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

TL;DR

This paper explores the properties of permutations of integer lattices with movement restrictions, revealing their connections to dynamical systems, matrix determinants, and multidimensional shifts of finite type.

Contribution

It introduces and analyzes the class of restricted permutations on Z^d, highlighting their relevance to dynamical systems and multidimensional shift spaces.

Findings

Permutations with restricted movement relate to dynamical properties.

These permutations connect to the calculation of determinants of bi-infinite matrices.

They serve as natural examples of multidimensional shifts of finite type.

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 $d\ge2$ these sets of permutations provide natural classes of examples of multidimensional shifts of finite type.