the sense of some desired statistical properties let us specify in more detail what we want to achieve and what our input for the problem at hand is. Consider a given invariant density $\rho(x)$ (corresponding to a finite measure) which arises in the asymptotic limit ($t \to \infty$) from the dynamics of some unknown $1-d$ fully chaotic single humped map. Consider also as given the first $m$ values $\{C(1), C(2),..,C(m)\}$ of the time autocorrelation function $C(n)$ (note that $C(0)$ is determined entirely through $\rho(x)$). We are seeking a map $f(x)$ which possesses the above statistical properties ($\rho(x), C(n) (n=0,...,m)$). We focus in the following on the representation (\[funx\]) for all admissible maps where the auxiliary functions $h_f(x)$ fulfill the requirements (\[hfprei\],\[hfprop\]). Using this expression for $f(x)$ it is guaranteed that the map we are looking for possesses the invariant density $\rho(x)$ (provided of course that $\mu(x)=\int_0^x \rho(t)~dt$ in (\[funx\]) exists). All our freedom in modelling the map $f(x)$ is now contained in the auxiliary function $h_f(x)$. We will use a suitable representation of $h_f(x)$ to parametrize the map $f(x)$. This parametrization is the starting point for a stochastic optimization procedure to obtain a map with the desired autocorrelation function $C(n)$ 