autocorrelation signal. Through a number of examples we have demonstrated that our approach possesses an enormous flexibility allowing for a large variety of qualitatively different behaviour of the density and correlation function. To our knowledge there is in general no unique map which belongs to a given density and correlation function. This fact has to be seen in the context of the present investigation as an advantage since it allows for a great flexibility and possible variety with respect to the underlying dynamical systems.

We would like to mention that the above-discussed prescribed behaviour of the autocorrelation data (exponential, oscillating, power law decay) is, strictly speaking, enforced only for the first five points included in our Monte-Carlo optimization. In principle it is imaginable that this behaviour represents a transient and the asymptotics of the correlation function might show a different behaviour. To determine exact asymptotic properties of certain dynamical systems is however not the issue of the present paper. Our goal is to extend the inverse problem by including the correlation function in terms of a few (experimentally) available data points, thereby enabling us to design a dynamical system with desired statistical properties. Furthermore our approach might be suggestive in 