Periods of periodic points of 1-norm nonexpansive maps

In this paper several results concerning the periodic points of 1-norm nonexpansive maps will be presented. In particular, we will examine the set R(n), which consists of integers p ?? 1 such that there exist a 1-norm nonexpansive map f: Rn ¿ Rn and a periodic point of f of minimal period p. The principal problem is to find a characterization of the set R(n) in terms of arithmetical and combinatorial constraints. This problem was posed in [12, section 4]. We shall present here a significant step towards such a characterization. In fact, we shall introduce for each n ¿ N a set T(n) that is determined by arithmetical and combinatorial constraints only, and prove that R(n) ¿ T(n) for all n ¿ N. Moreover, we will see that R(n) = T(n) for n = 1, 2, 3, 4, 6, 7, and 10, but it remains an open problem whether the sets R(n) and T(n) are equal for all n ¿ N.