This minicourse deals with the large eigenvalue limit for eigenfunctions of the laplacian, on a compact riemannian manifold M. More precisely, we will be interested in the quantum (unique) ergodicity problem, which asks about the weak limits of the probability measures
2 dx on M defined by the eigenfunctions phi, when the eigenvalue tends to infinity. Our approach will be purely focused on the dynamical, rather than arithmetic, aspects.
We will start with a brief introduction to microlocal analysis, which is a way to lift the problem to the cotangent bundle TM, and thus to relate it with a particular hamiltonian dynamical system, here the geodesic flow. We will discuss quantization procedures, and state the Egorov theorem, saying that the unitary flow induced by the laplacian converges, in some sense, to the geodesic flow, in the large eigenvalue limit (which is here equivalent to a "semiclassical" limit). This implies that the microlocal lifts of the measures
2 dx on TM converge to invariant measures of the geodesic flow.
We will then proceed to show how the qualitative properties of the geodesic flow can lead to various properties for eigenfunctions. For instance, eigenfunctions in completely integrable systems exhibit a completely different behaviour from those in ergodic systems. We will prove the Snirelman theorem, which concerns the case when the geodesic flow is ergodic with respect to the Liouville measure. It says that a "density one" subsequence of the measures
2 dx converges to the Liouvile measure. On a negatively curved manifold, Rudnick and Sarnak conjectured that the whole sequence actually converges, but this has remained an open problem so far.
The rest of the minicourse will be devoted to proving a recent result, joint with Stephane Nonnenmacher, giving a positive (and explicit) lower bound for the entropy of the limit measures. The result relies only on the Anosov property of the geodesic flow, it holds in variable negative curvature and arbitrary dimension. This bound implies, in particular, that less than half the mass of the measures
2 dx can go, in the limit, to a finite union of closed geodesics. The proof combines some simple ideas of ergodic theory with technical semiclassical estimates obtained from WKB methods.