Absract: Sinai billiard maps in dimension two have been known to be exponentially mixing (L.-S. Young) for almost two decades, and recent work of Demers and Zhang have shed new light on the spectrum of their transfer operators. The situation for the continuous time Sinai billiard is more delicate. I will present recent results and ongoing work on their spectrum.
Absract: We describe the Ruelle spectrum of the geodesic flow of compact hyperbolic manifolds (joint work with Dyatlov and Faure).
Absract: we study the existence of embedded eigenvalues in the continuous spectrum of the Neumann Laplace operator in a hyperbolic triangle with one cusp. We prove that generically in the space of such triangles there are no embedded eigenvalues. Joint with Chris Judge.
Absract: On exposera divers résultats et conjectures sur la théorie des résonances des surfaces de "congruence" qui sont obtenues en prenant des sous-groupes d'indice infini de SL2(Z). On motivera cet exposé par des problèmatiques de théorie des nombres (travaux de Bourgain, Gamburd, Sarnak).
Absract: We investigate the distribution of resonances for semiclassical quantum scattering systems for which the trapped set of the classical dynamics is a normally hyperbolic symplectic submanifold. We prove the presence of a resonance gap (or resonance free strip) related with the minimal rate of transverse hyperbolicity. This generalizes an old result of Gérard-Sjöstrand to the smooth setting. Such a gap has consequences on the dispersion of waves in various situations. Besides, following the strategy of Faure-Sjöstrand, this result can also be applied to recover the exponential mixing rate of contact Anosov flows proved a few years ago by Tsujii.
Absract: We consider the two most basic questions for Ruelle zeta functions, defined in terms of the closed orbits for Anosov, and more generally, for Axiom A flows: (i) To how large a domain can the zeta function be meromorphically extended? (ii) What can be said about the location of the zeros?
Absract: I will consider small pertubations of the geodesic flow on compact negatively curved surfaces. I will discuss some statistical properties satisfied by these families of perturbed flows and I will give applications of these results to the long time dynamics (below the Ehrenfest time) of the Schrödinger equation on such surfaces.
Johannes Sjöstrand (sous réserve)
Absract: On applique une méthode maintenant bien établie pour des opérateurs différentiels avec perturbations aléatoires (dévéloppée par M. Hager, W. Bordeaux-Montrieux, Sjöstrand) au cas des perturbations de grands blocs de Jordan. Par rapport à un travail de E.B. Davies et Hager, on trouve aussi la distribution angulaire des valeurs propres.
Absract: The talk concerns Ruelle transfer operators related to a given Markov partition for contact Anosov flows satisfying certain regularity conditions. In the classical case these operators depend on just one complex parameter and it is important to have information about their spectra when the (absolute value of this) parameter is large. However there are problems in hyperbolic dynamics that lead naturally to the study of Ruelle transfer operators depending on two complex parameters. We will discuss how to get strong spectral estimates of such operators under certain natural assumptions when the absolute values of both complex parameters are large.