Weil Representation, Deligne Sheaf and Proof of the Kurlberg-Rudnick Conjecture

Consider the two dimensional symplectic torus (T, ω) and an hyperbolic automorphism A of T. The automorphism A is known to be ergodic. In 1980, using a non-trivial procedure called quantization, the physicists J. Hannay and M.V. Berry attached to this automorphism a quantum operator ρ ~ (A) acting on a Hilbert space H~. One of the central questions of ”Quantum Chaos Theory”, in this model, is whether the operator ρ ~ (A) is ”quantum ergodic”? We consider the following two distributions on the algebra A = C∞(T) of smooth complex valued functions on T. The first one is given by the Haar integral: f 7−→ ∫