obtained via its defining formula $$C(n)=\int_{0}^{1}x f^{(n)}(x) d\mu(x) - \left(\int_{0}^{1}x d\mu(x)\right)^2
\label{corfct}$$ which requires a numerical integration. Here $f^{(n)}$ is the $n-th$ iterate of the map $f$.

To proceed with our central subject, the inverse problem, we assume that the first $m$ values of the autocorrelation function are given. This can be due to some experimentally given data for which a dynamical system should be modelled or due to the fact that a definite behaviour of the correlation function is required in the context of the control of a physical system. As discussed above each set $\{x_p(i),y_p(i),\alpha(i)\},~i=1,2,..,N$ respecting the constraint of monotony determines uniquely the autocorrelation function $C_{h_f}(n)$ of the corresponding map $f(x)$. One can now ask for the best set $\{x_p(i),y_p(i),\alpha(i),i=1,...,N\}$ in the sense that the resulting $C_{h_f}(n)$ possesses the least possible deviation from the given autocorrelation $C(n)$. As a measure for the above-mentioned deviation one can use a $\chi^2$-like cost function. We therefore introduce the following quantity: $$K[h_f]=\sqrt{\sum_{j=1}^{m} \left( \frac{C_{h_f}(j)-C(j)}{C(j)} \right)^2}
\label{chi2}$$ The functional $K[h_f]$ is a highly nonlinear function of the parameters $\{x_p(i),y_p(i),\alpha(i)\}$ and we are looking for the global minimum of this function. To perform the minimization of $K[h_f]$ we use a Monte-Carlo (MC) approach based 