  We consider fast oscillating perturbations of dynamical systems in regions
where one can introduce action-angle type coordinates. In an appropriate time
scale, a diffusion approximation of the first-integrals evolution is described
under the assumption that the set of resonance tori is small enough. If the
action-angle coordinates can be introduced just piece-wise , the limiting
diffusion process should be considered on an open book space. Such a process
can be described by differential operators, one on each page, supplemented by
some gluing conditions on the binding of the open book.
