, Volume 11, Issue 6, pp 1085-1116
Date: 29 Sep 2010

Entropy of Semiclassical Measures for Nonpositively Curved Surfaces


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 sequences of distributions associated to them and we study the entropic properties of their accumulation points, the so-called semiclassical measures. Precisely, 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, i.e.

$$h_{KS}(\mu,g)\geq \frac{1}{2} \int\limits_{S^*M} \chi^+(\rho) {\rm d} \mu(\rho),$$
where χ +(ρ) is the upper Lyapunov exponent at point ρ. The main strategy is the same as in Rivière (Duke Math J, arXiv:0809.0230, 2008) except that we have to deal with weakly chaotic behavior.