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)}.