2 GENERALIZATIONS OF THE PERRON-FROBENIUS THEOREM
an integer N N(D, V, || ||) such that px N whenever px is the minimal period
of a periodic point x G D of / .
In general, if D is a topological space and g : D D is a map, we shall say
that £ G D is a periodic point of g of minimal period p if gp($,) = £ and ^ ' ( 0 7^ £ for
0 j p. If £ has minimal period p and #m(£) = £, it is well-known that p|ra. As
is suggested by Eq. (1.1), the periodic points of an /i-nonexpansive map f : D D
play a central role in understanding the dynamics of the discrete dynamical system
z-/
f c
(x), fc0.
For general sets D C
Rn,
very little is known about the possible periods of
periodic points of Zi-nonexpansive maps / : D D. An upper bound has been
obtained by Misiurewicz [8], but this bound (N =
n!22
) seems to be conservative.
This is related to the fact that / may not have an extension F :
Rn

Rn
which is /i-nonexpansive. However, if
Kn
= {x G
Rn
I Xi 0, 1 i n}, (1.2)
and / :
Kn

Kn
is /i-nonexpansive and /(0) = 0, Akcoglu and Krengel [1]
have proved that the minimal period p of any periodic point of / satisfies p n\\
and Scheutzow [17] has shown that p lcm(l, 2,3,... ,n). In [10,11] Nussbaum
established various other constraints on possible periods and in [13], Nussbaum
and Scheutzow presented the idea of an "admissible" array as a natural object to
describe (generalizations) of the constraints found earlier in [10]. In this paper we
shall further analyze the admissible arrays, but first we need to recall some notation
and some further results from the literature.
The cone Kn induces a partial ordering on Rn by
x y if and only if Xi yi for 1 i n,
where Xi and yi denote the coordinates of x and y respectively. We shall write x y
if x y and x 7^ y\ and we shall write x « y if xi yi for 1 i n. We shall
use the notation x ^ y to mean that it is false that x y, and we shall say that
x
Rn
and y G
Rn
are incomparable or not comparable if x ^ y and y ^ x. A map
/ : D C
Rn

Rn
is order-preserving if f(x) f(y) for all x, y G D with x y. If
fi(x) denotes the
ith
coordinate of f(x), then / is called integral-preserving if
n n
^2fi(x) = J2^i for all xeD. (1.3)
i=\ i=l
DEFINITION
1.1. Let Kn denote the positive cone in Rn and u = (1,1,... , 1) G
Rn. Consider the following conditions on maps / : Kn Kn:
1. /(o) = 0,
2. / i s order-preserving,
3. / i s integral-preserving,
4. / is nonexpansive with respect to the /i-norm,
5. f(\u) = Xu for all A 0
and define the following sets of maps
jr^n) = {f
:Kn -Kn\f
satisfies (1), (2), (3) and (5)},
jr2(n)
=
y :Kn
-Rn\f
satisfies (1), (2) and (3)},
and
jr3(n) = {/ :Kn -+ Kn \ f satisfies (1) and (4)}.
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