On the action of semigroups in fiber bundles

Carlos José Braga Barros Departamento de Matemática Universidade Estadual de Maringá Cx. Postal 331 87.020-900 Maringá-Pr, Brasil

Luiz A. B. San Martin^{∗} Instituto de Matemática Universidade Estadual de Campinas Cx. Postal 6065 13.081-970 Campinas-SP, Brasil

Abstract

This paper studies the behavior of control sets for semigroup actions on principal bundles and their associated bundles. The emphasis is put on the description of those sets from their projections onto the base space and their intersections with the fibers.

1 Introduction

Semigroup actions on homogeneous spaces and fiber bundles have received quite some attention lately. This is in part due to the envisaged applications of this theory in dynamical systems, control theory or spectral theory of flows on vector bundles. One of the central ideas in these applications is the study of transitivity of the semigroup action, that is to say, the analysis of the control sets.

The present paper studies the control sets of semigroups acting of fiber bundles using topological methodology. The main object to be considered are the control sets for this action (see Definition 2.1 below). Our purpose is to describe these sets from their projections onto the base space and their intersections with the fibers. More specifically, let Q (M, G) be a principal bundle with base space M, total space Q and structure group G, and E (M, F, G, Q) a fiber bundle associated to Q with typical fiber F. Let also S_{Q} be a semigroup of automorphisms of Q. Then S_{Q} induces semigroups S_{E} and S_{M} acting on E and M respectively. We study here the control sets for the S_{E}-action on E. We first check that the projection onto M of the S_{E}-control sets are control sets for the S_{M}-action so that we are led to the analysis of the S_{E}-action over a control set. For this we suppose that the orbits S_{Q}q, q ∈ Q have nonempty interior. With this assumption, the intersection of S_{Q}q with the fiber through q identifies with a semigroup, say S_{q}, which has nonvoid interior in the structure group G. Roughly speaking, the intersection with a fiber of a control set for S_{E} is shown to be a control set for the action of S_{q} on the typical fiber F. So that we get a description of the control sets on E as soon as we have a procedure of bunching together the control sets on the different fibers. We present such a procedure for the invariant control sets, obtaining these sets as the union of invariant control sets on the fibers (see Theorem 4.4 below).

∗Research partially supported by CNPq grant n^{o} 301060/94-0

2 Preliminaries on control sets

In this section we recall the definition of control sets for semigroup actions and present the results that will be used throughout paper. We refer to [1],[3], for results on control sets for control systems and [4],[9],[10],[11], for control sets for semigroup actions.

Let S be a semigroup of diffeomorphisms acting on a manifold M.

Definition 2.1 A control set for S on M is a subset D ⊂ M satisfying

1. int D ≠ ∅,

2. D ⊂ cl(Sx) for all x ∈ D, and

3. D is maximal satisfying these two properties.

An invariant control set is a control set D which besides of 2) satisfies cl (Sx) = cl D for all x ∈ D. We note that the term invariant here is not properly used in the sense that invariant control sets are not always S-invariant (see [8] for an example of an invariant control set which is not S-invariant). However, in the context we work here, namely with the assumption that S has the accessibility property (i.e., int S (x) ≠ ∅ for all x ∈ M) invariant control sets are indeed S-invariant and closed subsets (see [2],[5]).

Suppose that both S and S^{−1} = {g^{−1} : g ∈ S} satisfy the accessibility property. Let D be a control set for S and define

D_{0} = {x ∈ D : x ∈ int (Sx) ∩ int (S^{−1}x)}.

Then we have the following result which is a slight extension of Proposition 2.2 of [10].

Proposition 2.2 Suppose D_{0} ≠ ∅. Then

1. D ⊂ int (S^{−1}x) for every x ∈ D_{0}.

2. D_{0} = int (S^{−1}x) ∩ int (Sx) for every x ∈ D_{0}

3. For every x, y ∈ D_{0} there exist g ∈ S with gx = y.

4. D_{0} is dense in D

5. D_{0} is S-invariant on D, in the sense that hx ∈ D_{0} if h ∈ S, x ∈ D_{0} and hx ∈ D.

Proof:

1. Take y ∈ D and x ∈ D_{0}. Since x ∈ int (S^{−1}x) ∩ D there exists g ∈ S such that gy ∈ int (S^{−1}x). Thus y ∈ g^{−1} int (S^{−1}x) ⊂ int (S^{−1}x).

