September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 1
42 Global Attractors of Non-autonomous Dissipative Dynamical Systems
the sets M
i
(i =
1.k). Assuming the contrary, we obtain that the set M
i
can be
presented in the form of a union of two closed nonempty disjoint positively invariant
sets M
0
i
and M
00
i
in V
i
. By virtue of Lemma 1.24 W
s
(M
0
i
) ∩ W
s
(M
00
i
) = ∅. Note,
that the sets M
0
i
and M
00
i
are asymptotically stable and W
s
(M
i
) = W
s
(M
0
i
) ∪
W
s
(M
00
i
). The sets M
0
i
and M
00
i
are open and positively invariant. That contradicts
the connectivity of the set V
i
.
Let now (X, f ) be a discrete dynamical system. Since the set M is uniformly
attracting, then for a sufficiently large number m we have f
m
(H) ⊂ H and, con-
sequently, f
m
(V
i
) ⊂ V
1
∪ V
2
∪ ··· ∪ V
k
. According to the connectivity of the sets
V
1
, V
2
, . . . , V
k
and the continuity of the mapping f, the sets f
m
(V
i
) i =
1, k
are connected and, consequently, for every number i there exists a unique num-
ber j such that f
m
(V
i
) ⊂ V
j
. Let as above M
i
:= M ∩ V
i
(i =
1, k), then
f
m
(M
i
) ⊆ f
m
(M) ∩ f
m
(V
i
) ⊆ M ∩ V
j
= M
j
. Since the set M is positively in-
variant, then f
m
(M) ⊆ M and, consequently, f
m
(M
i
) ⊆ M
j
. Thus, the mapping
f
m
permutes the sets M
1
, M
2
, . . . , M
k
and, consequently, there exists a number n
such that f
n
(M
i
) ⊂ M
i
(i =
1, k). Thus, every set M
1
, M
2
, . . . , M
k
is positively
invariant with respect to mapping f
n
. Then in view of Lemma 1.26 each of the sets
M
1
, M
2
, . . . , M
k
is asymptotically stable with respect to the discrete dynamical
system (X, f
n
) and W
s
(M) = W
s
(M
1
) ∪ W
s
(M
2
) ∪ ··· ∪ W
s
(M
k
). Note, that
f
n
(V
i
) ⊆ V
i
and V
i
⊂ W
s
(M
i
) (i =
1, k). Using the similar arguments as ones
above, for the case T = R
+
we obtain the connectivity of the set M
i
. Theorem is
completely proved.
Theorem 1.32 Let (X, Z
+
, f) be a locally dissipative dynamical system and J be
its Levinson center. If the space X is connected and locally connected, then the set
J is also connected.
Proof. If the space X is connected and locally connected, then by Theorems
1.17 and 1.31 the center of Levinson of the discrete dynamical system (X, Z
+
, f)
consists of finite numbers of components of the connectivity J
1
, J
2
, . . . , J
k
. In
the proof of Theorem 1.31 there has been established the existence of a number n
such that f
n
(J
i
) ⊆ J
i
(i =
1, k). Put P := f
n
and consider the discrete dynamical
system (X, P ) generated by the the positive powers of the mapping P . Note, that
the system (X, P ) is compactly dissipative and satisfies the following condition:
J = J
1
tJ
2
t···t J
k
. In addition, we have P (J
i
) ⊆ J
i
(i ∈
1, k). Thus, the center
of Levinson
˜
J of the discrete dynamical system (X, P ) is decomposable, but this
fact contradicts Theorem 1.29 because the space X is connected and, consequently,
indecomposable. The theorem is proved.
Remark 1.7 We note that only the connectedness of the space X, without local
connectivity, does not guarantee the connectivity of the center of Levinson J of the
discrete dynamical system (X, f ) (see
[
166
]
).