Graph Theoretical Criteria for the Wildness of Residue-class-wise Affine Permutations

A permutation of Z is called residue-class-wise affine if there is a positive integer m such that it is affine on residue classes (mod m). The smallest such integer is called the modulus of the respective permutation. A residue-class-wise affine permutation is called tame if the moduli of its powers are bounded, and wild otherwise. In this article, we describe two criteria for deciding whether a given residue-class-wise affine permutation is tame or wild. These criteria are based on properties of certain graphs assigned to residue-class-wise affine permutations, and they are suitable for computational purposes.