Classical limit of the quantized hyperbolic toral automorphisms

The canonical quantization of any hyperbolic symplectomorphismA of the 2-torus yields a periodic unitary operator on aN-dimenional Hilbert space,N=1/h. We prove that this quantum system becomes ergodic and mixing at the classical limit (N,N prime) which can be interchanged with the time-average limit. The recovery of the stochastic behaviour out of a periodic one is based on the same mechanism under which the uniform distribution of the classical periodic orbits reproduces the Lebesgue measure: the Wigner functions of the eigenstates, supported on the classical periodic orbits, are indeed proved to become uniformly speread in phase space.