Nalini Anantharaman

Precise counting results for closed orbits of Anosov flows

Abstract We study the problem of counting closed geodesics according to their lengths and under homological constraints on a compact surface of negative curvature. We show how to use Dolgopyats recent results to obtain a full asymptotic expansion, in addition to the leading term given by Lalley. We first state the properties of the stable and unstable leaves used by Chernov and Dolgopyat; then we introduce the usual transfer operators and we prove the result with the help of a dynamical ζ-function.

Entropy of Eigenfunctions

We study the high-energy limit for eigenfunctions of the Laplacian, on a compact negatively curved manifold. We review the recent result of Anantharaman–Nonnenmacher (Ann. Inst. Fourier 57(7):2465–2523, 2007) giving a lower bound on the Kolmogorov–Sinai entropy of semiclassical measures. The bound proved here improves that result in the case of variable negative curvature.

Entropy of Semiclassical Measures of the Walsh-Quantized Baker’s Map

Abstract.We study the baker’s map and its Walsh quantization, as a toy model of a quantized chaotic system. We focus on localization properties of eigenstates, in the semiclassical régime. Simple counterexamples show that quantum unique ergodicity fails for this model. We obtain, however, lower bounds on the entropies associated with semiclassical measures, as well as on the Wehrl entropies of eigenstates. The central tool of the proofs is an “entropic uncertainty principle”.

Patterson–Sullivan Distributions and Quantum Ergodicity

Abstract.This article gives relations between two types of phase space distributions associated to eigenfunctions

Counting geodesics which are optimal in homology

{\phi_{{{ir}}_{j}}}

Quantum ergodicity for the Anderson model on regular graphs

of the Laplacian on a compact hyperbolic surface XΓ: Wigner distributions

ENTROPY AND THE LOCALIZATION OF EIGENFUNCTIONS.

{\int_{S^{*}X_{\Gamma}} a\quad dW_{{{ir}}_{j}} =\langle Op(a)\phi_{{{ir}}_{j}},\phi_{{{ir}}_{j}}\rangle_{{{L}^2}(X_\Gamma)}}