Annales Henri Poincaré

, Volume 11, Issue 6, pp 1085–1116

Entropy of Semiclassical Measures for Nonpositively Curved Surfaces

Authors

    • Centre de Mathématiques Laurent Schwartz (UMR 7640)École Polytechnique
Article

DOI: 10.1007/s00023-010-0055-2

Cite this article as:
Rivière, G. Ann. Henri Poincaré (2010) 11: 1085. doi:10.1007/s00023-010-0055-2

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 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 gt 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.
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