Let Kn
={x∈ℝn[ratio ]xi[ges ]0
for 1[les ]i[les ]n} and suppose that
f[ratio ]Kn→Kn
is nonexpansive with respect to
the [lscr ]1-norm and f(0)=0. It is known that for every
x∈Kn there exists a periodic point
ξ=ξx∈Kn (so
fp(ξ)=ξ for some minimal positive integer
p=pξ) and fk(x)
approaches {fj(ξ)[ratio ]0[les ]j<p}
as k approaches infinity. What can be said about P*(n),
the set of positive integers p for which there exists a map f as above
and a periodic point ξ∈Kn of f of
minimal period p? If f is linear (so that f is a nonnegative, column
stochastic matrix) and ξ∈Kn is a periodic point
of f of minimal period p, then, by using the
Perron–Frobenius theory of nonnegative matrices, one can prove that p
is the least common multiple of a set S of positive integers the sum of
which equals n. Thus the paper considers a nonlinear generalization
of Perron–Frobenius theory. It lays the groundwork for a precise description of
the set P*(n). The idea of
admissible arrays on n symbols is introduced, and these arrays are used to
define, for each positive integer
n, a set of positive integers Q(n) determined
solely by arithmetical and combinatorial constraints. The paper
also defines by induction a natural sequence of sets P(n), and it is proved that P(n)⊂P*(n)⊂Q(n). The
computation of Q(n) is highly nontrivial in general, but in a sequel
to the paper Q(n) and P(n) are explicitly
computed for 1[les ]n[les ]50, and it is proved that
P(n)=P*(n)=Q(n) for
n[les ]50, although in general P(n)≠Q(n).
A further sequel to the paper (with Sjoerd Verduyn Lunel) proves that
P*(n)=Q(n) for all n. The
results in the paper generalize earlier work by Nussbaum and Scheutzow and place
it in a coherent framework.