Cheban D.N. Global Attractors Of Non-autonomous Dissipative Dynamical Systems

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The structure of the Levinson center with the condition of the hyperbolicity 157

Lemma 4.4 The sets R

(i = 1, 2, . . . , k) are locally maximal in Σ, i.e. for R

there exists a neighborhood U

of the set R

in Σ such that R

is a maximal closed

invariant set in U

Proof. Since R

∩ R

6= ∅ for all i 6= j, then there exist neighborhoods U

of the

sets R

such that

∩ U

= ∅ for i 6= j. Note, that in U

the set R

is maximal.

In fact, if we suppose that there exists a compact invariant set Λ

⊂ U

such that

* R

, then Λ

\ R

6= ∅. Let x ∈ Λ

\ R

. Since the sets α

and ω

are chain

recurrent

[

33, p.33

]

, then α

, ω

⊆ R

, i.e. x ∈ (W

) ∩ W

)) \ R

and,

consequently, x /∈ R(Σ). On the other hand, reasoning in the same way that in the

proof of Lemma 4.2, it is possible to show that x is chain recurrent, i.e. x ∈ R(Σ).

The obtained contradiction ﬁnishes the proof of the lemma. 

Lemma 4.5 The sets R

(i = 1, 2, . . . , k) are indecomposable.

Proof. Suppose that for some i the set R

can be represented in the form of a

union of two own closed invariant subsets A

and A

, i.e. R

= A

t A

. Since

∩A

= ∅ and R

∩R

= ∅ for all j 6= i, then there are neighborhoods U

and U

of the sets A

and A

respectively, such that

∩U

= ∅ and (U

∪U

) ∩R

= ∅

for all j 6= i. Let ε

> 0 be such that B(A

, ε

) ⊂

and B(A

, ε

) ⊂ U

We will take arbitrary points a

∈ A

, a

∈ A

and numbers 0 < ε < ε

and

t > 0. Since a

, a

∈ R

, then for the numbers ε and t there exists (ε, t, π)-chain

= x

, x

, . . . , x

= a

; t

, t

, . . . , t

} joining the points a

and a

. By Theorem

2.24 from

[

]

we can suppose that x

∈ R

(k = 0, 1, . . . , n). According to the

choice of the number ε and the invariance of the set A

, we have x

∈ A

for all

k = 0, 1, . . . , n, i.e. x

= a

∈ A

. The latter contradicts to our assumption. The

lemma is proved. 

4.2 The spectral decomposition of the Levinson’s center

Let h(X, S, π), (Y, S, σ), hi be a two-sided non-autonomous dynamical system with

the ﬁbers X

:= {x ∈ X | h(x) = y} (y ∈ Y ). And let every ﬁber X

be homeo-

morphic to V, where V is some n-dimensional manifold (for example, V = E

) and

p ∈ X

. Following the work

[

33, p.213

]

, we denote by W

(p) := {x ∈ X

| ρ(xt, pt) ≤

δ, t ≥ 0} and W

(p) := {x ∈ X

| ρ(xt, pt) ≤ δ, t ≤ 0} (δ > 0).

Deﬁnition 4.5 The compact invariant set Λ ⊆ X is said to be hyperbolic (or non-

autonomous dynamical system h(X, S, π), (Y, S, σ), hi has a hyperbolic structure on

Λ), if there are positive numbers N, ν, δ and γ such that

H1. W

(p) ∩ W

(p) = {p} for all p ∈ Λ, W

(p) and the sets W

(p) are sub-

manifolds from X

(y = h(p)), which are homeomorphic to the closed desk

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 4

158 Global Attractors of Non-autonomous Dissipative Dynamical Systems

in R

and R

n−k

respectively. In addition, if ρ(p, q) ≤ γ (p, q ∈ X

), then

(p) ∩ W

(p) 6= ∅;

H2. π

(p) ⊆ W

(π

p) for all t ≥ 0 and π

(p) ⊇ W

(π

p) for all t ≤ 0 and

and each p ∈ Λ;

H3. the manifolds W

(p) and W

(p) depend continuously on the point p ∈ Λ in

the distance of Hausdorﬀ;

H4. (a) ρ(p

t, p

t) ≤ Nexp(−νt)ρ(p

, p

) for all p

, p

∈ W

(p) and t ≥ 0;

(b) ρ(p

t, p

t) ≤ Nexp(νt)ρ(p

, p

) for all p

, p

∈ W

(p) and t ≤ 0.

