R}}^2/{{\bf Z}}^2$ to itself that is area-preserving and such that the linearization is at almost every point in ${{\bf T}}^2$ is $M$. This matrix has two eigenvalues equal to one. The map is continuous everywhere on the torus except on a one-dimensional discontinuity $D$ in ${{\bf T}}^2$. The behaviour of $f$ on $D$ does not affect a full measure set of $X$.

Parabolic area-preserving maps are not typical in the set of almost-everywhere linear maps on the torus [@Be]. However, their dynamical properties are of a particular interest, since such maps can be considered as an interpolating case between the hyperbolic maps ($|t|>2$) and the elliptic maps ($|t|<2$) in the area-preserving case. They are in some sense generalisations of interval exchange maps [@Ke75; @Ve78; @Ka80], piecewise rotations [@Go2; @ACP97] and interval translation maps to two dimensions. In fact, in Section \[secmaximal\] and the subsequent sections, our results use the fact that for rational coefficients of $M$ the map (\[eqmap\]) may be decomposed into a one-parameter family of 1 dimensional interval translation maps.

Hyperbolic area-preserving maps are characterized by a positive Liapunov exponents, and are often studied as model chaotic area-preserving dynamical systems (see eg. [@Be; @Ott94] and the Baker’s transformation). 