Multidimensional contracted rotations

We study the dynamics of multidimensional contracted rotations and address a problem posed by Y. Bugeaud and J-P. Conze in \textit{Acta 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 $f(x) = Ax +b\,\,(\textrm{mod}\,\mathbb{Z}^d)$ is asymptotically periodic for Lebesgue almost all $b\in\mathbb{R}^d$. We also include an example of a family of multidimensional contracted rotations $(d>1)$ not conjugate to the product of one-dimensional contracted rotations $(d=1)$, showing that our result cannot be reduced to or derived from the one-dimensional result of Bugeaud and Conze.