ERC Consolidator grant project IPFLOW




Mini worskhop in Jussieu/Orsay, December 10-11-12, 2018


program:

* Monday 10, 10h30-12h, salle  201 batiment 15-16, Jussieu: François Ledrappier (Paris 6): Entropies of negatively curved compact manifolds and rigidity problems.
* Tuesday 11, 10h30-12h, Salle 0A4 LMO Orsay: Alexander Strohmaier (Leeds Univ.): A relativistic trace formula.
* Tuesday 11, 13h30-15h, LMO salle 3L8: Frédéric Faure (Univ. Grenoble Alpes): Spectre de Ruelle du flot géodésique Anosov et quantification géométrique.
* Tuesday 11, 15h30-17h LMO salle 3L8: Stéphane Nonnenmacher (Univ. Paris Sud): Eigenmode delocalization on surfaces of variable negative curvature
* Wednesday 12, 10h30-12h, LMO salle 3L15 : Nguyen Thi Dang (Univ. Rennes): Topological mixing of the Weyl chamber flow.
* Wednesday 12, 13h30-15h, LMO salle 3L15 : Yannick Bonthonneau (Univ. Rennes): Taylor spectrum and tranversally hyperbolic multiflows.

Abstracts: 

F. Ledrappier: Entropies of negatively curved compact manifolds and rigidity problems.

This is a survey on rigidity problems on entropies.

A. Strohmaier: A relativistic trace formula

We argue that stationary globally hyperbolic spacetimes are a natural setting for the  trace-formulae. We prove a generalisation of the (Gutzwiller-Duistermaat-Guillemin) trace formula in this setting. On the spectral side one has the spectrum of the generator of the timelike Killing vector field on the kernel of the wave operator. On the geometric side one has the symplectic manifold of scaled lightlike geodesics. I will explain these notions and will sketch the proof if time permits.
(joint work with S. Zelditch)

F. Faure: Spectre de Ruelle du flot géodésique Anosov et quantification géométrique

On considérera un flot géodésique Anosov et plus généralement un flot de contact Anosov. On rappellera que le champ de vecteur qui génère le flot possède un spectre discret de Ruelle qui décrit la décroissance des corrélations dynamiques.  On montrera que la partie effective du champ de vecteur qui décrit ce spectre est un opérateur pseudo différentiel dans le cadre de la quantification géométrique dont le symbole  associé est une fonction sur la variété symplectique définie par la structure de contact et  à valeur endomorphisme dans le fibré des jets sur la direction instable. Cela donne une interprétation géométrique de travaux précédents. Comme conséquence on retrouve des propriétés du spectre de Ruelle comme sa  structure en bandes.
Travail en cours avec Masato Tsujii.

S. Nonnenmacher: Eigenmode delocalization on surfaces of variable negative curvature

On a compact Riemannian manifold of negative curvature, the eigenmodes of the Laplace-Beltrami operator enjoy some delocalization properties in the high frequency limit, due to the hyperbolicity of the geodesic flow. The (de)localization properties can be represented by semiclassical measures associated with sequences of eigenstates: those are probability measures on the unit cotangent bundle, invariant through the geodesic flow, which characterize the asymptotic localization properties of the eigenstates along the sequence.
Using a recent Fractal Uncertainty Principle, Dyatlov and Jin have shown that, on a compact surface of constant negative curvature, any semiclassical measure is supported on the full unit cotangent bundle. As a consequence, for any open subset of the surface, the L2 weights of the eigenstates on this subset are uniformly bounded from below by a positive constant: all eigenstates spread throughout the surface.
I will present an extension of this result to surfaces of variable negative curvature. The main technical difficulty is due to the fact that the foliations formed by the stable and unstable manifolds of the geodesic flow are no longer smooth, which makes the application of the Fractal Uncertainty Principle more delicate. (joint with S.Dyatlov and L.Jin)

N.T. Dang: Topological mixing of the Weyl chamber flow

First, I will introduce my topological dynamical system. I will give two examples in higher rank of space of Weyl chambers and introduce in these situations the Weyl
chamber flows I study. Then I will introduce the main topological property: topological mixing, give a brief summary of what was already known in the finite volume case or in the rank one case and state our result with Olivier Glorieux, a necessary and sufficient condition for topological mixing. I will then explain the geometric situation in higher rank, basing myself on the examples of the first part of my talk and introduce the main tools. In the last part of my talk, I will give the main ideas behind the proof of our main Theorem. This is a joint work with Olivier Glorieux
Y. Bonthonneau: Taylor spectrum and tranversally hyperbolic multiflows

I will report on a joint effort with C.Guillarmou, J.Hilgert and T.Weich. We considered a multiflow $\varphi_t$, $t\in \R^k$, acting on a compact manifold. Assuming that the multiflow is transversally Anosov, we found in the litterature a way to define a resonance spectrum on an appropriate anisotropic space. This is very much an ongoing project, and I will present the current state of affairs.