We consider the long time semiclassical evolution for the linear Schr?dinger equation. We show that, in the case of chaotic underlying classical dynamics and for times up to ?~(-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.
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