Elliptic area-preserving maps correspond (in an appropriate eigenbasis) to rigid rotation, where the presence of a discontinuity caused by $g$ will lead to very complicated dynamics [@As96; @As97; @Go1; @ACP97]. The elliptic–hyperbolic transition for the linear maps on the torus was studied by Amadasi and Casartelli [@AC91], while certain properties of linear parabolic maps were analyzed in [@Zyc98]. In particular, a generic parabolic map displays some sort of sensitive dependence on initial conditions. However, this property is not related to the Lyapunov exponents (which are always zero for parabolic maps), but is due to the discontinuity of the map.

Summary of main results
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We parametrize the set of parabolic area-preserving maps by $A$ and $\alpha$, namely $$\label{eqmapAalpha}
\begin{array}{l}
{\left(\begin{array}{c} {x'} \\ {y'} 
\end{array}\right)}=f{\left(\begin{array}{c} {x} \\ {y} 
\end{array}\right)}=
{\left(\begin{array}{c} {(1+A)x+\frac{A}{\alpha}y ~(\bmod 1)} \\ {-\alpha Ax+(1-A)y~(\bmod 1)} 
\end{array}\right)}
\end{array}.$$ Section \[secinvert\] characterises (in Theorem \[thmnoninvert\]) the open dense set of $(A,\alpha)$ such that the mapping (\[eqmapAalpha\]) is not invertible and discuss some properties of the invertible maps. Section \[secmaximal\] examines properties of the [*maximal invariant set*]{} [@Zyc98] $$X^+=\bigcap_{n=0}^{\infty} f^n(X)$$ which by the previous result is strictly smaller than $X$ for most $(A,\alpha)$. Note that any $A\subset X$ with $f(A)=A$ will have 