2. Suppose x ∈ D_{0} and y ∈ int (Sx)∩int (S^{−1}x). We have that int (Sx) = int (Sy) and int (S^{−1}x) = int (S^{−1}y) . Therefore y ∈ D_{0}. On the other hand, if x, y ∈ D_{0} we have by 1), y ∈ int (S^{−1}x) and x ∈ int (S^{−1}y). Thus, there exists g ∈ S such that gx = y. Since x ∈ int (Sx) we have that y = gx ∈ g int (Sx) ⊂ int (Sx). Therefore y ∈ int (Sx) ∩ int (S^{−1}x).

3. Follows immediately from D_{0} ⊂ int (Sx) ∩ int (S^{−1}x).

4. Take x ∈ D_{0}. Since int (Sx) and int (S^{−1}x) are open we have

cl D_{0} = cl(int (Sx)) ∩ int (S^{−1}x).

Now, D ⊂ int (S^{−1}x) by 1). On the other hand

D ⊂ cl(Sx) ⊂ cl(S int (Sx)) ⊂ cl(int (Sx)).

Therefore D ⊂ cl D_{0} and D_{0} is dense in D.

5. Take h ∈ S, x ∈ D_{0} and suppose that hx ∈ D. Since x ∈ int (Sx) we have hx ∈ int (Sx). We also have by 1) that hx ∈ int (S^{−1}x), and therefore hx ∈ D_{0}. □

We say that D_{0} is the set of transitivity in D, and D is named an effective control set if D_{0} ≠ ∅.

It is not difficult to show that D_{0} = {x ∈ D : x ∈ (int S)x} if S is a subsemigroup of a Lie group G with interior points in G and M is a homogeneous manifold of G. So that the above proposition implies [10, Prop.2.2].

As a complement to the above proposition we have the following statement which ensures the existence of effective control sets.

Proposition 2.3 Let x ∈ M be such that

x ∈ int (Sx) ∩ int (S^{−1}x) .

Then there exists a unique control set D such that x ∈ D_{0}, which is effective.

Proof: The subset int (Sx) ∩ int (S^{−1}x) satisfies properties 1) and 2) of Definition 2.1. Therefore if we apply the Principle of Maximality of Hausdorff to the subsets satisfying these properties and containing x, we get a maximal one, say D, which is a control set. This control set clearly contains x in its set of transitivity. Also, the uniqueness follows from the fact that different control sets do not overlap. □

At this point we mention that for semigroups generated by families of vector fields, which appear in the literature of control theory, there are more precise statements than the above ones: Let Ω be a family of vector fields on M and put

S_{Ω} = {X_{t_{1}}^{1} ∘ ⋯ ∘ X_{t_{k}}^{k} : X^{i} ∈ Ω, t_{i} ≥ 0}

where X_{t} stands for the flow of the vector field X. Then S_{Ω} is a semigroup of (local) diffeomorphisms on M. Under the well known Lie algebra rank condition both S_{Ω} and S_{Ω}^{−1} are accessible, and if this holds, property 2) of the definition of a control set imply that D is effective if it has nonempty interior (see e.g. [3]). In this case we have that D_{0} = int D.

3 Fiber bundles

In this section we study the control sets for semigroup actions on principal bundles and their associated bundles.

For the theory of principal bundles and their associated bundles we follow [6], and adhere to its notations. In particular, we consider here only locally trivial bundles.

Let Q(M, G) be a principal bundle having total space Q, base space M and structure group G. We denote by π : Q → M the canonical projection, and the right action of G on Q is denoted by R_{a}, or simpler by q.a or qa, q ∈ Q, a ∈ G. This action is free and transitive on the fibers, so that if we fix q ∈ π^{−1} (x) = Q_{x} in the fiber over x ∈ M, this fiber identifies with G by

i : a ∈ G ⟼ q.a ∈ π^{−1} (x) .  (1)

We remark that since Q is supposed to be locally trivial, it follows that this identification is actually a diffeomorphism.

Suppose that G acts on the left on a manifold F and consider the associated bundle to the principal bundle Q(M, G) with typical fiber F. This bundle will be denoted by E(M, F, G, Q), or simpler by E, the total space of the bundle.

We recall that the elements of E are equivalence classes with respect to the relation on Q × F given by (q, v) ∼ (q.a, a^{−1}v) for a ∈ G. We use the notation q · v, q ∈ Q, v ∈ F for the equivalence class of (q, v).

The canonical projection π : E ⟶ M of the associated bundle is related to that of the principal bundle by π(q · v) = π(q).

