cases where we can compute it, the maximal invariant set has positive measure. Nonetheless, there is numerical evidence ([@Zyc98]) that suggests that

-   There are $(\alpha,A)$ for which the Hausdorff dimension of $X^+$ is less than 2 (in particular, such that $\ell(X^+)=0$). By results here, this can only occur in cases where both parameters are irrational.

-   This is common for $A>0$ and $\alpha>0$.

Note that not all irrational parabolic maps have zero measure; for example Proposition \[propsimplecases\] shows that there are open regions in $(A,\alpha)$ with $\alpha<0$ where $\ell(X^+)>0$. For semirational cases ($\alpha=r/s$) we can reduce to maps of the form (\[map1d\]), however we cannot find factors that are rotations if $A$ is not rational. Results of suggest that the Hausdorff dimension of $X^+$ may vary between $1$ and $2$. The Hausdorff dimension is a rather subtle characteristic of a set. For example, two sets may be arbitrarily close in the sense of the Hausdorff metric, but their dimensions may differ by an arbitrarily large number. Therefore, small changes of the parameter $\alpha$, which do not change much the structure of the maximal invariant set (in sense of the Hausdorff metric) may induce wild fluctuations of its dimension. Note 