its first derivative at $x=x_s(i)$. We arrive then at the following generating formula for $h_f$: $$\begin{aligned}
h_{f,i}(x)=y_p(i-1) - &S_i& \alpha_R(i)(x_s(i)-x_p(i-1))^{1 - \alpha_L(i)}
 (x-x_p(i-1))^{\alpha_L(i)}~; \nonumber \\
 x &\in& [x_p(i-1),x_s(i)) \nonumber \\
h_{f,i}(x)=y_p(i) +
&S_i& \alpha_L(i)(x_p(i)-x_s(i))^{1 - \alpha_R(i)}
(x_p(i)-x)^{\alpha_R(i)}~; \nonumber \\
x &\in& [x_s(i),x_p(i))
\label{hfexpn}\end{aligned}$$ with: $$S_i=\frac{(y_p(i-1)-y_p(i))}
{\alpha_R(i) (x_s(i)-x_p(i-1)) + \alpha_L(i) (x_p(i)-x_s(i))}
\label{shel}$$ $h_f(x)$ given in eq.(\[hfexpn\]) can also be inverted analytically to obtain $h_f^{-1}$ defined in $[x_{\max},1]$. Comparing eqs.(\[hfexp\],\[hfexpn\]) we see immediately that the element of model II has 2 more parameters than the element of model I.

In the remaining part of this section we will discuss the algorithm which allows us to obtain dynamical systems possessing certain statistical properties based on the above ansatz for the auxiliary function $h_f(x)$. To this end we will concentrate on the parametrization of model I. One can then directly apply these ideas to the case of model II. We have determined the ansatz of $h_f(x)$ and therefore also of $f(x)$ in terms of the parameters $\{x_p(i),y_p(i),\alpha(i)\}$. Once the values of these parameters are chosen the map is completely specified. Due to its appearance (see eq.(\[funx\])) we automatically know its invariant density (measure) and expectation values of observables. The corresponding autocorrelation function can be 