There are the following mappings: A fixed q ∈ Q induces a bijection v ∈ F ⟼ q · v ∈ E_{x} where E_{x} = π^{−1}(x) and x = π(q), between the fiber over x and the typical fiber. Sometimes we also denote this mapping by q. Note that since the principal bundle Q is locally trivial it follows that this bijection between F and E_{x} is actually a diffeomorphism. On the other hand, a fixed v ∈ F defines a map v′ : Q ⟶ E by v′(q) = q · v. If the action of G on F is transitive this map is an onto submersion.

Let S_{Q} be a semigroup of local diffeomorphisms of Q commuting with the right action, that is, if ϕ ∈ S_{Q} then ϕ(q.a) = ϕ(q).a for all a ∈ G. We suppose that the elements of S_{Q} are defined in subsets of the form π^{−1}(U), with U an open set of M. Then S_{Q} induces a semigroup S_{M} of local diffeomorphisms of M. We use the same symbol ϕ to denote the diffeomorphism of M induced by ϕ ∈ S_{Q}. It is given by

ϕ(π(q)) = π(ϕ(q)).

Similarly, ϕ ∈ S_{Q} induces a local diffeomorphism, also denoted by ϕ, of the associated bundle E. It is given by the formula

ϕ (q · v) = ϕ (q) · v.

The corresponding semigroup of diffeomorphisms acting in E is denoted by S_{E}.

In case the semigroup S_{Q} is generated by a family Ω of vector fields on Q, its elements commute with the right action if and only if the vector fields are right invariant, i.e., R_{a∗} (X) = X for all X ∈ Ω and a ∈ G, where R_{a∗} stands for the differential of the right action. In this situation, the vector fields in Ω are projectable onto M and S_{M} is generated by these projections.

By the very definitions, the accessibility of S_{Q} imply that S_{M} is also accessible. On the other hand, we remark that if S_{Q} is generated by a family of right invariant vector fields, its accessibility is a consequence of the accessibility of S_{M} and S_{M}^{−1}, and the transitivity on Q of the group G_{Q} generated by S_{Q} (see [7]). In the sequel our basic assumption is that both S_{Q} and S_{Q}^{−1} are accessible. However, since we wish to work over control sets of S_{M}, the following stronger version of accessibility is needed.

Definition 3.1 Let D be an effective control set for S_{M}. The semigroup S_{Q} is said to be accessible over D if for some, and hence for all, q ∈ π^{−1} (D_{0}), int (S_{Q}q) ∩ π^{−1} (D) ≠ ∅. Similarly, S_{E} is accessible over D if int (S_{E}u) ∩ π^{−1} (D_{0}) ≠ ∅.

Given q ∈ Q, the intersection S_{Q} (q) ∩ Q_{x}, x = π (q) can be seen as subset of G by putting

S_{q} = {a ∈ G : ∃ϕ ∈ S_{Q}, ϕ (q) = q.a}.  (2)

It is readily seen that S_{q} is a subsemigroup of G if S_{q} ≠ ∅. The subset

T_{q} = {a ∈ G : ∃ϕ ∈ S_{Q}, q.a = ϕ (q) ∈ int (S_{Q}q)}  (3)

which is identifies to int (S_{Q}q) ∩ Q_{x} is also a semigroup in G. Since the identification (1) is a diffeomorphism we have that T_{q} ⊂ int S_{q}, where the interior of S_{q} is taken in G. The following lemma ensures that in presence of accessibility T_{q} is not empty.

Lemma 3.2 Let D ⊂ M be an effective control set for S_{M} and suppose that S_{Q} is accessible over D. Let q ∈ π^{−1} (D_{0}). Then T_{q} ≠ ∅ so that int (S_{q}) ≠ ∅.

Proof: Take ϕ ∈ S_{Q} such that ϕ (q) ∈ int (S_{Q}q) ∩ π^{−1} (D). Since x = π(q) ∈ D_{0}, there exists ψ ∈ S_{Q} such that the corresponding diffeomorphism of M takes π(ϕ(q)) back into x, so that ψϕ (q) belongs to the same fiber as q. By the choice of ϕ we have that ψϕ (q) ∈ int (S_{Q}q) showing that T_{q} has interior points as claimed. □

An immediate consequence of this lemma is that the accessibility of S_{Q} over D implies that S_{Q}^{−1} is also accessible over D. In fact, T_{q}^{−1} = {a^{−1} :

a ∈ T_{q}} identifies with int (S_{Q}^{−1}q) ∩ Q_{x} so that int (S_{Q}^{−1}q) is not empty if T_{q} ≠ ∅.