Remark 4.1 a. If the set Λ ⊆ X is hyperbolic in usual sense

[

33, 261

]

, then

under some additional conditions of smoothness, the set Λ will also be hyperbolic in

the sense of the deﬁnition above. The inverse statement, generally speaking, is not

true.

b. For a discrete autonomous dynamical system there was introduced a close

notion (axiom A

) in the work

[

]

Denote by M(Σ) a closure of all the recurrent points of the set Σ. The following

statement takes place.

Theorem 4.1 Let Y be a compact minimal set, Σ be a compact invariant set

from X and let one of the following two conditions be fulﬁlled:

a. the number of minimal sets in Σ is ﬁnite;

b. if the set Σ contains an inﬁnite number of minimal sets, then on the set M(Σ)

(except, maybe, a ﬁnite number of isolated minimal sets) the non-autonomous

dynamical system h(X, S, π), (Y, S, σ), hi has a hyporbolic structure.

Then the relation ∼ decomposes the set R(Σ) on ﬁnite number of diﬀerent classes

of equivalence.

Proof. If the set Σ contains only a ﬁnite number of minimal sets, then the statement

of the theorem coincide with the lemma 4.1

Let now the condition b. of the theorem be fulﬁlled. And we suppose that the

theorem is not true. Then there is a denumerable family of classes of equivalence

: k = 1, 2, . . . }. Since the sets R

are closed, invariant and the set Σ is compact,

then every set R

contains at least one compact minimal subset M

⊆ R

. Since the

set Y is minimal, then h(M

) = Y for all k = 1, 2, . . . . Let y ∈ Y and p

∈ M

∩X

According to the compactness of the set Σ, we can suppose that the sequence {p

}

is convergent. Put p := lim

k→+∞

and we note that p ∈ Λ :=

∪{M

: k = 1, 2, . . . }.

Without loss of generality we can suppose that the set Λ is hyperbolic. Let γ > 0 be

the number ﬁguring in the condition H1. for the set Λ. The sequence {p

} ⊆ Λ is

convergent; therefore, starting from some k

, the manifolds W

) and W

)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 4

The structure of the Levinson center with the condition of the hyperbolicity 159

are a nonempty intersection for all k

, k

≥ k

. Let now k

6= k

and k

, k

≥ k

Chose points x

∈ W

) ∩ W

), x

∈ W

) ∩ W

) and consider the

homoclinic contour K := H(x

) ∪ H(x

). By Corollary 2.19

[

]

and taking in

consideration that on an ω-limit (α-limit) set every two points are equivalent in the

sense of the relation ∼, we obtain the equivalence of every two points of homoclinic

contour K, i.e. the set K belongs to one class of equivalence. Thus, beginning with

, the points p

and p

are equivalent for all k

, k

≥ k

. On the other hand,

the point {p

} was chosen from the diﬀerent classes of equivalence. The obtained

contradiction proves our theorem. 

Theorem 4.2 Under the conditions of Theorem 4.1 on the center of Levinson J

of the dynamical system h(X, S, π), (Y, S, σ), hi the following assertions hold:

(1) the relation ∼ decomposes the set R(J) on ﬁnite number of diﬀerent classes of

equivalence, i.e. R(J) = R

t R

t ··· t R

;

(2) the sets R

(i = 1, 2, . . . , k) are closed, invariant, indecomposable, locally max-

imal and in the collection {R

, R

, . . . , R

} there is no r (r ≥ 1) cycles;

(3) J = ∪{W

) : i =

1, k}, where W

) := {x ∈ X : lim

t→−∞

ρ(xt, R

) = 0}

(i =

1, k).

Proof. The ﬁrst and the second statements of Theorem 4.2 directly follow from

Lemmas 4.3-4.5 and Theorem 4.1. Now we will prove the third statement. Let

x ∈ W

) (i =

1, k), then by the compact dissipativity of the dynamical system

(X, S, π) the set H(x) is compact and, consequently, x ∈ J.

Inverse. Let x ∈ J. In view of the compactness and invariance of the set J, the

sets α

and ω

are nonempty, compact, indecomposable and chain recurrent

[

]

Since under the conditions of Theorem 4.1 R(J) = R

t···tR

and the sets

(i =

1, k) are disjoint, closed and invariant, then there are i, j ∈ {1, 2 . . . , k}

such that α

⊆ R

and ω

⊆ R

. Thus x ∈ W

). The theorem is proved. 

