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

Klaus Schmidt¹ ; Gabriel Strasser¹

¹ Mathematics Institute University of Vienna Oskar-Morgenstern-Platz 1 A-1090 Wien, Austria

Studia Mathematica, Tome 235 (2016) no. 2, pp. 137-170

Résumé

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.

Détail

DOI : 10.4064/sm8498-8-2016

Keywords: investigate dynamical properties set permutations mathbb restricted movement permutations mathbb mathbf mathbf lies every mathbf mathbb prescribed finite set mathsf subset mathbb permutations occur example restricted orbit equivalence boyle tomiyama kammeyer rudolph rudolph calculation determinants certain bi infinite multi diagonal matrices these sets permutations provide natural classes multidimensional shifts finite type