Multidimensional contracted rotations

Abstract

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 $f(x)=Ax+b(\text{mod}{\mathbb{Z}}^d)$ 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.

10 pages, 2 figures