Remark 4.2 Theorems 4.1 and 4.2 are true also in the case, if we replace the

condition of the minimality of the set Y by the following condition: the set Y

contains only ﬁnite number of minimal sets.

4.3 One-dimensional systems with hyperbolic center

In this section we study the structure of the Levinson center for the scalar dissipative

equation u

= f(t, u) with recurrent (almost periodic, periodic) right-hand side.

Although the problem under study concerns the equation u

= f(t, u), its solution

is given within the framework of general non-autonomous dynamical system.

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 4

160 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Let Λ ⊆ X be a hyperbolic set of non-autonomous dynamical system

h(X, S, π), (Y, S, σ), hi, (X, h, Y ) be a ﬁnite-dimensional vector bundle with the ﬁber

, and |·| be a Riemannian metric on (X, h, Y ), that is compatible with the metric

ρ on X.

Remark 4.3 a. If p ∈ Λ is such that k = n (k = 0), then the set H(p) is

exponentially stable on S

−

b. For n = 1 the hyperbolic set H(p) is exponentially stable either on S

or on

−

Everywhere in this section, we will suppose that the set Y is compact minimal

and n = 1.

Lemma 4.6 If the compact minimal set M of X is hyperbolic, then h|

: M → Y

is a homeomorphism (or, what is the same thing, M

:= M ∩X

consists of exactly

one point for each y ∈ Y ).

Proof. Let M ⊆ X be a compact minimal subset of X. If M is hyperbolic,

then, by Remark 4.3 b., we can assume that M is exponentially stable on S

and,consequently, M is uniformly stable in the positive direction, i.e. for each ε > 0

there exists a γ(ε) > 0 such that ρ(m

, m

) < γ (m

, m

∈ M and h(m

) = h(m

))

implies the inequality ρ(m

t, m

t) < ε for all t ≥ 0. To complete the proof of the

Lemma it is suﬃcient to refer to Theorem 3 from

[

331, p.110

]

. 

Let Σ be a nonempty compact invariant set in X, {M

| α ∈ A} by the family

of all minimal subsets of Σ, and M(Σ) :=

∪{M

| α ∈ A}.

Lemma 4.7 If M(Σ) is hyperbolic, then Σ contains a ﬁnite number of minimal

sets M

, M

, . . . , M

and, consequently, M(Σ) = M

t··· tM

Proof. Let us assume the contrary, i.e. let there exist a countable family {M

: k =

1, 2, . . . } of distinct minimal sets in Σ. Let y ∈ Y and p

∈ M

∩X

(k = 1, 2, . . . ).

Since p

∈ M(Σ), we can assume that {p

} is convergent. Let p

→ p, then

p ∈ M(Σ). Since M(Σ) is hyperbolic, we can assume that (see Remark 4.3 b.)

H(p) is exponentially stable on S

, and, consequently, there exists a k

∈ N such

that ρ(p

t, p

t) ≤ N exp(−νt)ρ(p

, p

) for all k

, k

≥ k

and t ≥ 0. It follows

from the last inequality that ρ(p

t, p

t) → 0 as t → +∞ for all k

, k

≥ k

. But

this is not possible because by Lemma 4.6 p

and p

are jointly recurrent. This

contradiction proves the Lemma 4.7. 

Lemma 4.8 Under the conditions of Lemma 4.7, for each point x ∈ Σ there exist

recurrent points p

and p

(h(p

) = h(p

) = h(x)) such that the following conditions

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 4

The structure of the Levinson center with the condition of the hyperbolicity 161

are fulﬁlled:

a. lim

t→+∞

ρ(xt, p

t) = 0 and b. lim

t→−∞

ρ(xt, p

t) = 0. (4.1)

Proof. Let x ∈ Σ, ω and α-limit sets ω

and α

are nonempty, compact and

contain the minimal sets M

and M

, respectively. Since Y is minimal, it follows

that h(M

) = h(M

) = Y , and, since M(Σ) is hyperbolic, the sets M

and M

consist of exactly one point each for an arbitrary y ∈ Y .

