Spectral deviations for the damped wave equation

We prove a Weyl-type fractal upper bound for the spectrum of the damped wave equation, on a negatively curved compact manifold. It is known that most of the eigenvalues have an imaginary part close to the average of the damping function. We count the number of eigenvalues in a given horizontal strip deviating from this typical behaviour; the exponent that appears naturally is the `entropy' that gives the deviation rate from the Birkhoff ergodic theorem for the geodesic flow. A Weyl-type lower bound is still far from reach; but in the particular case of arithmetic surfaces, and for a strong enough damping, we can use the trace formula to prove a result going in this direction

Old and new about treeability and the Haagerup property for measured groupoids

This is mainly an expository text on the Haagerup property for countable groupoids equipped with a quasi-invariant measure, aiming to complete an article of Jolissaint devoted to the study of this property for probability measure preserving countable equivalence relations. We show that our definition is equivalent to the one given by Ueda in terms of the associated inclusion of von Neumann algebras. It makes obvious the fact that treeability implies the Haagerup property for such groupoids. For the sake of completeness, we also describe, or recall, the connections with amenability and Kazhdan property (T).Comment: 38 page

Amenability and exactness for dynamical systems and their C*-algebras

In this survey, we study the relations between amenability (resp. amenability at infinity) of C*-dynamical systems and equality or nuclearity (resp. exactness) of the corresponding crossed products.Comment: 16 pages, Ams-Tex, minor grammatical change