We study the high-energy asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of Anosov type. To do this, we look at families of distributions associated to them on the cotangent bundle 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 for the geodesic flow is bounded from below by half of the Ruelle upper bound; that is, where is the upper Lyapunov exponent at point .
"Entropy of semiclassical measures in dimension 2." Duke Math. J. 155 (2) 271 - 335, 1 November 2010. https://doi.org/10.1215/00127094-2010-056