ABSTRACT.

Let Kn = {x e Mn | x{ 0, 1 i n} and suppose that / : Kn —

Kn is nonexpansive with respect to the /i-norm, ||x||i =

YM=I

xii a n d /(0) = 0- It

is known (see [1]) that for every x G ifn there exists a periodic point £ = £x G i^n

(so

/p(£)

= £ for some minimal positive integer p = p%) and

fk(x)

approaches

{P(0 | 0 j p} as /c tends to infinity. In a previous paper [13] the set P2(n)

of positive integers p for which there exists a map / as above and a periodic point

£ G

Kn

of minimal period p was related to the idea of "admissible arrays" and a set

Q(n) determined by certain arithmetical and combinatorial constraints. In a sequel

to this paper [14] it is proved that P2(n) = Q(n) for all n, but the computation of

Q(n) is highly nontrivial. Here we derive a variety of theorems about admissible

arrays and use these theorems to compute Q(n) explicitly for n 50 and prove

that P(n) — Pi{n) — Q(n) for n 50, where P(n) is a naturally occurring set

defined below.

Received by the editor September 3. 1996, and in revised form March 18, 1997.