Exponential mixing and | ln h | time scales in quantized hyperbolic maps on the torus

We study the behaviour, in the simultaneous limits (h) over bar --> 0, t --> infinity, of the Husimi and Wigner distributions of initial coherent states and position eigenstates, evolved under the quantized hyperbolic toral automorphisms and the quantized baker map, We show how the exponential mixing of the underlying dynamics manifests itself in those quantities on time scales logarithmic in h. The phase space distributions of the coherent states, evolved under either of those dynamics, are shown to equidistribute on the torus in the limit (h) over bar --> 0, for times t between 1/2 \ln (h) over bar\/gamma and \ln (h) over bar\/gamma, where gamma is the Lyapounov exponent of the classical system. For times shorter than 1/2 \ln (h) over bar\/gamma , they remain concentrated on the classical trajectory of the center of the coherent state, The behaviour of the phase space distributions of evolved position eigenstates, on the other hand, is not the same fur the quantized automorphisms as for the baker map. In the first case, they equidistribute provided t --> infinity as (h) over bar --> 0, and as long as t is shorter than \ln (h) over bar\/gamma. In the second case, they remain localized on the evolved initial position at all such times.