space and in frequency space. Consistent with previous results for 4D maps we find that: (i) Trapping takes place close to regular structures outside the Arnold web. (ii) Trapping is not due to a generalized island-around-island hierarchy. (iii) The dynamics of sticky orbits is governed by resonance channels which extend far into the chaotic sea. We find clear signatures of partial transport barriers. Moreover, we visualize the geometry of stochastic layers in resonance channels explored by sticky orbits.Comment: 20 pages, 11 figures. For videos of 3D phase-space slices and time-resolved animations see http://www.comp-phys.tu-dresden.de/supp

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

Eigenfunction asymptotics and spectral Rigidity of the ellipse

This paper is part of a series concerning the isospectral problem for an ellipse. In this paper, we study Cauchy data of eigenfunctions of the ellipse with Dirichlet or Neumann boundary conditions. Using many classical results on ellipse eigenfunctions, we determine the microlocal defect measures of the Cauchy data of the eigenfunctions. The ellipse has integrable billiards, i.e. the boundary phase space is foliated by invariant curves of the billiard map. We prove that, for any invariant curve $C$, there exists a sequence of eigenfunctions whose Cauchy data concentrates on $C$. We use this result to give a new proof that ellipses are infinitesimally spectrally rigid among $C^{\infty}$ domains with the symmetries of the ellipse

Articles publicats per investigadors de l'ETSEIB indexats al Journal Citation Reports: 2011

Informe que recull els 296 treballs publicats per 220 investigadors de l'Escola Tècnica Superior d'Enginyeria Industrial de Barcelona (ETSEIB) en revistes indexades al Journal Citation Reports durant l’any 2011Preprin

On the length spectrum of analytic convex billiard tables

Billiard maps are a type of area-preserving twist maps and, thus, they inherit a vast num-ber of properties from them, such as the Lagrangian formulation, the study of rotational invariant curves, the types of periodic orbits, etc. For strictly convex billiards, there exist at least two (p, q)-periodic orbits. We study the billiard properties and the results found up to now on measuring the lengths of all the (p, q)-trajectories on a billiard. By using a standard Melnikov method, we find that the first order term of the difference on the lengths among all the (p, q)-trajectories orbits is exponentially small in certain perturba-tive settings. Finally, we conjecture that the difference itself has to be exponentially small and also that these exponentially small phenomena must be present in many more cases of perturbed billiards than those we have presented on this work

The frequency map for billiards inside ellipsoids

The billiard motion inside an ellipsoid Q Rn+1 is completely integrable. Its phase space is a symplectic manifold of dimension 2n, which is mostly foliated with Liouville tori of dimension n. The motion on each Liouville torus becomes just a parallel translation with some frequency ! that varies with the torus. Besides, any billiard trajectory inside Q is tangent to n caustics Q 1 ; : : : ;Q n, so the caustic parameters = ( 1; : : : ; n) are integrals of the billiard map. The frequency map 7! ! is a key tool to understand the structure of periodic billiard trajectories. In principle, it is well-defined only for nonsingular values of the caustic parameters. We present four conjectures, fully supported by numerical experiments. The last one gives rise to some lower bounds on the periods. These bounds only depend on the type of the caustics. We describe the geometric meaning, domain, and range of !. The map ! can be continuously extended to singular values of the caustic parameters, although it becomes “exponentially sharp” at some of them. Finally, we study triaxial ellipsoids of R3. We compute numerically the bifurcation curves in the parameter space on which the Liouville tori with a fixed frequency disappear. We determine which ellipsoids have more periodic trajectories. We check that the previous lower bounds on the periods are optimal, by displaying periodic trajectories with periods four, five, and six whose caustics have the right types. We also give some new insights for ellipses of R2