www.scielo.br/cam
Chain control sets and fiber bundles
CARLOS J. BRAGA BARROS*
Departamento de Matemática, Universidade Estadual de Maringá 87020-900 Maringá, PR, Brasil
E-mail: [email protected]
Abstract. LetSbe a semigroup of homeomorphisms of a compact metric spaceMand suppose thatFis a family of subsets ofS. This paper gives a characterization of theF-chain control sets as intersection of control sets for the semigroups generated by the neighborhoods of the subsets in F. We also study the behavior ofF-chain control sets on principal bundles and their associated
bundles.
Mathematical subject classification:93B03, 93B05.
Key words:semigroups, control sets, chain control sets, principal bundles, associated bundles.
1 Introduction
The concept of chain control set for control systems was introduced by Colonius and Kliemann [4], [7]. These sets appear as a tool for the analysis of the asymp-totic properties of control systems (see [4], [5], [6], [7]). Extending this notion for general classes of semigroups Braga Barros and San Martin [2] defined chain control sets for a family of subsets of a semigroup acting on a homogeneous space. In that paper chain control sets were characterized as intersection of con-trol sets for the semigroups generated by the neighborhoods of the subsets in the family. This paper shows a similar result for semigroups of homeomorphisms of a compact metric space. The approach is different from that of [2] because the semigroup may have empty interior in the group of homeomorphisms. An important fact which permits our generalization of [2] is that the hypothesis H
#460/98. Received: 30/X/98. Accepted: 11/XI/03.
defined there can be extended to semigroup of homeomorphisms acting on a compact metric space.
Apart from this general characterization of chain control sets we also study chain control sets on fiber bundles. The action of semigroups in fiber bundles arises naturally in many contexts. For instance in nonlinear control systems the linearized flow evolves on a fiber bundle over the state space of the system (see [4], [7]). The action of semigroups of diffeomorphisms on fiber bundles were studied by Braga Barros and San Martin [3]. In [3] the control sets were described from their projections onto the base space and their intersections with the fibers.
In this paper we pursue the same kind of results for chain control sets. We show that a chain control set in the total space of a fiber bundle projects inside a chain control set in the base space. On the other hand, we also show that a chain control set in the fiber is contained in a chain control set in the total space. Assuming that the structural group of a principal bundle is compact we use the characterization of chain control sets as intersections of control sets to show that the inverse image by the projection of a chain control set in the base space of the fiber bundle is a chain control set in the total space.
We believe that these topological results are useful for further developments of the dynamical aspects of control systems.
2 F-chain control sets
In this section we recall the concept of a chain control set associated with a family of subsets of a semigroup. We refer to [2] for the definition of chain control sets for semigroup actions and corresponding results on homogeneous spaces of Lie groups.
For a semigroup of homeomorphisms of a compact metric space it is shown, under certain conditions, that a chain control set is the intersection of control sets for the semigroups generated by the neighborhoods of the subsets in the family. We begin by assuming thatSis a semigroup of homeomorphisms of a compact metric spaceM. From now on, and in the whole paper we assume thatS and
We fix a distancedonMand define chain control sets for a familyFof subsets ofS.
Definition 1. LetF be a family of subsets ofS. Takex, y ∈ M,a realε >0 andA ∈ F. A (S, ε, A)-chain fromx toy consists of pointsx0 = x, x1,. . .,
xn−1, xn = y inM andφ0, . . . , φn−1 ∈ A such thatd(φj(xj), xj+1) < εfor
j =0, . . . , n−1. A subsetE⊂Mis called aF-chain control set if it satisfies
1. int(E)= ∅.
2. For everyx, y ∈ E, there exists a(S, ε, A)-chain fromx toy, for every
ε >0 andA∈F.
3. Eis maximal satisfying these properties.
It is easy to see that the notion of F-chain control set does not change if an equivalent distance is considered. Also, it follows from the third condition that two F-chain control sets are either disjoint or coincident. Furthermore, an application of Zorn’s Lemma shows that any subset satisfying the first two conditions in the Definition 1 is contained in aF-chain control set.
Next, we show that effective control sets (for the theory of control sets we refer to [11], [7]) are contained inF-chain control sets. We need to impose the following conditions onF.
