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
W c f
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
where H^oo := maxi;
{|^| | z = (zu ... ,zn) G
Note that the sets ^ ( n ) , i = 1,2,3, are closed under composition of functions.
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
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.
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.
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).
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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