Entropy of quantum limits for symplectic linear maps of the multidimensional torus

Gabriel Riviere (1)

1 CMLS - Centre de Mathématiques Laurent Schwartz

In the case of a linear symplectic map A of the 2d-torus, semiclassical measures are A-invariant probability measures associated to sequences of high energy quantum states. Our main result is an explicit lower bound on the entropy of any semiclassical measure of a given quantizable matrix A in Sp(2d,Z). In particular, our result implies that if A has an eigenvalue outside the unit circle, then a semiclassical measure cannot be carried by a closed orbit of A.

Mots clés: Quantum chaos; Kolmogorov-Sinai entropy; semiclassical measures

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Systèmes dynamiques [math.DS]

Gabriel Riviere. Entropy of quantum limits for symplectic linear maps of the multidimensional torus. International Mathematics Research Notices, 2011, 2011 (11), pp.2396-2443. ⟨10.1093/imrn/rnq155⟩. ⟨hal-00464803⟩