A series expansion for the time autocorrelation of dynamical variables

We present here a general iterative formula which gives a (formal) series expansion for the time autocorrelation of smooth dynamical variables, for all Hamiltonian systems endowed with an invariant measure. We add some criteria, theoretical in nature, which enable one to decide whether the decay of the correlations is exponentially fast or not. One of these criteria is implemented numerically for the case of the Fermi-Pasta-Ulam system, and we find indications which might suggest a sub-exponential decay of the time autocorrelation of a relevant dynamical variable.