On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold

In this article, we study the internal stabilization and control of the critical nonlinear Klein-Gordon equation on 3-D compact manifolds. Under a geometric assumption slightly stronger than the classical geometric control condition, we prove exponential decay for some solutions bounded in the energy space but small in a lower norm. The proof combines profile decomposition and microlocal arguments. This profile decomposition, analogous to the one of Bahouri-G\'erard on $\R^3$, is performed by taking care of possible geometric effects. It uses some results of S. Ibrahim on the behavior of concentrating waves on manifolds

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

Analyticity in spaces of convergent power series and applications

We study the analytic structure of the space of germs of an analytic function at the origin of \ww C^{\times m} , namely the space \germ{\mathbf{z}} where \mathbf{z}=\left(z\_{1},\cdots,z\_{m}\right) , equipped with a convenient locally convex topology. We are particularly interested in studying the properties of analytic sets of \germ{\mathbf{z}} as defined by the vanishing locus of analytic maps. While we notice that \germ{\mathbf{z}} is not Baire we also prove it enjoys the analytic Baire property: the countable union of proper analytic sets of \germ{\mathbf{z}} has empty interior. This property underlies a quite natural notion of a generic property of \germ{\mathbf{z}} , for which we prove some dynamics-related theorems. We also initiate a program to tackle the task of characterizing glocal objects in some situations