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It is easy to show that (a) if |α| > L, then |α1| < |α|, (b) if |α| ≤ L, then |α1| ≤ L. Hence the orbit (1.1) is ultimately periodic for every α ∈ Z[i]. The proof of the two easy assertions stated above is given in the lecture notes [3]. An integer π ∈ Z[i] is said to be periodic if there is a positive integer k for which J(π) = π. Let P be the set of periodic points. From the assertions (a) and (b) we see that if π ∈ P, then |π| ≤ L. Repeating the expansion defined above, we obtain

Additive functions with respect to expansions over the set of Gaussian integers