Let x be non-recurrent (in the contrary case Lemma is obvious) and M

= {p

}

(i = 1, 2). Then there exist t

→ +∞ and t

→ −∞ such that xt

→ p

and

→ p

. By Lemma 4.6 p

→ p

(i = 1, 2) as n → +∞. Hence

ρ(xt

, p

) → 0 (4.2)

a. Let M

be exponentially stable on S

. Then (4.2) implies (4.1a). Now we

show that (4.1b) holds. Indeed, if M

is exponentially stable on S

, then M

exponentially stable on S

−

. If we assume the contrary, then, by (4.2) ρ(xt, p

t) → 0

as t → +∞. It follows from Lemma 4.6 that p

= p

= p. Let ε = ρ(x, p) > 0 and

γ(ε) > 0 be chosen for ε from the condition of uniform stability of H(p) on S

. Then

it follows from (4.2) that for suﬃciently large n we have ρ(xt

, p

) < γ(ε) and,

consequently, ρ(x(t + t

), p

(t + t

)) < ε for all t ≥ 0. In particular, ρ(x, p) < ε,

which contradicts the choice of ε. This contradiction proves the exponential stability

of the set M

on S

−

. From (4.2) it follows (4.1b).

b. We will show that no M

⊆ ω

cannot be exponentially stable on S

−

. Indeed,

if we assume the contrary, then it follows from (4.2) that ρ(xt, p

t) → 0 as t → −∞

and, consequently, α

= M

⊆ ω

. Thus M

= M

= α

⊆ ω

. Let p := p

= p

∈

∩ X

= M

∩ X

and ε = ρ(x, p) > 0. Further, reasoning in the same manner

as at the end of the preceding item, we obtain a contradiction, which proves the

exponential stability on S

of the minimal set M

⊆ ω

. The Lemma is completely

proved. 

Under the conditions of Lemma 4.7, the family Σ consists of only a ﬁnite number

of distinct minimal sets M

, M

, . . . , M

and M(Σ) = M

∪M

∪···∪M

. Let y ∈ Y

and {γ

(y)} = M

∩ X

. We will suppose that M

, M

, . . . , M

are ordered such

that γ

(y) < γ

(y) < ··· < γ

(y). Let us set µ

:= inf Σ

an ν

:= sup Σ

. It is

obvious that µ

, ν

∈ Σ

Lemma 4.9 Under the conditions of Lemma 4.7 the equalities µ

= γ

(y) and

= γ

(y) hold for all y ∈ Y .

Proof. We prove, for example, the ﬁrst equality (the second equality is proved in the

same manner). If we suppose that µ

6= γ

(y) for certain y ∈ Y , then µ

< γ

(y).

By Lemma 4.8, there exist recurrent points p

and p

from M(Σ), such that (4.1a)

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 4

162 Global Attractors of Non-autonomous Dissipative Dynamical Systems

and (4.1b) hold. We show that p

, p

≤ γ

(y). In fact, from (4.1) it follows the

existence of t

→ +∞ and t

→ −∞ such that µ

→ p

and µ

→ p

Moreover, by Lemma 4.6, γ

(y)t

→ γ

(y) (i = 1, 2). Since µ

< γ

(y), it follows

that µ

< γ

(y)t

and, consequently, p

≤ γ

(y). Analogously, p

≤ γ

(y).

Thus, p

= p

= γ

(y). Since H(γ

(p)) = M

is hyperbolic, it follows that M

exponentially stable either on S

or on S

−

. Further, reasoning in the same manner

as in Lemma 4.8, we obtain a contradiction. The lemma is proved. 

Theorem 4.3 Let Σ ⊆ X be a nonempty compact invariant subset of X and

M(Σ) ⊆ Σ be hyperbolic. Then the following statements are valid:

(1) Σ consists of only a ﬁnite number of distinct minimal sets M

, M

, . . . , M

and

M(Σ) = M

∪ M

∪ ··· ∪ M

;

(2) each minimal set M

(i = 1, 2, . . . , k) is homeomorphic to Y and, in particular,

is almost periodic (periodic) if Y has this property;

(3) for each point x ∈ Σ there exist points p

and p

from M(Σ) such that relations

(4.1) are fulﬁlled;

(4) if M

, M

, . . . , M

are ordered such that γ

(y) < γ

(y) for all y ∈ Y (where

{γ

(y)} := M

∩X

) if and only if i < j, then ∂Σ = M

∪M

, where ∂Σ is the

boundary of Σ.

Proof. This assertion follows from Lemmas 4.6-4.9. 

