The ``Quantum unique ergodicity" problem for anosov geodesic flows: an approach by entropy
Seminar Room 1, Newton Institute
Let $M$ be compact, negatively curved Riemannian manifold, and let $(\psi_n)$ be an orthonormal basis of eigenfunctions of the Laplacian on $M$. The Quantum Unique Ergodicity problem concerns the behaviour of the sequence of probability measures $|\psi_n(x)|^2 dx$ on $M$, or, more precisely, of their "microlocal" lifts to the tangent bundle $TM$. The limits of convergent subsequences must be invariant probability measures of the geodesic flow (sometimes called "quantum invariant measures"), and it is known that a very large subsequence converges to the Liouville measure on the unit tangent bundle. A conjecture of Rudnick and Sarnak says that this should actually be the only possible limit. E. Lindenstrauss proved the conjecture recently, in the case when $M$ is an arithmetic surface (of constant negative curvature) and $(\psi_n)$ is a common basis of eigenfunctions for the Laplacian and the Hecke operators. However, very little is known in the non-arithmetic case.
In the general case of an Anosov geodesic flow, I present an attempt to bound from below the metric entropy of "quantum invariant measures". I actually prove the following: if the $L^p$ norms of the $\psi_n$s do not grow too fast with $n$, then the corresponding quantum invariant measures cannot be entirely carried on a set of zero topological entropy.