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Entropy of chaotic eigenstates

These notes present a recent approach to study the high-frequency eigenstates of the Laplacian on compact Riemannian manifolds of negative sectional curvature. The main result is a lower bound on the Kolmogorov-Sinai entropy of the semiclassical measures associated with sequences of eigenstates, showing that high-frequency eigenstates cannot be too localized. The method is extended to the case of semiclassical Hamiltonian operators for which the classical flow in some energy range is of Anosov type, and to the case of quantized Anosov diffeomorphisms on the torus.