Definition 2. Let F be a family of subsets of a semigroup S. We say that F satisfies property Pl (respectively Pr) if for every φ, ψ ∈ S, φ = 1 and everyA∈F, there exists a positive integernsuch thatφnψ ∈A(respectively
ψ φn∈A).
Let’s denote bySAthe semigroup generated by a subsetA⊂S.
Proposition 1. Assume thatFis a family of subsets of a semigroupSsatisfying the propertiesPl andPr and takeA∈F. LetDbe an effective control set for
S. ThenD⊂cl(SAx)for anyx ∈D.
definition of effectiveness. Suppose first thaty∈D0. SinceD⊂int(S−1y)for everyy ∈D0(see [3], Proposition 2.2), there existsψ ∈Ssuch thaty =ψ (x). Also, there existsφ ∈S− {1}such thatφ (y)=y. ByPlwe haveφnψ∈A, and
φnψ (x) =y, so thaty ∈ SAx. For an arbitraryy takez ∈ D0. We have seen thatz ∈ SAx. We have to show thaty ∈ cl(SAz). Takeφ ∈ S− {1}such that
φ (z)=zand a sequenceψm∈Ssuch thatψm(z)→y. ByPl,ψmφn∈SAfor large enoughn, and sinceψmφn(z)= ψm(z), there exists a sequenceρk ∈ SA
withρk(z)→y.
As a consequence we have.
Corollary 1. Suppose thatFsatisfies bothPlandPr. Then any effective control set forSis contained in aF-chain control set.
We can define effectiveF-chain control sets.
Definition 3. LetFbe a family of subsets ofS. AF-chain control set is called
effectiveif it contains an effective control set forS.
We observe that for a familyFsatisfying bothPlandPraF-chain control set
Eis effective if and only if the subsetE0= {x ∈E :x ∈int(Sx)∩int(S−1x)} is not empty. In fact, if we assume thatx ∈int(Sx)∩int(S−1x)then there exists an effective control setDsuch thatx∈D0(see [3], Proposition 2.3).
LetM be a compact metric space. Assume thatS is a semigroup of home-omorphisms ofM and suppose that d is a metric onM. In the groupG of homeomorphisms ofMwe consider the metric of uniform convergence
d′(φ, ψ )= sup x∈M
d(φ (x), ψ (x))
which is right invariant underG. For a subsetA⊂Swe put
B(A, ε)= {φ ∈G: ∃ψ ∈A, d′(φ, ψ ) < ε}.
We denote bySε,Athe subsemigroup ofGgenerated byB(A, ε).
Hypothesis H: There exist real numbers c > 0 and η > 0 satisfying the following property: for allx ∈ Mandy ∈Bη(x),there existsφ ∈Gsuch that
φ (x)=yandd(φ (x), x)≥cd′(φ,1M). In general this condition is not satisfied.
Example 1. Consider the compact metric space
M=
∞
n=1
Cn∪ {0}
whereCn are the circlesCn = {(x, y)∈ R2:x2+y2 =1/n}, with the metric inherited from the standard metric ofR2. All homeomorphismsφ ofM have the property thatφ (0)=0, and therefore the hypothesis H is not satisfied at 0. In fact, ifφ (0) = 0 then the connected component containingφ (0)is a circle. The connected component containing 0 is the set{0}. And clearly this set in not homeomorphic to a circle.
However, we observe that in [2] it was shown that hypothesis H holds for the action of a Lie groupGon a large class of homogeneous spacesG/H, namely those for which there is a compact subgroupK ⊂Gwhich acts transitively on
G/H.This class of homogeneous spaces includes the flag manifolds (Furstenberg boundaries) of semi-simple Lie groupsG.
Now, we can relate(S, ε, A)-chains with reachability ofSε,A.
Proposition 2. Takex, y ∈MandA⊂S. Then
1. There exists an(S, ε, A)-chain fromxtoyify ∈Sε,Ax. We also have that there is a(S, ε′, A)-chain fromxtoyfor everyε′ > εify ∈cl(Sε,Ax).
2. Suppose that the hypothesis H is satisfied and takeεsuch that0< ε < η. Assume that x0, . . . , xn ∈ M andφ0, . . . , φn−1, determine a(S, ε, A)
-chain from x0toxn. Then there existsψ ∈ Sε′,A such thatψ (x0) = xn, whereε′=ε/c.
