in the context of the control of a physical system [@Bollt; @Gora].

Restricting ourselves to the first part of the problem, i.e. the construction of a map with a given invariant density (the so called inverse Frobenius-Perron problem (IFPP)) we have recently found a general and for practical purposes very helpful representation of the solution of the IFPP within the class of smooth complete and unimodal maps [@PiSchmeDia]. However the combined problem (both the invariant density and time autocorrelation function are given) is much more complicated. Some progress in this direction can be made if one uses the class of piecewise linear Markov maps as a basis for this search. Constructing such a map with a given autocorrelation function one can then obtain the proper invariant density performing a suitable conjugation (coordinate) transformation [@BaraDa]. A substantial disadvantage of this method is that the changes of the autocorrelation function in the course of the conjugation transformation are not fully under control. Another rather anaesthetic aspect is the fact that in many cases the solution found to the inverse problem does not fulfill certain smoothness criteria.

The purpose of this letter is to present a new approach to the inverse problem including 