1. INTRODUCTION

3

Prom results of Crandall and Tartar [2] it follows that, if / G ^(n) then / is

nonexpansive with respect to the /i-norm. Therefore, we have

f i ( n ) c f

2

W c f

3

W .

If / satisfies (1), (2) and (3), one can easily show that / satisfies (5) if and only

if / is "sup-norm-decreasing", i.e., if and only if for all x G

Kn,

\\f(x)\\oo\\x\U

where H^oo := maxi;

n

{|^| | z = (zu ... ,zn) G

Mn}.

Note that the sets ^ ( n ) , i = 1,2,3, are closed under composition of functions.

DEFINITION

1.2. Let p be a positive integer. For j — 1,2,3, we shall write

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

of / of minimal period p.

Because Fi(n) C ^{n) C ^(n) we have, by definition,

Pi(n) C P2(n) C P3(n).

If Sn denotes the symmetric group on n symbols and a an element of Sn then,

by permutation of the coordinates, a induces a linear map a that belongs to T\ (n)

and it is easy to see that £ = (1,2,3,... , n) G Kn is a periodic point of minimal

period p equal to the order of a as an element of symmetric group Sn.

Thus P\{n) contains the set of all orders of elements of Sn. However, as the

next theorem shows Pi(n) is, in general, larger than the set of orders of elements

0fS

n

.

DEFINITION 1.3. Let n be a positive integer and £ = { l , 2 , . . . , n } . I f S c E ,

then lcm(S) denotes the least common multiple of the elements of S and gcd(S')

denotes the greatest common divisor of the elements of S. Furthermore |5| denotes

the cardinality of the set S.

THEOREM

1.4. (See Nussbaum [10].) If p± G Pi(rai) and p2 G Pi(n2), then

lcm(pi,p2) G P\{ni + 712). If Pi G Pi(m) for 1 i r, then rlcm(pi,p2,... ,pr) £

Pi (rra).

By using Theorem 1.4, and recalling that 2 G Pi (3) and 3 G Pi (3), we conclude

that 12 = 21cm(2,3) G Pi (6); but every element of 56 has order p 6.

A modification of the argument given in [10] to prove Theorem 1.4 can be used

to prove that a similar result holds for P2 (n). A proof of the following theorem will

be given in Chapter 8.

THEOREM

1.5. Ifpi G P2{n\) andp2 G P2{n2), then lcm(pi,p2) G P2(ni+n2).

If Pi £ P2{m) for 1 i r, then rlcm(pi,p2,... ,pr) G P2(rm).

Using Theorem 1.4 and 1.5 and the fact that fori = 1,2, Pi(l) = {1}, one can

also see that Pi(n) C Pi(n + 1) for all n 1 and that if p G Pi(n) and d|p, then

dePi(n) (i = l,2).

DEFINITION

1.6. We define inductively, for each n 1, a collection of positive

integers P(n) by P(l) = {1} and, for n 1, p G P(n) if and only if either

(A) p = lcm(pi,p2), where pi G P(ni), p2 G P(n2) and ni and n2 are positive

integers with n — n\ + n2 or

(B) n = rra for integers r 1 and ra 1 and p = rlcm(pi,p25 • • • ,Pr), where

Pi G P(ra) for 1 i r.