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Entropy and the localization of eigenfunctions. (English) Zbl 1175.35036
The authors considers a compact manifold with an Anosov geodesic flow and studies the large eigenvalue limit of the eigenfunctions of the Laplacian. The main result is a general lower bound on the Kolmogorov-Sinai metric entropy in the measure theoretic sense, from which it follows that many of the ergodic components of the semiclassical invariant measure have positive entropy. A corollary of this is that eigenfunctions cannot concentrate entirely on a closed geodesic.
As a particular case, these results apply to manifolds with negative curvature. The theorems of this paper give a partial answer to the quantum unique ergodicity conjecture of Rudnick and Sarnack, according to which, for negatively curved manifolds, the semiclassical invariant measure should be Liouville.
The author also gives an interesting conjecture about a general and explicit entropy lower bound.

35J10 Schrödinger operator, Schrödinger equation
35B15 Almost and pseudo-almost periodic solutions to PDEs
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
94A17 Measures of information, entropy
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