Before proceeding we mention that if q and q′ = q.a are in the same fiber then the semigroups S_{q′} and S_{q} are related by

S_{q′} = a^{−1}S_{q}a.

In fact, take c ∈ S_{q′} and let ϕ ∈ S_{Q} be such that ϕ (q′) = q′.c. Then ϕ (q) .a = ϕ (q.a) = q.ac, that is, ϕ (q) = q.aca^{−1}. This shows that aca^{−1} ∈ S_{q}, and hence that S_{q′} ⊂ a^{−1}S_{q}a. The reverse inclusion is shown the same way by writing q = q′.a^{−1}. With the same arguments it follows that T_{q′} = a^{−1}T_{q}a.

Given a fiber bundle E (M, F, G, Q) associated to Q we have that S_{Q} induces a semigroup S_{E} of diffeomorphisms of E. We shall assume in the sequel that the action of G on the typical fiber is transitive. In this case, the map q ⟼ q · v mentioned above is an onto submersion so that the accessibility of S_{Q} implies that of S_{E}.

We are interested in the control sets for S_{E}. For this we note that for each q ∈ Q, the semigroup S_{q} acts on the typical fiber F. So the idea for studying control sets on E is by breaking down the action of S_{E} into the action of S_{M} on the base space and the action of S_{q} on F with q running through the different fibers of Q. We start by looking at the behavior of the control sets under projections.

Proposition 3.3 Let E be the above fiber bundle with projection π : E → M and S_{Q}, S_{E} and S_{M} the above semigroups. Let D ⊂ E be a control set for S_{E}. Then

1. there exists a unique control set C ⊂ M for S_{M} and such that π(D) ⊂ C. If D is invariant then C is also invariant.

2. If D is effective then C is also effective and π (D_{0}) ⊂ C_{0}.

Proof:

1. Let x, y ∈ π(D), and take u, v ∈ D such that π(u) = x and π(v) = y. By definition there is a sequence ϕ_{n} ∈ S_{E} such that ϕ_{n}(u) → v. Therefore π(ϕ_{n}(u)) = ϕ_{n}(π(u)) → π(v) = y. Hence y ∈ cl(S_{M}x). Since x and y were arbitrary this shows that π (D) satisfies the second condition for a control set. Moreover, π is an open map hence π (D) has nonempty interior so that it is contained in a control C for S_{M}. The uniqueness of C is due to the fact that the intersection of different control sets is empty and if D is invariant then the invariance of C follows form π(ϕ(u)) = ϕ(π(u)) for every u ∈ E.

2. Take v ∈ D_{0}. Then v ∈ int (S_{E}v) ∩ int (S_{E}^{−1}v). This implies that

π (v) ∈ int (S_{M}π (v)) ∩ int (S_{M}^{−1}π (v)) which shows that π (v) ∈ C_{0}. Hence C_{0} is not empty and C is effective. □

This proposition defines a mapping D ⟶ C from the control sets in E into the control sets in M. We show next that in the case E is compact and in presence of accessibility every effective control set in M is in the image of this mapping.

Proposition 3.4 Let C ⊂ M be an effective control set for S_{M}. Suppose that the typical fiber F is compact and that S_{Q} is accessible over C. Then there exists an effective control set D ⊂ E for S_{E} with π(D) ⊂ C and π (D_{0}) ⊂ C_{0}.

Proof: Take x ∈ C_{0} and q ∈ Q_{x}. By Lemma 3.2 T_{q} is an open and nonvoid semigroup in G. On the other hand, the compactness of F implies that there exists an effective control set for T_{q} on F (see e.g. Section 2 of [9]). Hence there are a ∈ T_{q} and v ∈ F such that av = v. Of course q.a ∈ int (S_{Q}q), and since the map q ∈ Q ↦ q ·v ∈ E is open we have that qa·v ∈ int (S_{E} (q · v)). But qa · v = q · av = q · v so that q · v ∈ int (S_{E} (q · v)). Arguing the same way with S_{Q}^{−1} instead of S_{Q}, we conclude that

q · v ∈ int (S_{E} (q · v)) ∩ int (S_{E}^{−1} (q · v)) .

