6

GENERALIZATIONS OF THE PERRON-FROBENIUS THEOREM

DEFINITION

1.12. If / : D c W1 - Mn, we shall write / G Qi{n) (respectively,

/ £

£/2(n))

^

a n

d

on^y

if ^ is a lower semilattice (respectively, lattice), f(D) C D

and f is a lower semilattice homomorphism (respectively, a lattice homomorphism)

of D.

DEFINITION

1.13. Let p be a positive integer. For j = 1,2, we shall write

p G Qj (n) if and only if there exists a map / G Qj (n) and a periodic point £ G Kn

of / of minimal period p.

From Theorem 1.10 it follows that P${n) C Qi(n) and from Theorem 1.11 it

follows that P2(n) C Q2(^)« The following theorem describes the situation more

precisely.

THEOREM

1.14. (See [13]). For eten/ positive integer n the following inclu-

sions hold: P\(n) C Q2W C Q(n) and

P(n) C Pi(n) C P2(n) C P3(n) C Qi(n) C Q(n).

REMARK

1.15. From the results in Chapter 7 it follows that, in general, P(n) 7^

Q(ri). However, in a subsequent paper, together with M. Scheutzow [14], it is proved

that actually

P2(n) = Q(n) for n 1.

Therefore the set Q(n) is a central object of study. Also note that as a consequence

of Theorem 1.5, the set Q(n) satisfies rules A and B of Definition 1.6.

From Theorem 1.14 and Remark 1.15, it follows that it is of interest to study

the complement of P(n) in Q(n). Since Q(n) is rather difficult to compute, the

goal of this paper is to analyze how well P(n) approximates Q(n). In the following

theorems we outline the results that we shall prove in this paper.

THEOREM

A. If S C {1,2,... , n} is a set such that lcm(S) has at most three

prime factors, then lcm(S) G Q(n) if and only if lcm(S) G P(n).

THEOREM

B. If S C {1,2,... ,n} is a set such that lcm(S) has four prime

factors and gcd(S) 1, then lcm(S) G Q(n) if and only if lcm(S) G P(n).

THEOREM

C. For every positive integer n less than or equal to 50 we have

Q(n) = P(n).

THEOREM

D. There exist integers n and sets S such that lcm(S) G Q(n) while

lcm(S) £ P{n). In particular, q =

23

x

72

x 11 x 13 G Q(78) and q G P(79) but

g0P(78) .

The difficulty in proving these results is that the definition of Q(n) is indirect.

Theorems A and B give necessary and sufficient conditions for special sets S to be

array-admissible for n and are very interesting but, unfortunately, cannot be the

whole story, as follows from Theorem D.

An interesting consequence of our results concerns the possibility of extending

an /1-nonexpansive map / : D — » D defined on a closed subset D of Kn to the whole

of Kn as an ^-nonexpansive map. For example, let po = (0,1,1), p\ = (1,0,1),

p2 = (3/2,1/2,0) and p3 = (1/2,3/2,0) and D = {po,Pi,jp2,P3}. H we define

a map / : D -» D by f(pi) = p

i + 1

for i = 0,1,2 and f(p3) = p0, then / is

order- and integral-preserving and nonexpansive in the

/1-norm.

Furthermore, /