translation map*]{}

Introduction
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We consider a class of maps of the torus $X=[0,1]^2$ of the form $$\label{eqmap}
\begin{array}{l}
x'=ax+by ~(\bmod ~1)\\
y'=cx+dy ~(\bmod ~1)
\end{array}$$ where $(x,y)\in[0,1]^2$. This can be thought of as a map $f=g\circ M$ where $$M={\left(\begin{array}{cc} {a} & {b} \\ {c} & {d}
\end{array}\right)}$$ (we also write $M=(a,b;c,d)$ for convenience) and $g(x)=x-\lfloor x\rfloor$ is a map that takes modulo $1$ in each component. Although such maps are linear except at the discontinuity induced by the map $g$, their dynamical behaviour can be quite complicated. Depending on the eigenvalues $\lambda_{1,2}$ of $M$ we refer to the map as elliptic ($\lambda_1=\overline{\lambda}_2\neq \lambda_1$), hyperbolic ($\lambda_1>\lambda_2$) or parabolic ($\lambda_1=\lambda_2$). In this paper we focus on the area-preserving parabolic case with determinant $ad-bc=1$ and trace $a+d=2$. Such maps arise naturally on examining linear maps with a periodic overflow. In particular, suppose that one would like to iterate a matrix using a digital representation with very small discretization error but a finite range which we set to be $(0,1)$. If the calculation overflows such that the fractional part remains, we will get the map (\[eqmap\]) (compare, for example, with [@ACP97]).

In this case $f$ is a piecewise continuous map of ${{\bf T}}^2={{\bf 