Quantum effective evolution

For large values of $ \nu\in\mathbb{R}$, the evolution $ \hat{F}_{\nu}^{t}\varphi$ , $ t\in\mathbb{N}$, of  $ \varphi\in C^{\infty}\left(S^{1}\right)$ does not decay to a unique equilibrium state but evolves in a finite dimensional linear "effective space", due to a large (but finite) number of resonances with large modulus. This is like a "finite dimensional effective quantum evolution".
The following movies show the evolution of an initial wave packet in space x, and its Husimi distribution in phase space (x,xi). One observes  well that the Husimi distribution evolves on the trapped set K.
For $ \nu=80$:

evolution of f(x)Husimi evolution

For lower values of $ \nu\in\mathbb{R}$, the number of resonnances is reduced and for large time a single resonance  dominates (with the largest modulus). Here is an example  for $ \nu=10$:


evolution of f(x)Husimi evolution