Theorem 4.4 Let h(X, S, π), (Y, S, σ), hi be compact dissipative and J be its

Levinson center. If M(J) is hyperbolic, then the following statements are valid:

(1) J contains only a ﬁnite number of the minimal sets M

, M

, . . . , M

, each of

which is homeomorphic to Y , and M(J) = M

∪ M

∪ ··· ∪ M

;

(2) for each point x ∈ J there exist points p

and p

in M(J) such that relations

(4.1) are fulﬁlled;

(3) if x /∈ J, then

(a) ρ(xt, γ

(y)t) → 0 as t → +∞ if x < γ

(y), where y = h(x), and

(b) ρ(xt, γ

(y)t) → 0 as t → +∞ if x > γ

(y).

(4) ∂J = M

∪M

and ∂J is uniformly asymptotically stable in the positive direction

and, in particular, k is odd.

Proof. This statement follows in essence from Theorem 4.4 and properties of the

Levinson center. 

Remark 4.4 Although Theorems 4.3 and 4.4 are proved for one-dimensional

systems, yet in fact we have always used only a consequence of one-dimensionality

(see Remark 4.3 b.). Therefore, Theorems 4.3 and 4.4 are valid with insigniﬁcant

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 4

The structure of the Levinson center with the condition of the hyperbolicity 163

changes for n ≥ 2 also if besides the hyperbolicity of M we require that for each

m ∈ M the set H(m) is exponentially stable either on S

or on S

−

. In addition,

the minimal sets M

, M

, . . . , M

in Theorems 4.3 and 4.4, generally speaking, not

homeomorphic to Y but are only distal covering of Y of ﬁnite multiplicity, which

also ensures their almost periodicity (periodicity) if Y is almost periodic (periodic).

Using the known connection between non-autonomous dynamical system and

diﬀerential equations, we can obtain statements, analogous to Theorems 4.3 and

4.4, for diﬀerential equations.

Let us consider the diﬀerential equation

= f(t, u), (4.3)

where f ∈ C

0,2

(R × R

, R

) and C

0,2

(R × R

, R

) is the space of all continuous

functions f : R × R

→ R

, continuously diﬀerentiable twice in u ∈ R

, equipped

with the topology of uniform convergence of functions and their derivatives up to

second order inclusive on compacts from R × R

. Along this equation (4.3) we

consider the family of equations

= g(t, u), (4.4)

where g ∈ H(f) :=

| τ ∈ R}, and f

(t, u) := f(t+τ, u), the bar denoting closure

in f ∈ C

0,2

(R × R

, R

). Let us suppose that the condition of non-local extend-

ability is fulﬁlled for each equation (4.4). Let ϕ(t, v, g) denote the solution of (4.4)

that passes through v ∈ R

for t = 0. Let Y := H(f) and (Y, R, σ) be a dynamical

system of translations on Y. Further, let X := R

× Y. Then we can deﬁne the

dynamical system (X, R, π) by the rule π(τ, (v, g)) := (ϕ(τ, v, g), g

). Finally, we

set h := pr

: X → Y. Then we obtain the non-autonomous dynamical system

h(X, R, π), (Y, R, σ), hi. Applying Theorems 4.3 and 4.4 to the so-constructed

dynamical system, we get the corresponding statements for equation (4.3). For

example, we formulate the statement that follows from Theorem 4.4. Fist of all, let

us recall certain notion related to equation (4.3).

We recall that the equation (4.3) is called dissipative if the non-autonomous

dynamical system, generated by it, is dyssipative. Let J ⊂ R

× H(f) be the

Levinson center of (4.3) and Σ ⊆ J.For each point (v, g) ∈ Σ let us consider the

equation in variations for (4.4) along the solution ϕ(t, v, g), i.e., the linear equation

= g

(t, ϕ(t, v, g))w. (4.5)

The family of equation (4.5), where (v, g) ∈ Σ, is said to satisfy the exponential

dichotomy on R

[

132

]

and the corresponding constants can be chosen independent

of (v, g) ∈ Σ.

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164 Global Attractors of Non-autonomous Dissipative Dynamical Systems

Theorem 4.5 Let f ∈ C

0,2

(R × R

, R

) and f, f

and f

be jointly recurrent

(almost periodic, τ-periodic) in t ∈ R uniformly in x on compacts from R. Suppose

that the equation (4.3) is dissipative, J ⊂ R × H(f), be its Levinson center, and

the family of equation (4.5), where (v, g) ∈ M(J), satisfy the exponential dichotomy

condition on R. Then for each g ∈ H(f), (4.4) has only a ﬁnite number of uniformly

compatible solutions

[

300

]

: These are q

, q

, ··· , q

(k is an odd number and depends

only of f ); all the remaining solutions, bounded on R, converge to the indicated

solutions as t → +∞ or −∞. But if a certain solution of (4.4) is bounded on R

only (since (4.3) is dissipative, all solutions of (4.4) is bounded on R

), then it

converges to one of the solutions q

and q

as t → +∞.

