can even be investigated analytically. The interplay between the dynamical and statistical behaviour becomes more transparent if we consider the inverse problem, i.e. given the statistical behaviour of a system how could we extract relevant information on the possible dynamics. This subject has recently been addressed and discussed in the literature [@InvPro; @Dia96; @PiSchmeDia; @BaraDa; @Bollt; @Gora] and from a communications point of view in refs. [@Abel; @Broom]. The purpose is then to construct an one-dimensional fully chaotic (and ergodic) dynamical law for a given invariant density or/and time autocorrelation function. Given the latter we can on the one hand calculate all expectation values, i.e. statistical averages, of measurable observables depending exclusively on the single dynamical variable. On the other hand the given autocorrelation function provides us with valuable information on the dynamics of the trajectories of the system. It represents an important quantity strongly related to the physics described by the dynamical system. The construction of a dynamical system with prescribed statistical properties can have the background of either some experimentally given data for which a dynamical system should be modelled or it can be justified by the fact that a definite behaviour of the correlation function is required 