Therefore Proposition 2.3 implies that q · v belongs to an effective control set, say D, for S_{E}. Since the projection of q · v is x ∈ C_{0} we get from the previous proposition that π (D) ⊂ C and π (D_{0}) ⊂ C_{0}. □

Concerning the intersection of a control set with a fiber, we have Theorem 3.5 Let D ⊂ E be an effective control set for S_{E}. Take x ∈ M such that D_{0} ∩ E_{x} ≠ ∅ and let q ∈ Q_{x}. Then there exists an effective control set A for S_{q} in F such that

D_{0} ∩ E_{x} = q · A_{0}.

Proof: Define the subset B_{0} ⊂ F by D_{0} ∩ E_{x} = q · B_{0}. We have that B_{0} is open in F because D_{0} is open in E and q defines a diffeomorphism between F and E_{x}. Take v_{1}, v_{2} ∈ B and put u = q·v_{1} and w = q·v_{2} with u, w ∈ D_{0}∩E_{x}. Since u, w ∈ D_{0} we have by Proposition 2.2 that there is ϕ ∈ S_{E} such that ϕ(q·v_{1}) = q·v_{2}. Clearly, this map fixes the fiber over x so that in the principal bundle level, ϕ (q) = q.a for some a ∈ G. Since ϕ is in the semigroup we have that a ∈ S_{q}. However ϕ(q · v_{1}) = ϕ(q) · v_{1} = (q.a) · v_{1} = q · av_{1}, hence av_{1} = v_{2}. This shows that B_{0} ⊂ S_{q}x for all x ∈ B. Therefore there is a control set A for S_{q} such that B_{0} ⊂ A, i.e., D_{0} ∩ E_{x} ⊂ q · A.

We have that A is effective and that D_{0}∩E_{x} ⊂ q·A_{0}. In fact, take v ∈ B_{0}. Then v ∈ A and for every v_{1} ∈ B_{0} there are ϕ ∈ int S_{E} and ψ ∈ int S_{E}^{−1} such that ϕ (q · v) = q · v_{1} and ψ (q · v) = q · v_{1}. Each one of these maps fixes the fiber over x. Hence we can write ϕ (q) = q.a and ψ (q) = q.b with a ∈ T_{q} and b ∈ T_{q}^{−1}. Clearly, both a and b map v into v_{1} so that B_{0} ⊂ int S_{q}v ∩int S_{q}^{−1}v and

v ∈ int (S_{q}v) ∩ int (S_{q}^{−1}v) .

Hence A is effective and B_{0} ⊂ q · A_{0}.

The reverse inclusion follows if we prove that for any z ∈ q · A_{0} there exists u ∈ D_{0} such that z ∈ S_{E}u and u ∈ S_{E}z. In fact in this case we have that

D ⊂ cl S_{E}u ⊂ cl S_{E}z

because u ∈ S_{E}z, and reciprocally, for any w ∈ D, u ⊂ cl S_{E}w so that

z ∈ S_{E}u ⊂ cl S_{E}w.

Therefore the maximality property in the definition of a control set ensures that z ∈ D. Furthermore, the fact that z ∈ S_{E}u, u ∈ D_{0} implies, by Proposition 2.2, that z ∈ D_{0}.

Now, given z = q · v ∈ q · A_{0} take u ∈ D_{0} ∩ E_{x}. Then u ∈ q · A_{0} so that u = q · v_{1} with v_{1} ∈ A_{0}. Hence there are a, b ∈ S_{q} such that av = v_{1} and bv_{1} = v. By the definition of S_{q}, we have that ϕ (q) = q.a and ψ (q) = q.b for some ϕ, ψ ∈ S_{Q}. Therefore if we consider the corresponding mappings ϕ, ψ ∈ S_{E}, we have

ϕ (z) = ϕ (q · v) = ϕ (q) · v = q · av = q · v_{1} = u

and

ψ (u) = ψ (q · v_{1}) = ψ (q) · v_{1} = q · bv_{1} = q · v = z

concluding the proof. □

Joining together the previous results we can show that control sets for T_{q} on the fibers are contained in control sets in the bundle.

Proposition 3.6 Let E be the above fiber bundle and suppose that for q ∈ Q, T_{q} ≠ ∅. Let also B be an affective control set for T_{q} in F. Then there exists an effective control set D for S_{E} in E such that q · B_{0} = D_{0} ∩ E_{x}.