The sequencesx0, x1 = φ0(x0), . . . , xk = φk−1(xk−1) = y andψ0, . . . , ψk−1 determine an(S, ε, A)-chain fromxtoy. In fact,
d(ψi−1(xi−1), xi) = d(ψi−1(xi−1), φi−1(xi−1))
≤ d′(ψi−1, φi−1) < ε
fori = 0, . . . , k−1. Consequently there exists a(S, ε, A)-chain from x to
y. Now, suppose thaty ∈ cl(Sε,Ax). There exists a sequenceφn ∈ Sε,A such thatφn(x)converges toy. Takeε′ > εand letn0 be such thatd(φn0(x), y) <
ε′ −ε. As before, there exist a (S, ε, A)-chain fromx to φn0(x). Let y0 =
x, . . . , yn = φn0(x0) ∈M, ψ0, . . . , ψn−1 ∈ Abe a(S, ε, A)-chain fromx to
φn0(x). Thusd(ψi(yi), yi+1) < ε fori =0, . . . , n−1. Therefore, the chain
z0 = x, z1 = y1, . . . , zn−1 = yn−1, zn =y andψ0, . . . , ψn−1 ∈ Adetermine an(S, ε, A)-chain fromxtoy. In fact,
d(ψn−1(yn−1), y) ≤ d(ψn−1(yn−1), φn0(x))+d(φn0(x), y)
< ε′
andd(ψi−1(yi−1), yi) < ε < ε′fori=1, . . . , n−1.
2. Since d(φi(xi), xi+1) < ε < η, the hypothesis H implies that there is
ψi ∈Gsuch that
d(ψi(φi(xi)), xi+1)=d(ψi(φi(xi)), φi(xi))≥cd′(ψi,1M)
i=0, . . . , n−1,and therefored′(ψi,1M) < ε/c=ε′. Defineτi =ψiφi. We have
d′(τi, φi)=d′(ψiφi, φi)=d′(ψi,1M) < ε′
because d′ is right invariant. Therefore, τ
i ∈ B(A, ε′). On the other hand,
τi(xi)=ψiφi(xi)=xi+1,andxn=τn−1. . . τ0(x0).
Proposition 3. Suppose that the hypothesis H is satisfied and assume thatF
satisfiesPlandPr. LetDbe an effective control set forSonM. Then, for each
ε >0 andA ∈ F there exists a control setDε,A for the perturbed semigroup
Proof. In fact, forx, y ∈D,we will show thaty ∈Sε,Ax, forεsmall enough. By Proposition 1, there exists,φ ∈ SA such thatφ (x) is neary. On the other hand, by the definition ofSA we have φ = φ1. . . φn withφi ∈ A. Using H, there isγ withd′(γ φ, φ)=d′(γ ,1) < εandγ φ (x)=yso thatγ φ∈Sε,Aand
y∈Sε,Ax.
The next theorem characterizes the F-chain control sets as intersections of control sets for the perturbed semigroups.
Theorem 1. LetS be a semigroup of homeomorphisms of a compact metric spaceM. Suppose that the action ofSonM satisfies H, and consider a family
Fof subsets ofSsatisfyingPl andPr. LetDbe an effective control set forSon
M, and forε >0andA∈F, denote byDε,AtheSε,Acontrol set containingD. Then
E=
ε,A
Dε,A
is the onlyF-chain control set containingD.
Proof. Since D ⊂ E we have int(E) = ∅. Also, for any x, y ∈ E, y ∈
cl(Sε′,Ax)for allε′>0 andA∈F. Therefore, by the Proposition 2 there exists
a(S, ε, A)-chain fromx toyfor allε > 0 andA∈ F, which shows thatEis chain transitive. It remains to verify the maximality ofE. Takex /∈Eandy ∈E
and suppose that for everyε >0 andA ∈F there are(S, ε, A)-chains fromx
toy and fromy tox. Since the action ofS onM satisfies H andSε,A ⊂Sε′,A
forε′ > ε, Proposition 2 shows that y ∈ Sε,Ax and x ∈ Sε,Ay for all ε > 0 andA∈ F. However,y ∈Dε,A, so thatx ∈Dε,Afor allε, Acontradicting the assumption thatx /∈E. Hence, there is no chain either fromxtoyor fromy to
x, which shows the maximality ofE.
