We consider small perturbations of a simple completely integrable system with many degrees of freedom: a collection of independent one-degree-of-freedom oscillators (in the perturbed system the individual oscillators are no longer independent). We show that the long-time behavior of such a system, even in the case of purely deterministic perturbations, should, in general, be described as a stochastic process. The limiting stochastic process is a Markov process on an open book space corresponding to the collection of first integrals of the non-perturbed system.