can be viewed as a result of the additional flexibility within model II which allows to independently adapt the left and right derivative in a given interval thereby joining them smoothly together (see the above description for the ansatz of $h_f$ in model II). The prescribed and optimized autocorrelation data for model I are illustrated in subfigure 2(b): They show a deviation of $129 \%$ which is predominantly due to the inability of reproducing the single point $C(1)$ within the approach of model I. Although hardly visible in fig.2(b) the prescribed and optimized data coincide very well for $\{C(n),2\le n \le5\}$. The optimized correlation function for model II leads to a much better approximation to the prescribed exponential decay and yields an error of only $5.4 \%$ (see fig.2(d)).

As a third example we use an invariant density obeying a power law with an exponent $\beta=-0.5$ while the autocorrelation function is chosen to possess a decaying oscillatory behaviour. The corresponding minimization results are shown in fig.3. Both maps (figs.3(a,c)) show an almost horizontal derivative at $x=1$ and a strong cusp at $x_{\max}$. For the map of model II the left derivative at the maximum is almost vertical. Again the appearance 