Permutations of $\mathbb {Z}^d$ with restricted movement
Volume 235 / 2016
Studia Mathematica 235 (2016), 137-170
MSC: 37A35, 37B10, 37B50.
DOI: 10.4064/sm8498-8-2016
Published online: 14 October 2016
Abstract
We investigate dynamical properties of the set of permutations of $\mathbb {Z}^d$ with restricted movement, i.e., permutations $\pi $ of $\mathbb {Z}^d$ such that $\pi (\mathbf {n})-\mathbf {n}$ lies, for every $\mathbf {n}\in \mathbb {Z}^d$, in a prescribed finite set $\mathsf {A}\subset \mathbb {Z}^d$. For $d=1$, such permutations occur, for example, in restricted orbit equivalence (cf., e.g., Boyle and Tomiyama (1998), Kammeyer and Rudolph (1997), or Rudolph (1985)), or in the calculation of determinants of certain bi-infinite multi-diagonal matrices. For $d\ge 2$ these sets of permutations provide natural classes of multidimensional shifts of finite type.
