of the map of model II is much smoother compared to the map of model I. Regarding the autocorrelation function we have the opposite situation compared to the previous example. Within model I (fig.3(b)) we obtain a relative error of $22 \%$ for the autocorrelation data while model II (fig.3(d) provides an error of $45 \%$. Obviously model I is advantageous in the present case (at least for the present number of grid points chosen for the ansatz of $h_f$): it nicely reproduces the oscillations of the correlation function.

Finally we study the case that the autocorrelation function decays algebraically with the exponent $\gamma=-2.5$, i.e. we encounter the case of long-range correlations. The invariant density is chosen to be uniform. Both resulting maps (fig.4(a,c)) possess a large but finite derivative at $x=1$. The map of model I possesses an almost vertical derivative at a single point on its right branch. In contrast to our second example (see above) this point does not coincide with the fixed point. It is conjectured that this single point with almost vertical derivative is responsible for the observed power law decay of the correlation function (fig4(b)) (see also ref.[@Hor]). Analogously the map of model II 