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 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.
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