ENTROPY OF SEMICLASSICAL MEASURES

ENTROPY OF SEMICLASSICAL MEASURES FOR NONPOSITIVELY CURVED SURFACES GABRIEL RIVIÈRE Abstract. We study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of nonpositive sectional curvature. To do this, we look at se- quences of distributions associated to them and we study the entropic properties of their accumu- lation points, the so-called semiclassical measures. Precisely, 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 ?. The main strategy is the same as in [17] except that we have to deal with weakly chaotic behavior. 1. Introduction Let M be a compact, connected, C∞ riemannian manifold. 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 H(x, ?) := ??? 2 x 2 . This quantity corresponds to the classical kinetic energy in the case of the absence of potential.

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• quantum unique


• geodesic flow


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• flow gt over


• semiclassical measure


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• liouville measure