Remark 4.5 A statement, analogous to Theorem 4.5, holds also for the diﬀerence

equations.

4.4 The dissipative cascades

Let M be a diﬀerentiable manifold, U be an open subset of M, f : U → f(U ) ⊆ M

be a diﬀeomorphism of the class C

(r ≥ 1), Λ ⊂ U be a maximal compact invariant

set of the diﬀeomorphism f . Suppose that U is a domain of attraction for Λ, i.e.

(Λ) = U. In this case, the dynamical system generated by the positive powers

of the diﬀeomorphism f is dissipative and its Levinson center J = Λ.

Lemma 4.10 Let x

∈ U . If ω

is a compact and hyperbolic set of the diﬀeo-

morphism f : U → M, then for every x ∈ ω

and δ > 0 there is p ∈ P er(f) such

that ρ(x, p) < δ and Σ

⊆ B(ω

, δ), where P er(f) is a set of all the periodic points

of the diﬀeomorphism f, Σ

is a trajectory of the point p.

Proof. Let δ > 0. We may take the number δ to be so small such that B(ω

, δ) ⊂

U(ω

), where

U(ω

) is a neighborhood of the hyperbolic set ω

ﬁguring in the

theorem of Anosov about the family of ε-trajectories

[

261, p.220

]

. In the cited

above theorem we will chose a number ε (ε <

) that corresponds to the number

. Then for

there is l > 0 such that f

∈ B(ω

) for all n ≥ l. If x ∈ ω

then there is a number n

≥ l such that f

∈ B(x,

). Put x

= f

. There

exists a number N > 0 such that f

∈ B(x,

). Consider the space X

{0, 1, . . . , N −1} with discrete topology, the homeomorphism of the shift τ : X

→

, τ(k) = k + 1 mod(N ), and the mapping ϕ : X

→ M deﬁned by the equality

ϕ(k) = f

(k = 0, 1, . . . , N − 1). Note, that ϕ : X

→ B(ω

) ⊂

U(ω

Further: P (ϕ ◦ τ, f ◦ ϕ) := sup

0≤k≤N−1

ρ(ϕ(τ(k)), f(ϕ(k))) ≤ ρ(x

, f

). Since

ρ(x

, f

) ≤ ρ(x

, x) + ρ(x, f

) < ε, then P (ϕ ◦ τ, f ◦ ϕ) < ε. According to

Anosov’s theorem of the family of ε-trajectories, there exists a continuous mapping

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 4

The structure of the Levinson center with the condition of the hyperbolicity 165

ψ : X

→

U(ω

) such that ψ ◦ τ = f ◦ ψ and P (ϕ, ψ) <

. To ﬁnish the

proof of the lemma it is suﬃcient to note that ψ(0) ∈ P er(f), ρ(ψ(0), x) <

and

ψ(k) ∈ B(ω

, δ) for all k = 0, 1, . . . , N −1. 

Corollary 4.1 If the set Σ is locally maximal and ω

⊆ Σ, then under the

conditions of Lemma 4.10, in the set Σ there exists at least one periodic point f.

Corollary 4.2 Let Λ be a locally maximal compact invariant set. If Σ ⊆ ω

⊆ Λ

is a compact minimal set, which does not contain periodic trajectories, then under

the conditions of Lemma 4.10 in the set Λ there exists an inﬁnite number of diﬀerent

hyperbolic points p

→ p ∈ Σ ⊆ ω

, moreover, Σ

⊆ B(ω

, ε

) and ε

↓ 0.

Deﬁnition 4.6 The set M ⊆ X of the dynamical system (X, S, π) is called quasi-

minimal, if there is a stable in the sense of Poisson point p ∈ M such that M = H(p).

Theorem 4.6 Let f : U → M be a diﬀeomorphism of the class C

(k ≥ 1) and Λ

be a compact invariant set of the diﬀeomorphism f. If the set M

(Λ) is hyperbolic,

then:

(1) the relation ∼ decomposes the set R(Λ) on ﬁnite number of classes of equiva-

lence, i.e. R(Λ) = R

t R

t ··· t R

;

(2) the sets R

(i =

1, k) are closed, invariant, indecomposable, locally maximal

and in the collection {R

, R

, . . . , R

} there is no r-cycles, where r ≥ 1.

