$A\subset X^+$ and so in this sense the set is maximal.

We say a map (\[eqmapAalpha\]) is [*semirational*]{} if $\alpha$ is rational. It is [*rational*]{} if both $A$ and $\alpha$ are rational. If $f$ is not semirational we say it is [*irrational*]{}. Note that the map is rational if and only if $a$, $b$, $c$ and $d$ are rational.

We investigate the two dimensional Lebesgue measure $\ell(\cdot)$ of this set, showing in Theorem \[thmposmeas\] that for rational parabolic maps we have $\ell(X^+)>0$. We give explicit lower and upper bounds for $\ell(X^+)$ depending on $A$ and $\alpha$.

We discuss two particular examples of rational maps where we can compute $\ell(X^+)$, namely with $A=\frac{1}{2}$ and $\alpha=1$ or $\frac{1}{2}$. In the former case we show that $\ell(X^+)=\frac{1}{2}$ and in the latter case we show that $\ell(X^+)=\frac{1}{2}-\frac{\sqrt{3}\pi}{72}-\frac{1}{8}\ln 3\sim
0.28710$. In some special cases, e.g., $-1\leq \alpha<0,~ |A| \le 1 $ and $ A = 1, ~ 0< \alpha < 1$ , we can compute $X^+$, and thus we can obtain exact values of $\ell(X^+)$. Other than these cases we have not been able to compute exact values of $\ell(X^+)$ other than by numerical approximation, even for rational $A$ and $\alpha$. We suspect that for 