ENTROPY OF SEMICLASSICAL MEASURES IN DIMENSION

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profil-urra-2012

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ENTROPY OF SEMICLASSICAL MEASURES IN DIMENSION 2 GABRIEL RIVIÈRE Abstract. We study the high-energy asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surfaceM of Anosov type. To do this, we look at families of distributions associated to them on the cotangent bundle T ?M and we derive entropic properties on their accumulation points in the high-energy limit (the so-called semiclassical measures). We show that the Kolmogorov-Sinai entropy of a semiclassical measure µ for the geodesic ﬂow gt is bounded from below by half of the Ruelle upper bound, i.e. hKS(µ, g) ≥ 1 2 ∫ S?M ?+(?)dµ(?), where ?+(?) is the upper Lyapunov exponent at point ?. 1. Introduction In quantum mechanics, the semiclassical principle asserts that in the high energy limit, one should observe classical phenomena. Our main concern will be the study of this property when the classical system is said to be chaotic. Let M be a compact C∞ Riemannian surface. For all x ?M , T ?xM is endowed with a norm ?.?x given by the metric over M . The geodesic ﬂow gt over T ?M is deﬁned as the Hamiltonian ﬂow corresponding to the Hamiltonian H(x, ?) := ??? 2 x 2 .

lebesgue measure

arithmetic quantum

geodesic ﬂow

ﬂow gt over

semiclassical measure

kolmogorov-sinai entropy

distribution µ

nonpositive curvature

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profil-urra-2012

Nombre de lectures

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