Wildness of iteration of certain residue-class-wise affine mappings

A mapping is called residue-class-wise affine if there is a positive integer m such that f is affine on residue classes (mod m). The smallest such m is called the modulus of f. In this article it is shown that if the mapping f is surjective but not injective, then the set of moduli of its powers is not bounded. Further it is shown by giving examples that the three other combinations of (non-)surjectivity and (non-)injectivity do not permit a conclusion on whether the set of moduli of powers of a mapping is bounded or not.