Entropy of semiclassical measures in dimension 2

Abstract

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 $\mu$ for the geodesic flow $g^t$ is bounded from below by half of the Ruelle upper bound; that is, ${\displaystyle h_{\mathrm{KS}}(\mu ,g)\ge \frac{1}{2}{\int }_{S^{*}M}{\chi }^{+}(\rho ) d\mu (\rho ),}$ where ${\chi }^{+}(\rho )$ is the upper Lyapunov exponent at point $\rho$.