On the distribution of recurrence times in nonlinear systems

A recent result on the recurrence properties of the sequence m=min_pZ |Mm–p| for irrational , together with a rather old, but little-known result by Florek and Slater on the recurrence properties of the sequence m mod 1 with respect to connected intervals in the interval [0, 1] show that integrable Hamiltonian systems with two degrees of freedom have at most three different recurrence times with respect to an arbitrary Poincaré section of the invariant tori in phase space. We discuss a possible extension of this result to arbitrary integrable systems with any number of degrees of freedom and propose the recurrence time spectrum as a new quantity for characterising simple and complex behaviour of general nonlinear systems.