10 DISCRETE DYNAMICAL SYSTEMS
to a compact set there is a convergent subsequence of the
{Tkiyj}
which we
label the same. Since the limit of this sequence must belong to
CJ(S),
we ob-
tain a contradiction. Therefore, u(B) attracts B. Lemma 2.1.1 implies UJ(B) is
invariant. This completes the proofs of the assertions concerning
UJ(B).
The same type of argument proves the assertions for a(B) and the lemma is
proved.
2.2. Stability of invariant sets and asymptotically smooth maps.
Our objective in this section is to study the relationships between asymptotic
stability and uniform asymptotic stability of compact invariant sets. This will
motivate the introduction of an important class of mappings, namely, asymptot-
ically smooth maps.
Suppose T: X —• X is continuous and J is an invariant set. The set J is
stable if, for any neighborhood V of J, there is a neighborhood U of J such that
TnU
C V for all n 0. The set J attracts points locally if there is a neighborhood
W of J such that J attracts points of W, The set J is asymptotically stable (a.s.)
if J is stable and attracts points locally. The set J is uniformly asymptotically
stable (u.a.s.) if J is stable and attracts a neighborhood of J.
If J attracts points locally, then it need not be that J is stable. This is easily
seen by choosing T to be the time one map of an appropriate flow in the plane.
Clearly, J u.a.s. implies J is a.s. Without further hypotheses on T, J being
a.s. may not imply that J is u.a.s. It is not trivial to give an example for which
a.s. does not imply u.a.s. However, there are examples in Cooperman [1978] for
linear maps and Brumley [1970] (see also Hale [1974]) for linear neutral delay
differential equations. We content ourselves with remarking that these examples
have the property that the cj-limit set of every point is zero and yet the rate of
approach to zero is not uniform in t for every point in X. For a linear map T, it
is not difficult to show that u.a.s. implies \Tn\ &An, for all n, where A E [0,1)
and A ; is a constant. Therefore, for such maps, a.s. does not imply u.a.s.
We now study the stability concepts in more detail. An equivalent definition
of stability is contained in
LEMMA 2 . 2 . 1 . An invariant set J is stable if and only if, for any neighbor-
hood V of J, there is a neighborhood V C V of J such that TV C V.
PROOF. If J is stable and V is a neighborhood of J, then there is a neigh-
borhood W of J such that
TnW
C V. Let V = {jn
QTnW.
Then V satisfies
the stated properties. The converse is clear.
The following result relates various types of attractivity properties for stable
J.
LEMMA 2.2.2 . If J is a compact invariant set which is stable, then the
following statements are equivalent
(i) J attracts points locally.
(ii) There is a bounded neighborhood W of J such thatTW C W and J attracts
compact sets ofW.
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