the interval $(x_{\max},1]$ (see (\[genhf\])). Here we will investigate two different choices for the element $h_{f,i}$. In both cases continuity of $h_f$ is guaranteed. The first case, refered to in the following as model I, is obtained by choosing the order of $h_{f,i}$ at the left point $x_p(i-1)$ in the interval $[x_p(i-1),x_p(i))$. This parametrization allows us to fit the derivative of the map at $x=0$ and has the advantage of using a minimal set of parameters in describing the element $h_{f,i}$. The disadvantage of this choice is that one cannot fit independently the order of the maximum of the map at the right point $x_p(N)$ of the last interval $[x_p(N-1),x_p(N)]$. This is the reason for considering also a second case (model II). Here we use for the element $h_{f,i}$ an expression representing expansions around both limiting points $x_p(i-1),x_p(i)$ which match together at some point of the interval $[x_p(i-1),x_p(i)]$. An additional parameter is therefore required in order to satisfy continuity for $h_{f,i}$ and its first derivative in $(x_p(i-1),x_p(i))$. Due to this fact model II uses 2 more parameters than model I for the description of a single element $h_{f,i}$.

Let us now discuss the two models in more detail. For model 