74 research outputs found
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 boundComment: 42 pages, 3 figures. Compared to the second version, I have removed
the proof in the case of surfaces of nonpositive curvature and I have written
it in a different article (arXiv:0911.1840
Perturbation of the semiclassical Schr\"odinger equation on negatively curved surfaces
We consider the semiclassical Schr\"odinger equation on a compact negatively
curved surface. For any sequence of initial data microlocalized on the unit
cotangent bundle, we look at the quantum evolution (below the Ehrenfest time)
under small perturbations of the Schr\"odinger equation, and we prove that, in
the semiclassical limit and for typical perturbations, the solutions become
equidistributed on the unit cotangent bundle.Comment: 48 pages. Compared with version 1, we consider slightly different
families of perturbations in order to simplify the expositio
Dispersion and controllability for the Schr\"odinger equation on negatively curved manifolds
We study the time-dependent Schr\"odinger equation on a compact riemannian manifold on which the
geodesic flow has the Anosov property. Using the notion of semiclassical
measures, we prove various results related to the dispersive properties of the
Schr\"odinger propagator, and to controllability
Eigenmodes of the damped wave equation and small hyperbolic subsets
We study stationary solutions of the damped wave equation on a compact and
smooth Riemannian manifold without boundary. In the high frequency limit, we
prove that a sequence of -damped stationary solutions cannot be
completely concentrated in small neighborhoods of a small fixed hyperbolic
subset made of -damped trajectories of the geodesic flow. The article
also includes an appendix (by S. Nonnenmacher and the author) where we
establish the existence of an inverse logarithmic strip without eigenvalues
below the real axis, under a pressure condition on the set of undamped
trajectories.Comment: 24 pages. With an appendix by S. Nonnenmacher and the author. In this
new version, we modified an uncorrect exponent in the statement of Theorem
A.1 from the appendi
Spectral analysis of Morse-Smale flows I: construction of the anisotropic spaces
We prove the existence of a discrete correlation spectrum for Morse-Smale
flows acting on smooth forms on a compact manifold. This is done by
constructing spaces of currents with anisotropic Sobolev regularity on which
the Lie derivative has a discrete spectrum
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