Permutations of ${\mathbb{Z}}^d$ with restricted movement

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 $A\subset {\mathbb{Z}}^d$. For $d=1$, such permutations occur, for example, in restricted orbit equivalence, 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 examples of multidimensional shifts of finite type.