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Summary: ENTROPY AND THE LOCALIZATION OF EIGENFUNCTIONS.
NALINI ANANTHARAMAN
Abstract. We study the large eigenvalue limit for the eigenfunctions of the
Laplacian, on a compact manifold of variable negative curvature or more
generally, assuming only that the geodesic flow has the Anosov property. We
prove that the Wigner measures associated to eigenfunctions cannot concen-
trate entirely on sets of small topological entropy under the action of the
geodesic flow, such as, for instance, closed geodesics.
1. Introduction, statement of results
We consider a compact Riemannian manifold M of dimension d 2, and as-
sume that the geodesic flow (gt
)tR, acting on the unit tangent bundle of M, has a
"chaotic" behaviour; this refers to certain asymptotic properties of the flow when
time t tends to infinity: ergodicity, mixing, hyperbolicity... Here we mean that the
geodesic flow has the Anosov property. The name "quantum chaos" expresses the
belief that the chaotic properties of the flow should still be visible in the correspond-
ing quantized dynamical system: that is, according to the Schr¨odinger equation, the
unitary flow exp(i t
2 ) tR
acting on the Hilbert space L2
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