were able to derive a general representation of all ergodic and chaotic complete unimodal maps with a given invariant density. This corresponds to the general solution of the so called inverse Frobenius-Perron problem [@DynSta; @InvPro; @BaraDa; @Gross; @Ghik; @Pala; @Hunt] within this class of maps. We review here the basic aspects of our approach.

As a starting-equation for the construction of the map we use the Frobenius-Perron equation: $$\rho(y)\,|dy|=\sum\limits_{x_i=f^{-1}(y)}\rho(x_i)|dx_i|
\label{froperequ}$$ where the summation runs over all preimages of $y$ (for unimodal maps $i=L(left),R(right)$). For any given complete unimodal $f(x)$ the right preimage of $y$ is determined if the left preimage is given and vice versa. The essential feature of our approach is the following: A prescribed relation between the two preimages reduces the number of independent differentials on the rhs of eq.(\[froperequ\]) and allows to integrate the Frobenius-Perron equation. Such a relation is given by the function $h_f(x)$ which maps the position of the left preimage onto the position of the right one:

$$\begin{aligned}
h_f&:&[0,x_{\max}]\longrightarrow[x_{\max},1] \nonumber \\
x_R&=&h_f(x_L)\;\;\;\;\;\;\mbox{with}\;\;f(x_L)=f(x_R)
\label{hfprei}\end{aligned}$$

where $x_{\max}$ is the position of the maximum of the map. $h_f(x)$ is a monotonously decreasing function on the defining interval and it is differentiable with the exception of a finite 