ENTROPY OF SEMICLASSICAL MEASURES FOR QUANTIZED CAT MAPS

ENTROPY OF SEMICLASSICAL MEASURES FOR QUANTIZED CAT-MAPS GABRIEL RIVIÈRE Abstract. In the case of semiclassical measures of the quantized cat-map, we give a simpli- fied proof of Anantharaman-Nonnenmacher's result for semiclassical measures on a riemannian compact manifold of constant negative curvature [2]. We show that for any hyperbolic matrix A in SL(2,Z) and any semiclassical measure µ associated to it, the Kolmogorov-Sinai entropy is bounded from below, i.e. hKS(µ,A) ≥ ?+ 2 , where ?+ is the positive Lyapunov exponent of A. Thanks to Faure-Nonnenmacher-de Bièvre construction in [8], this bound is optimal. 1. Introduction In the case of manifolds of negative curvature, the Quantum Unique Ergodicity Conjecture states that all eigenfunctions of the Laplacian equidistribute on M in the high energy limit [13]. In [1], Anantharaman proved that for a compact riemannian manifold M of Anosov type, the Kolmogorov-Sinai entropy of any semiclassical measure associated to a sequence of eigenfunctions of ∆ is positive. Her result proves in particular that eigenfunctions of the Laplacian cannot concentrate only on a closed geodesic in the large eigenvalue limit. After that, in the case where M is a compact manifold of constant curvature K = ?1 [2], Anantharaman and Nonnenmacher proved the following explicit bound 1 on the Kolmogorov-Sinai entropy of a semiclassical measure µ: (1) hKS(µ, g) ≥ d? 1 2

• quantum states


• anantharaman-nonnenmacher's result


• principle


• result than


• semiclassical measures


• main simplification


• ?? ?


• cat-maps


• weyl quantization


• than