Degli Esposti M., Graffi S., Isola S.
Classical Limit of the Quantized Hyperbolic Toral Automorphisms
(100K, AmS-TeX)

ABSTRACT.  The canonical quantization of any hyperbolic symplectomorphism
$A$ of the 2-torus  yields a periodic unitary operator on a $N$-dimensional
Hilbert space,
$N=\frac1{h}$. We prove that this quantum system becomes ergodic and mixing at
the classical limit ($N\to\infty $, $N$ 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 spread in phase space.