many $\alpha>0$ and $A>0$ (both irrational) then $\ell(X^+)=0$, whereas $\ell(X^+)>0$ for $\alpha$ rational. Section \[secnonlinexample\] briefly discusses examples of nonlinear parabolic maps, and we conclude with some remarks on open problems in Section \[secdiscuss\].

Invertibility of linear maps on the torus {#secinvert}
=========================================

Consider a matrix $M$ corresponding to a parabolic map on the torus (\[eqmap\]). Apart from the special [*horocyclic*]{} [@CFS82] cases $${\left(\begin{array}{cc} {1} & {B} \\ {0} & {1}
\end{array}\right)}~~~~{\left(\begin{array}{cc} {1} & {0} \\ {B} & {1}
\end{array}\right)},
\label{horo}$$ any such matrix can be written in the form $$M=M_{A,\alpha}={\left(\begin{array}{cc} {1+A} & {A/\alpha} \\ {-\alpha A} & {1-A}
\end{array}\right)},
\label{mapM}$$ where $A$ and $\alpha$ are real parameters. Observe that the cases $\alpha, A \rightarrow 0$ (with $A/\alpha$ held constant) and $\alpha\rightarrow\infty$, $A\rightarrow 0$ (with $\alpha A$ held constant) correspond to the cases (\[horo\]). Observe that $$M_{A,\alpha}^{-1}=M_{-A,\alpha},$$ and if $S(x,y)=(y,x)$ is transposition then $S^2=I$ and $$SM_{A,\alpha}=M_{-A,1/\alpha} S.$$ Moreover, in the absence of the rounding $g$ we have $$M_{A,\alpha}^p=M_{pA,\alpha}$$ for any $p\in{{\bf Z}}$.

Noninvertibility of generic parabolic maps
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The next lemma gives necessary and sufficient conditions that (\[eqmap\]) is invertible (ignoring points that land on the discontinuity). We say the map is invertible if there is a full measure subset 