Proof. By [1, Lemma 3.1] there is only one closed invariant control set, sayC, forSin a compact metric spaceMif and only ifC =x∈Mcl(Sx)= ∅. Define
Cε,A=
x∈Mcl(Sε,Ax). Thus, it is enough to show thatDε,A∩Cε,A= ∅. Pick
x∈ M. SinceD⊂int(S−1y)for everyy ∈D0we can chooseφ ∈Ssuch that
φ (x)∈D0. ButF satisfies propertyPl and thereforeφn ∈Afor some integer
n(takeφ = ψ in the definition of Pl). We have that φn(x) ∈ D because D is invariant. SinceD ⊂ Dε,A, andDε,A is a control set forSε,A it follows that
Dε,A⊂cl(Sε,Ax).
3 Fiber bundles
In this section we study the behavior ofF-chain control sets on principal bundles and their associated bundles. We refer to [9] for the theory of fiber bundles.
We start by settling some notation. LetGbe a topological group. Suppose thatGacts effectively and on the right on a topological spaceQ.We denote by
Q(M, G)the principal bundle with total spaceQ, base spaceMand structural groupG. We denote byπQ:Q→Mthe canonical projection.
LetSQbe a semigroup of homeomorphisms ofQcommuting with the right action, i.e.,Qφ (q ·a) = Qφ (q)·a, a ∈ GifQφ ∈ SQ. The semigroupSQ induces a semigroupSM of homeomorphisms of M. In fact, if y ∈ M and
y=πQ(q)we define an elementMφ∈SM as
Mφ (y)=πQ(Qφ (q)),
ifQφ∈SQ.
Suppose that the structural group G acts on the left and transitively on the topological groupF. We consider the fiber bundle associated to the principal bundleQ(M, G)with typical fiberF. This bundle is denoted byE(M, F, G, Q),
or simply byE, the total space of the bundle. Since we will be interested in the study of chain control sets we assume, in the paper, thatE, QandMare metric spaces.
The elements ofEare equivalence classes with respect to the relation onQ×F
given by(q, v)∼(qa, a−1v), a∈G. We use the notation[q, v],q ∈Q, v∈F for an element ofE.
The canonical projection πE : E → M on the fiber bundle is defined as
LetQφbe a homeomorphism ofQcommuting with the right action ofG. Then
Qφ induces the homeomorphism of E defined byEφ ([q, v]) = [Qφ (q), v]. Therefore the semigroupSQof homeomorphisms ofQinduces a semigroupSE of homeomorphisms ofEand
SE([q, v])= [SQ(q), v]
Givenq ∈Qwe define the subset
Sq =SQ(q)∩πQ−1(x), x=πQ(q)
Through the identification of the fiber overxwithGviaa ∈G −→q·a ∈
πQ−1(x),Sqcan be viewed as a subset ofG
Sq = {a∈G: ∃Qφ ∈SQ, Qφ (q)=q·a}
It follows immediately thatSqis a subsemigroup ofGifSq = ∅. We observe thatSqis the subsemigroup acting on the typical fiber.
The semigroupsSQ,SEandSqwere also considered in [3].
Suppose thatF is a family of subsets ofSQ. The familyF induces a family FM in the semigroupSM, a family FE inSE and a family Fq inSq. In fact, for eachA ∈ F we defineAM = {Mφ ∈ SM : ∃Qφ ∈ AandMφ (πQ(q)) =
πQ(Qφ (q))},AE = {Eφ ∈SE : ∃Qφ ∈AandEφ ([q, v])= [Qφ (q), v]}and
Aq= {a∈G: ∃Qφ ∈AandQφ (q)=q·a}. Thus we can define the induced familiesFM = {AM :A∈F},FE = {AE :A∈F}andFq= {Aq :A∈F}.
Proposition 5. LetF be a family of subsets ofSQsatisfying bothPl andPr. ThenFM,FE andFqalso satisfyPl andPr.
