Nicolas Roy's Curriculum Vita - Institut für Mathematik - Humboldt ...

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Published papers Listof publications 1 [7] N. Roy and V. Humilière, The geometry of the space of fibrations. To appear in Ann. Global Anal. Geom. (2009) [6] N. Roy, F. Faure and J. Sjöstrand, Semi-classical approach for Anosov diffeomorphisms and Ruelleresonances. Open Math. Journal (2008),vol. 1,35–81 [5] N. Roy, A semi-classical K.A.M. theorem. Comm. Partial Differential Equations 32 (2007), no. 5, 745–770 [4] N. Roy and F. Faure, Ruelle-Pollicottresonances for real analytic hyperbolicmaps. Nonlinearity 19(2006), no. 6, 1233–1252 [3] N. Roy, Intersections of Lagrangian submanifolds and the Mel’nikov 1-form. J. Geom. Phys. 56(2006), no. 11,2203–2229 [2] N. Roy, Regular deformations of completely integrable systems. J. Symplectic Geom. 3 (2005), no. 1, 1–16 [1]N.Roy,Thegeometryofnondegeneracyconditionsincompletelyintegrablesystems. Ann. Fac. Sci. Toulouse Math. (6)14(2005), no. 4, 705–719 Doctoral Thesis [0] N. Roy, Sur les déformations des systèmes complètement intégrables classiques et semiclassiques. Grenoble (France), September2003 Submited articles N. Roy, A Weinstein’stubular neighbourhood for Lagrangian fibrations. ArXiv 0905.0594 Articlesin preparation N. Royand V. Humilière,The geometryof the spaceof completelyintegrablesystems 1 pdffiles availableatØØÔÛÛÛÑØÙÖÐÒÖÓÝ 4
Summaries of published articles [0] Sur lesdéformationsdessystèmescomplètementintégrablesclassiquesetsemi-classiques Inthisthesis,westudyseveralaspectsofperturbationsofcompletelyintegrablesystems. The first part deals with completely integrable Hamiltonian systems (classical mechanics) and their deformations. This work as developped further and gave rise to the articles [1,2] described below. The second part concerns perturbations of semiclassical completely integrable systems. We introduce the theory of Pseudodifferential operators with small parameter on the torus. The particular algebraic structure of the torus provides a weel-defined notion of total symbol for these operators. The main results of this part were improved and are presented in[5]. [1] Thegeometryofnondegeneracy conditions incompletelyintegrablesystems In K.A.M-like theorems one needs to impose nondegeneracy conditions on the completely integrable Hamiltonian H0 in order to insure that the set of Diophantine tori (which persist after a small perturbation H0 + εK) has large measure. These conditions are usualy given in action-angle coordinates, but it is possible to formulate them geometrically, by considering completely integrable systems defined on a symplectic manifold by a fibration in Lagrangian tori and a Hamiltonian function constant on the fibers. In this article, we give a geometric definition for several nondegeneracy conditions, we explain the different implication relation that exist between them, and we show the unicity of the fibration for non-degenerate Hamiltonians. [2] Regular deformationsofcompletelyintegrable systems Given a symplectic manifold (M, ω) and a fibration M → B in lagrangian tori, we show thatanyfamily φ ε ofsymplectomorphims canbedecomposed inauniquewayas φ ε = Φ ε ◦ ϕ ε ,with Φ ε afamilyoffiber-preservingsymplectomorphimsand ϕ ε afamilyofHamiltonian symplectomorphims. This normal form is used in a second part to study deformations of non-degenerate regular completely integrable hamiltonian. Namely, we make explicit the spaceofperturbationsKofacompletelyintegrableHamiltonian H0,suchthattheperturbed Hamiltonian H0 + εK is“completelyintegrable up toO ε 2 ”. [3] Intersectionsof Lagrangian submanifoldsand the Mel’nikov1-form Mel’nikov’s theory/method deals with Hamiltonian systems H0 ∈ C∞ R2 with a hyperbolic point x0 and their perturbations Hε := H0 + εK, where H1 (t) ∈ C∞ R2 × R depends on the time in a periodic way. We assume that the stable and the unstable manifolds of x0 coincide, thus forming together a manifold of homoclinic points. Meln’ikov’s method provides a way to detect homoclinic points in the perturbed system Hε, in which the stable andunstable manifolddonolongercoincide. Namely,themethodpredictthatthesehomoclinic points are given by the zeros of a function (called “Mel’nikov function”) and that this function admitsan integral form, in which appearsonlythe flow of H0. This method extends to perturbations of Hamiltonian systems having two transversally hyperbolic periodic orbits linked by a heteroclinic manifold. Unfortunately, the most of the works on this topic deal always with models, written in a coordinate system and with an explicit splitting between periodic and transversal varaibles. Besides the fact that these models are non-generic, they alsohide the geometrical characterof the method. In this article, we develop the general theory of intersections of pairs of familiies of La- grangian submanifolds N ± ε , with N+ 0 = N− 0 , and constrained to live in an auxiliary family of submanifolds N ± ε ⊂ Pε ⊂ M. We show that the intersections N + ε ∩ N − ε which survive when ε = 0 are detected by a 1-form on N0, called Mel’nikov 1-form. We show that it is the appropriate tool for studying heteroclinic points between two transversally hyperbolic 5
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