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

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

2010-03-26

arxiv.org/abs/1003.5061v1

Int. Math. Res. Notices 2011 (2011) 2396-2443

Mathematics

Dynamical Systems

25 pages

Scientific paper

10.1093/imrn/rnq155

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.

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