theorem, $$\begin{aligned}
\ell(X^+) &=& \int_{B\in[1-\frac{1}{s})}\int_{s\in X_B^+}
\,d\theta\,ds
\\
&=& \int_{B\in[1-\frac{1}{s})}\ell(X_B^+)\,ds.\end{aligned}$$ Upon integrating the estimates in Theorem \[thmxb+meas\] we obtain the result. [\
]{}

We can characterise the dynamics on the lines $L_B$ in the following way:

For any rational map, almost all $B$ and all $x\in L_B$, $\omega(x)$ consists of a set with positive one-dimension measure in $L_B$. Conversely, for a countable set of $B$ and all $x\in L_B$, the trajectory going through $x$ is eventually periodic.

[[**Proof**]{} ]{}In the proof of Theorem \[thmposmeas\], the reduced map $S_B$ will be an irrational rotation for all irrational $B$ and a rational rotation for all rational $B$. The result follows. [\
]{}

Density of irrational $B$ simply implies the following result for the original map.

For any rational parabolic map, the preimages of the discontinuity are dense in $X^+$.

We do not have a precise analytical expression for $\ell(X^+)$ even for rational parabolic maps. Numerically one can approximate the measure and obtain various values of $\ell(X^+)$; see for example Table \[tablevals\]. In certain simple cases one can obtain exact values of $\ell(X^+)$ by constructing the maximal invariant set explicitly.

\[propsimplecases\] One can compute $$\ell(X^+)= \left\{
\begin{array}{ll}
1 & \mbox{ if the map 