1. INTRODUCTION
5
of completeness we present some more definitions to explain the techniques and to
introduce sets of integers Qi(n), i = 1,2.
If £, y G Mn, we define x Ay and x V y in the standard way:
x A y := 2, ^i = min{x;, ^ } for 1 i n
xV y := w, Wi = max{a^, ^ } for 1 i n.
If V C
Mn,
V is called a iower semilattice if x Ay €V whenever x G 7 and y G F ;
V is called a lattice if x A y G V and x V y G F whenever x eV and y G V\
A finite lower semilattice (respectively, finite lattice) is a lower semilattice (lat-
tice) with finite cardinality. If A C
Mn,
there is a minimal (in the sense of set
inclusion) lower semilattice V D A and a minimal lattice Y D A; V is called the
lower semilattice generated by A, and Y the lattice generated by A. If \A\ oo, it
follows that \V\ oc and \Y\ oc. If V is a lower semilattice, a map h : V V
is called a lower semilattice homomorphism of V if
h(x Ay) = h(x) A h(y) for all x,y G V.
If Y is a lattice, a map h : Y Y is a lattice homomorphism if
/i(x Ay) = h(x) A h{y) and /i(x Vy) = h(x) V /i(y) for all x, y G V.
If W C Mn is a lower semilattice (lattice), h : W W is a lower semilattice (lattice)
homomorphism of W and £ G VF is a periodic point of minimal period p of h, we
let V denote the finite lower semilattice (lattice) generated by
A = {W{£)\0jp}.
From the definitions it follows that h(V) C V and
/ip(x)
= x for all x e V. In
particular, /i|y is a lower semilattice (lattice) homomorphism, h\V is one-to-one,
onto and
(h\V)-1
=
hp~1\V
is also a semilattice (lattice) homomorphism of V.
The relevance of these ideas in our situation is indicated by the following the-
orems.
THEOREM
1.10. (See Scheutzow [17]). Suppose that f G ^ ( n ) and that £ G
Kn is a periodic point of f of minimal period p. Let A = {P(£) \ 0 j p}. IfV
denotes the finite lower semilattice generated by A, then f(V) C V, f\V is a lower
semilattice homomorphism ofV,
fp(x)
= x for all x G V and (/|V)
- 1
/
P _ 1
| V
is a lower semilattice homomorphism ofV.
In the next theorem, recall that a norm || || on Rn is called strictly monotonic if
I Ml II2/II whenever 0 x y. The Zp-norms are strictly monotonic for 1 p oc;
the Zoo-norm is not strictly monotonic.
THEOREM
1.11. (See Nussbaum [9]). Suppose that f :
Kn
-
Kn
is an order-
preserving map with /(0) = 0 and f is nonexpansive with respect to a strictly
monotonic norm \\ ||. Assume that £ G Kn is a periodic point of f of minimal
period p, let A = {/J(£) | 0 j p] and let L denote the the finite lattice generated
by A. It then follows that f(L) C L, f\L is a lattice homomorphism of L, fp(x) = x
for all x G L and (/|L) - 1 = fp~1\L is a lattice homomorphism of L.
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