In this paper we will examine the asymptotic behaviour of the iterates of linear maps A : Rn ¿ Rn that are nonexpansive (contractive) with respect to a classical p-norm on Rn. As a main result it will be shown that if 1 ?? p ?? 8 and p /= 2, there exists an integer q ?? 1 such that the sequence (Akqx)k is convergent for each x ¿ Rn. Moreover the integer q is the order, or twice the order, of a permutation on n letters.