We consider the long time semiclassical evolution for the linear Schrödinger 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.
Semiclassical approximation and noncommutative geometry