Proof. In fact, takeA ∈ F. There existsnsuch thatQφ (Qρ)n =σ ∈ Aif
Qφ, Qρ∈SQ. Thus
Mσ (πQ(q))=πQ((Qφ (Qρ)n)(q))=(Mφ (Mρ)n)(πQ(q))
and thereforeMφ (Mρ)n ∈A
M. For the familyFE we have
andEφ (Eρ)n ∈ AE. Now, take a, b ∈ Sq and Aq ∈ Fq. Then there ex-ist Qφ, Qψ ∈ SQ such that Qφ (q) = q ·a and Qψ (q) = q ·b. Thus
Qφ (Qψ )n(q)=Qφ (q·an)=q·ban. SinceQφ (Qψ )n∈Awe haveban∈Aq. We argue in the same way for thePr property.
The following theorem shows that chain control sets in the total space of a fiber bundle project into chain control sets in the base of the bundle.
Theorem 2. LetEbe a fiber bundle with projectionπ :E → M. LetF be a family of subsets inSQ. Suppose thatE is compact, and letH ⊂ E be a FE-chain control set. Then there exists a FM-chain control setB ⊂ M such thatπ(H )⊂B.
Proof. Since int(H ) = ∅ andπ is an open map, π(H ) has nonempty inte-rior. Take ε > 0, AM ∈ FM and x′, y′ ∈ π(H ). Pick x, y ∈ H such that
π(x) = x′ and π(y) = y′. Let’s show that there exists an (SM, ε, AM)) -chain from x′ to y′. Since E is compact, π is uniformly continuous so that there is δ > 0 such that d(π(z), π(z′)) < ε ifd(z, z′) < δ, z, z′ ∈ E. Let
x0=x, x1, . . . , xn−1, xn =y inEtogether withEφ0, Eφ1, . . . , Eφn−1inAE form a (SE, δ, AE)-chain from x to y. Since d(xj, Eφj(xj)) < δ we have thatd(π(xj), π(Eφj(xj))) = d(π(xj), Mφj(π(xj))) < ε which shows that
π(xi), Mφi determine a(SM, ε, AM))-chain fromx′toy′. Sinceπ(H )satisfies properties 1 and 2 in the definition of aFM-chain control set it is contained in a
FM-chain control set.
We recall that we are assuming thatSQandSQ−1have the accessibility property. However, since we wish to work over effective FM- chain control sets, the following version of accessibility is needed.
Definition 4. LetF be a family of subsets of SQsatisfying both Pl andPr. Suppose thatH is an effectiveFM-chain control set inMand takeH0= {x ∈
H :x ∈int(SMx)∩int(SM−1x)}. The semigroupSQis said to beaccessible over
H0if for everyq ∈πQ−1(H0), int(SQq)∩πQ−1(H0)= ∅.
Lemma 1. LetH ⊂Mbe an effectiveFM-chain control set and suppose that
Proof. TakeQφ ∈SQsuch thatQφ (q)∈int(SQq)∩πQ−1(H0). There exists an effective control setDforSM such thatx =πQ(q)∈D0. ThusπQ(Qφ (q))=
Mφ (πQ(q))∈D0and there existsQψ ∈SQsuch thatMψ (πQ(Qφ (q)))=x, so thatQψ Qφ (q)belongs to the same fiber asq. Now,Qψ Qφ (q)∈int(SQq) hence if we leta∈Gbe such thatQψ Qφ (q)=q·athena∈int(Sq)showing
the lemma.
As a converse of Theorem 2, we have
Theorem 3. LetEbe a fiber bundle with projectionπ :E→M. LetF be a family of subsets inSQ satisfying bothPl andPr. Suppose thatEis compact,
B ⊂ M is an effective FM-chain control set and that SQ is accessible over
B0. Then there exists an effectiveFE-chain control set H ⊂ Eand such that
π(H )⊂B.
Proof. Takex∈B0= {x∈B :x ∈int(SMx)∩int(SM−1x)}. Thenxbelongs to an effective control setCcontained inB ([3, Proposition 2.3]). We have shown in [3, Proposition 3.4] that there exists an effective control setD ⊂ E forSE withπ(D)⊂C. Corollary 1 implies thatDis contained in aFE-chain control set, sayH. Using the fact that chain control sets do not overlap and the Theorem 2 there exists aFM-chain control setB ⊂Msatisfyingπ(E)⊂B.
