Semiclassical approximation and noncommutative geometry

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 ${\mathrm{\hslash }}^{-2+ϵ},ϵ>0$, 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.