  We consider the long time semiclassical evolution for the linear
Schr\"odinger equation. We show that, in the case of chaotic underlying
classical dynamics and for times up to $\hbar^{-2+\epsilon},\ \epsilon>0$, the
symbol of a propagated observable by the corresponding von Neumann-Heisenberg
equation is, in a sense made precise below, precisely obtained by the
push-forward of the symbol of the observable at time $t=0$. The corresponding
definition of the symbol calls upon a kind of Toeplitz quantization framework,
and the symbol itself is an element of the noncommutative algebra of the
(strong) unstable foliation of the underlying dynamics.
