consuming part of the algorithm is the calculation of the autocorrelation function which has to be done for each MC step.

First we consider a simple example for which the maps we are looking for are well-known. We use an exponential decay as a prescribed behaviour for the autocorrelation signal while the invariant density of the dynamical process is assumed to be uniform in $[0,1]$. A corresponding family of maps, the nonsymmetric tent maps, has been studied in some detail in ref.[@Mori]. They are the nonsymmetric tent maps. The resulting maps of our Monte-Carlo optimization are illustrated for model I in fig.1(a) and for model II in fig.1(c). They show only minor differences, i.e. the outcome of approach I is almost the same as for approach II, and both are to a very good approximation an asymmetric tent map. Both models lead also to a very good approximation of the corresponding prescribed correlation function with an error of only $2.2\%$.

For the second example the two models lead to quantitatively different results. The invariant density is supposed to be linear while the autocorrelation function shows an exponential decay similar to the previous case. The results of the stochastic minimization for 