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On quantum ergodicity for linear maps of the torus. (English) Zbl 1042.81026

Summary: We prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus (“cat maps”). We show that there is a density one sequence of integers so that as \(N\) tends to infinity along this sequence, all eigenfunctions of the quantum propagator at inverse Planck constant \(N\) are uniformly distributed. A key step in the argument is to show that for a hyperbolic matrix in the modular group, there is a density one sequence of integers \(N\) for which its order (or period) modulo \(N\) is somewhat larger than \(\sqrt{N}\).

MSC:

81Q50 Quantum chaos
11L07 Estimates on exponential sums
11Z05 Miscellaneous applications of number theory
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
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