this case are illustrated in figure 2. The two obtained dynamical systems (figs.2(a,c)) have a very different appearance reflecting the fact that our modelling procedure allows for various dynamical systems with the same invariant density and autocorrelation. Let us briefly address the main features of the resulting maps. The map of model I possesses an almost marginal unstable fixed point at the origin and an almost vertical derivative at $x=1$. Additionally it possesses two obvious points of noncontinuous derivatives located on the left and right branch of the map, respectively. The one on the left branch has a nonvanishing right derivative which, via the function $h_f(x)$, is mapped to a point with almost vertical left derivative in the right branch of $f(x)$. The reader should note that the latter point coincides with the nonzero fixed point of the map ! The map of model II (fig.2(c)) is almost straighlined on its left branch and shows an almost vertical right derivative at the maximum and an almost vertical left derivative at $x=1$. In general the map of model II is ’smoother’ than the map of model I, which is an overall tendency to be observed in any of our examples. It 