Entropies des flots magnétiques
Entropy and localization of eigenfunctions
Entropy of eigenfunctions of the Laplacian in dimension 2
We study asymptotic properties of eigenfunctions of the Laplacian on compact Riemannian surfaces of Anosov type (for instance negatively curved surfaces). More precisely, we give an answer to a question of Anantharaman and Nonnenmacher [4] by proving that the Kolmogorov-Sinai entropy of a semiclassical measure for the geodesic flow is bounded from below by half of the Ruelle upper bound. (This text has been written for the proceedings of the Journées EDP (Port d’Albret-June, 7-11 2010))
Entropy of Geodesic Flow and Exponent of Convergence of Some Dirichlet Series.
Ergodic geodesic flows on product manifolds with low dimensional factors.
Ergodic theory of interval exchange maps.
A unified introduction to the dynamics of interval exchange maps and related topics, such as the geometry of translation surfaces, renormalization operators, and Teichmüller flows, starting from the basic definitions and culminating with the proof that almost every interval exchange map is uniquely ergodic. Great emphasis is put on examples and geometric interpretations of the main ideas.
Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces.
Estimation de l'entropie des systèmes lagrangiens sans points conjugués
Expanding maps on Cantor sets and analytic continuation of zeta functions
Explicit Teichmüller curves with complementary series
We construct an explicit family of arithmetic Teichmüller curves , , supporting -invariant probabilities such that the associated -representation on has complementary series for every . Actually, the size of the spectral gap along this family goes to zero. In particular, the Teichmüller geodesic flow restricted to these explicit arithmetic Teichmüller curves has arbitrarily slow rate of exponential mixing.
Exponential mixing for the Teichmüller flow
We study the dynamics of the Teichmüller flow in the moduli space of abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Hölder observables. A geometric consequence is that the action in the moduli space has a spectral gap.