Francesco Bonechi, Stephan De Bievre
Exponential mixing and log h time scales in quantized 
hyperbolic maps on the torus
(92K, latex)

ABSTRACT.  We study the behaviour, in the simultaneous limits \hbar going to 0, 
t going to \infty, 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 \hbar. The phase space distributions of the coherent 
states, evolved under either of those dynamics, are shown to 
equidistribute on the torus in the limit \hbar going to 0, for 
times t between |\ln\hbar|/(2\gamma) and 
|\ln\hbar|/\gamma, where \gamma is the Lyapounov exponent of 
the classical system. For times shorter than |\ln\hbar|(2\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 for the quantized 
automorphisms as for the baker map. 
In the first case, they equidistribute provided t goes to \infty 
as \hbar goes to 0, and as long as t is shorter than 
|\ln \hbar|/\gamma}. In the second case, they remain localized on 
the evolved initial position at all such times.