LetdE be a metric on the total spaceE of the fiber bundle with typical fiber
F. Let also dF be a right invariant metric on the fiberF. Consider a metric
dM on the base spaceM such that the canonical projection πE : E → M is Lipschitizian, that is, there exists a constantc >0 such that for everyx, y ∈E
we havedE(x, y)≤cdM( πE(x), πE(y)).
The characterization of the chain control sets as intersection of control sets of the perturbed semigroup gives us the following result.
control set onE. Therefore there are as many effectiveFE-chain control sets in
EasFM-chain control sets inM.
Proof. We first show that((SM)ε,AM)E ⊂(SE)cε,AE, that is,Eφ∈(SE)cε,AE if
Mφ∈(SM)ε,AM. It is enough to show thatEφ ∈B(AE, cε)ifMφ∈B(AM, ε). This is true because
d′(Eγ , Eφ) = sup [q,v]∈E
dE(Eγ ([q, v]), Eφ ([q, v]))
= sup [q,v]∈E
dE([Qγ (q), v],[Qφ (q), v])
≤ c sup [q,v]∈E
dM(πQ(Qγ (q)), πQ(Qφ (q)))
= cd′(Mγ , Mφ).
By the Theorem 1 we haveB =ε,AM Dε,AM whereDε,AM is a control set for
(SM)ε,AM. Thusπ
−1(B) = ε,AMπ
−1(D
ε,AM). Now, [3, Proposition 3.7 and Proposition 3.9] implies that π−1(Dε,AM) is a control set for ((SM)ε,AM)E ⊂
(SE)cε,AE and thereforeπ
−1(B) is a F
E chain control set in E, again by the characterization of chain control sets as intersections of control sets.
The next theorem shows that aFq-chain control set in a fiber of a bundle is contained in aF-chain control set in the total space.
Theorem 5. LetE be the bundle associated to the principal bundle Qwith projectionsπE andπQ,respectively. Suppose that F is a family of subsets of
SQ. Takeq ∈ πQ−1(x),x ∈ M. Assume thatH is aFq-chain control set onF. Then
1. AnyFq-chain control set inπQ−1(x)is contained in aF-chain control set inQ.
2. [q, H]is contained in aFE-chain control set inE.
Proof.
1. Let H be a Fq-chain control set inπQ−1(x). Pick z, z′ ∈ H. Then for everyε > 0 andAq ∈ Fq there existsx0 = z, x1, . . . >, xn−1, xn = z′ inπQ−1(x)and >a0, a1, . . . , an−1 ∈ Aq such that >d(xjaj, xj+1) < εfor
>x0, . . . , xn and Qφj, j = 0, . . . , n−1 determine a >F-chain from
ztoz′.
2. Take[q, v]and[q, v′] in[q, H]. SinceH is aFq-chain control set, for every ε > 0 and Aq ∈ Fq there exist v0 = v, . . . , vn = v′ inF and
a0, . . . , an−1 ∈ Aq such that dF(ajvj, vj+1) < εforj =0, . . . , n−1. LetQφj ∈Abe defined asQφj(q)=qaj. Then[q, v0], . . . ,[q, vn]and
Eφj,j =0, . . . , n−1 determine a(SE, ε, A)-chain from[q, v]to[q, v′]. In fact,
dE(Eφj([q, vj]),[q, vj+1]) = dE([qaj, vj],[q, vj+1])
= dE([q, ajvj]),[q, vj+1])
= dF(ajvj, , vj+1)
< ε.
Remark. Since we can construct the invariant control sets in the total space of an associated bundle in terms of the invariant control sets in the fibers (see [3]) we could ask if this is also true for the chain control sets which contains the invariant control set. We have tried to answer this question using the characterization of chain control sets as intersections of control sets (Theorem 1). The proof does not work because the semigroup associated (in the fiber) of the perturbed semigroup in the total space does not coincide with the perturbed semigroup of the fiber and there is no useful inclusion between them. We observe that Theorem 5 can be regarded as a lemma in this expected description of the chain control sets in fiber bundles.
4 Acknowledgment
I would like to thank an anonymous referee for suggestions to improve the paper, in particular for providing Example 1.
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