- 99-381 Francesco Bonechi, Stephan De Bievre
- Exponential mixing and log h time scales in quantized
hyperbolic maps on the torus
Oct 11, 99
(auto. generated ps),
of related papers
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.