Multidimensional contracted rotations

We study the dynamics of multidimensional contracted rotations and address a problem posed by Y. Bugeaud and J-P. Conze in $\mathit{\text{Acta}}\mathit{\text{Arithmetica}}$ in 1999. More precisely, we show that if $A$ is an invertible linear contraction of ${\mathbb{R}}^d$, then the map $f:[0,1{)}^d\to [0,1{)}^d$ defined by $f(x)=Ax+b(\text{mod}{\mathbb{Z}}^d)$ is asymptotically periodic for Lebesgue almost all $b\in {\mathbb{R}}^d$.