Proof. This statement follows from Theorem 4.1. 

Corollary 4.3 Let M be a diﬀerentiable manifold and f : M → M be a diﬀeo-

morphism of the class C

(k ≥ 1). If the set of the chain recurrent points R(f ) of

the diﬀeomorphism f is compact and hyperbolic, then:

(1) the relation ∼ decomposes the set R(f) on ﬁnite number of classes of equiva-

lence, i.e. R(f) = R

t R

t ··· t R

;

(2) the sets R

(i =

1, k) are closed, invariant, quasi-minimal, locally maximal and

in the collection {R

, R

, . . . , R

} there is no r (r ≥ 1)-cycles;

(3) if, in addition, in the set M there exists an inﬁnte number of diﬀerent minimal

sets, then in the set M there is a nontrivial (non periodic) minimal set.

Proof. The formulated assertion follows from Theorem 4.6, Theorems 7.5 (and its

discrete analog) 7.8 from

[

]

and Lemma 6.4 from

[

261

]

. 

Theorem 4.7 Let f : U → M be a dissipative diﬀeomorphism and J be its

Levinson center. If one of the following two condition is fulﬁlled:

a. in the set Λ there is only a ﬁnite number of minimal sets;

September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 4

166 Global Attractors of Non-autonomous Dissipative Dynamical Systems

b. in the set Λ there is an inﬁnite number of diﬀerent minimal sets and the set

(Λ) is hyperbolic

[

261

]

then:

(1) the relation ∼ decomposes the set of the chain recurrent points R(f) of the

diﬀeomorphism f on ﬁnite number of classes of equivalence, i.e. R(f ) = R

t ··· t R

;

(2) the sets R

(i = 1, 2, . . . , k) are closed, invariant, indecomposable, locally max-

imal and in the collection {R

, R

, . . . , R

} there is no r (r ≥ 1)-cycles;

(3) J = ∪{W

) : i ∈

1, k}.

Proof. This statement follows from Theorem 4.2. In fact, if the set M

(Λ) 6= ∅

and it is hyperbolic, then from the results

[

261

]

(see Theorems 0.1, 5.4 and the

theorem from the ﬁrst appendix) follows, that on the set M(Λ), except, maybe,

a ﬁnite number of isolated minimal sets, a dynamical system generated by the

positive powers of the diﬀeomorphism f has a hyperbolic structure in the sense of

our deﬁnition. 

Remark 4.6 a. Theorem 4.7 is an analog of the well-known theorem of Smale

about spectral decomposition

[

261

]

for A-diﬀeomorphisms.

b. The analog of the theorem 4.7 is true also for the continuous dynamical system

(ﬂows).

Theorem 4.8 Let f : U → M be a dissipative diﬀeomorphism and let its center of

Levinson contain an inﬁnite number of minimal sets. If the set M

(Λ) is hyperbolic,

then the statements 1.-3. of Theorem 4.6 hold. In addition, in the set Λ there is an

inﬁnite number of hyperbolic periodic trajectories.

Proof. According to Theorem 4.7 it is suﬃcient to show, that in the set Λ there is

an inﬁnite number of diﬀerent hyperbolic periodic trajectories. Denote by

U ⊂ U a

neighborhood of the set M

(Λ) such that every compact, invariant with respect to

f set Λ

⊂

U is hyperbolic. By Theorem 2

[

261, p.224

]

there is such a neighborhood.

According to

[

226, p.52

]

a metric space 2

is compact. Since in the set Λ there is

an inﬁnite number of the minimal sets {M

} and M

⊆ R = R

t R

t ··· t R

then we can suppose that M

⊆ R

(i = 1, 2, . . . ). Taking into consideration the

compactness of the space 2

, we may suppose that the sequence {M

} is convergent

in the metric of Hausdorﬀ. In particular, beginning from a number i

, we have

⊆

U for i ≥ i

. In view of the choice of the set

U, all the minimal sets M

(i ≥ i

) are hyperbolic. The following two cases are possible:

a. every minimal set M

(i ≥ i

) is periodic and in this case the theorem is proved.

b. between the sets M

(i ≥ i

) there is a nontrivial minimal set M.