number of points. It obeys the equations:

$$\begin{aligned}
h^\prime_f(x)<0 &&\;\;\;\;x\in[0,x_{\max}] \nonumber \\
h_f(0)=1  &&\;\;\;\;h_f(x_{\max})=x_{\max}
\label{hfprop}\end{aligned}$$

In terms of $h_f$ we obtain the general solution to IFPP within the class of unimodal maps as: $$f(x)=\mu^{-1}\left(1-|\mu(x)-\mu(H_f(x))|\right)
\label{funx}$$ where $H_f(x)$ is: $$H_f(x)=\left\{ \begin{array}{l@{\quad;\quad}l}
        h_f(x)&0 \leq x<x_{\max}\\
        h^{-1}_f(x)&x_{\max}\leq x \leq 1\\
        \end{array}\right.
\label{genhf}$$ and $\mu(x)=\displaystyle{\int_0^x} \rho(y) dy$ is the corresponding invariant measure. All unimodal maps with prescribed invariant density $\rho(x)$ are given by (\[funx\]), $h_f(x)$ taking on all possible functional forms obeying (\[hfprei\]) and (\[hfprop\]) (for more details on the above derivation as well as its relation to the standard conjugation procedure we refer the reader to [@PiSchmeDia]).

The above shows that fixing the invariant measure is a relatively weak constraint in the framework of the inverse problem and that there is still a considerable freedom to model the mapping. In the next section we will use this freedom and suitably parametrize $h_f(x)$ such that, by tuning appropriately the corresponding parameters, we get a map with an autocorrelation function possessing a minimum deviation (to be defined below) from a desired correlation function.

Stochastic optimization as a tool for the construction of a map
===============================================================

Before presenting our search method for the optimal map in 