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ENTROPY AND THE LOCALIZATION OF EIGENFUNCTIONS. NALINI ANANTHARAMAN
 

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 (g t ) t#R , 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
) # t#R
acting on the Hilbert space L 2 (M) -- where # stands

  

Source: Anantharaman, Nalini - Centre de Mathématiques Laurent Schwartz, École Polytechnique

 

Collections: Mathematics