(fig.4(c)) shows a very well-pronounced vertical derivative at the cusp. It is interesting to observe that in this case both models I and II provide very satisfactory approximations to the prescribed autocorrelation data (fig4(b,d)), i.e. a relative error of only $13 \%$.

Summary
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We have introduced a stochastic Monte-Carlo based approach to the inverse problem which uses both the invariant density as well as a finite number of points of the autocorrelation function as prescribed statistical quantities. A key ingredient for this approach was a recently found representation for one-dimensional complete chaotic and single-humped dynamical system in terms of an auxiliary function $h_f$. This representation is a formal solution of the inverse Frobenius-Perron problem and provides the dynamical system $f(x)$ explicitly as a function of the given measure and the function $h_f$. $h_f$ therefore reflects the freedom of changing the map without changing its invariant measure (density). In order to quantify the freedom available for the determination of $h_f$, i.e. to parametrize its functional space, we have introduced two different models which allow to vary $h_f$ extensively. The parameters involved are then used within our stochastic minimization procedure to obtain a correlation function with least deviation from a prescribed 