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.
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