Entropy of semiclassical measures in dimension 2

We study the high-energy asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface M 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 g t is bounded from below by half of the Ruelle upper bound; that is, h KS ( μ , g ) ≥ 1 2 ∫ S * M χ + ( ρ ) d μ ( ρ ) , where χ + ( ρ ) is the upper Lyapunov exponent at point ρ .