Gabriel Riviere

Entropy of semiclassical measures in dimension 2

We study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of Anosov type. We show that the Kolmogorov-Sinai entropy of a semiclassical measure for the geodesic flow is bounded from below by half of the Ruelle upper bound

Entropy of Semiclassical Measures for Nonpositively Curved Surfaces

AbstractWe study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of nonpositive sectional curvature. To do this, we look at sequences of distributions associated to them and we study the entropic properties of their accumulation points, the so-called semiclassical measures. Precisely, we show that the Kolmogorov–Sinai entropy of a semiclassical measure μ for the geodesic flow gt is bounded from below by half of the Ruelle upper bound, i.e.

Delocalization of Slowly Damped Eigenmodes on Anosov Manifolds

h_{KS}(\mu,g)\geq \frac{1}{2} \int\limits_{S^*M} \chi^+(\rho) {\rm d} \mu(\rho),

EIGENMODES OF THE DAMPED WAVE EQUATION AND SMALL HYPERBOLIC SUBSETS

where χ+(ρ) is the upper Lyapunov exponent at point ρ. The main strategy is the same as in Rivière (Duke Math J, arXiv:0809.0230, 2008) except that we have to deal with weakly chaotic behavior.

Long-Time Dynamics of the Perturbed Schrödinger Equation on Negatively Curved Surfaces

We study the long time dynamics of the Schrödinger equation on Zoll manifolds. We establish criteria under which the solutions of the Schrödinger equation can or cannot concentrate on a given closed geodesic. As an application, we derive some results on the set of semiclassical measures for eigenfunctions of Schrödinger operators: we prove that adding a potential

TWO-MICROLOCAL REGULARITY OF QUASIMODES ON THE TORUS

{V \in C^{\infty} (\mathbb{S}^{d})}