Permutations of Zd with restricted movement

We investigate dynamical properties of the set of permutations of Z(d) with restricted movement, i.e., permutations pi of Z(d) such that pi (n) n lies, for every n is an element of Z(d), in a prescribed finite set A subset of 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 >= 2 these sets of permutations provide natural classes of multidimensional shifts of finite type.