Proof: Take q·v ∈ B_{0} and put u = q·v. Then as in the proof of Proposition 3.4 there exists a control set, say D, for S_{E} such that u ∈ D_{0}. The above theorem ensures then that D_{0} ∩ E_{x} = q · B_{0}. □

Remark: In the above proposition we are assuming the existence of a control set for the semigroup T_{q} on the fiber F so we do not require compactness of the fibre, as in Proposition 3.4, where this assumption is made to ensure the existence of a control set. Also, it is not clear that effective control sets for S_{q} are contained in effective control sets in the bundle. This is because the interior of S_{q} may contain boundary points of S_{Q}, i.e., int S_{q} may be larger than T_{q}.

With the above proposition we conclude our results about general control sets on fiber bundles. Later on we shall get more precise statements about invariant control sets. In order to finish this section we consider the case where one of the semigroups S_{q} acts transitively on the typical fiber, or equivalently, the intersection of a control set with the fiber is the whole fiber. In this case we get transitivity of the semigroup above the control sets.

Proposition 3.7 Assume that S_{M} is accessible, let C ⊂ M be an effective control set, and suppose that for some x ∈ C_{0} there exists q ∈ Q_{x} such that S_{q} acts transitively on F. Then for any p ∈ Q such that π (p) ∈ C_{0} we have that S_{p} is transitive on F. Moreover, S_{E} is accessible over C, D = π^{−1} (C) is an effective control set on E and S_{E} is transitive above C_{0}.

Proof: Pick y ∈ C_{0}. Then there are ϕ, ψ ∈ S_{M} such that ϕ (x) = y and ψ (y) = x. By considering the corresponding actions on the principal bundle Q put p = ϕ (q). Then ψ (p) is in the fiber of q so that there is a ∈ G such that ψ (p) = q.a. Let us show that S_{p} is also transitive on F. Take v_{1}, v_{2} ∈ F. Then there exists b ∈ S_{q} such that bav_{1} = v_{2}, because S_{q} is transitive. This element of S_{q} is given by θ (q) = q.b. We have that ϕθψ belongs to S_{Q}. But

ϕθψ (p) = ϕθ (q.a) = ϕ (q.ba) = p.ba

so that ba ∈ S_{p}. Hence v_{2} ∈ S_{p}v_{1} showing the transitivity of this semigroup. Since y ∈ C_{0} was arbitrary and S_{p.a} = a^{−1}S_{p}a we have the first part of the proposition. From this we get immediately that S_{E} is transitive on π^{−1} (C_{0}), that is, π^{−1} (C_{0}) ⊂ S_{E}u for all u ∈ π^{−1} (C_{0}). In particular, S_{E} is accessible over C. Taking closures and using the equalities cl C = cl C_{0} and π^{−1} (cl C_{0}) = cl π^{−1} (C_{0}) we have

π^{−1} (C) ⊂ π^{−1} (cl C) ⊂ cl π^{−1} (C_{0}) ⊂ cl S_{E}u

for all u ∈ π^{−1} (C_{0}). On the other hand, if π (v) ∈ C then S_{E}v∩π^{−1} (C_{0}) ≠ ∅ so the above inclusion implies that π^{−1} (C) ⊂ cl S_{E}v. This shows that π^{−1} (C) is an effective control set, say D′, for S_{E}. However, the previous proposition ensures the existence of a control set D such that E_{x} ⊂ D_{0} and π (D) ⊂ C, that is, D ⊂ π^{−1} (C). Hence we have that

D ⊂ π^{−1} (C) ⊂ D′

which shows that D = π^{−1} (C). □

This proposition applies in particular to principal bundles, in which case the assumption that S_{q} is transitive amounts to say that S_{q} coincides with the structure group G. In case G is compact and connected this assumption is automatic because in such a group G itself is the only of its semigroups with nonempty interior. Hence we have

Corollary 3.8 Let Q be such that its structure group G is compact and connected. Let C be an effective control set on the base M and assume that S_{Q} is accessible over C. Then π^{−1}(C) is an effective control set for S_{Q} on Q and S_{Q} is transitive above C_{0}.

In this corollary it is not needed to require that G is connected as soon as we know that the total space of the bundle is itself connected. In fact, we have Proposition 3.9 With the set up as in the above corollary, suppose that π^{−1} (C_{0}) is connected instead of assuming that G is connected. Then the same result holds.

Proof: Let q ∈ Q be such that π (q) ∈ C_{0}. We have that T_{q} is a semigroup with nonvoid interior in the compact group G. Hence G_{0} ⊂ T_{q} where G_{0} stands for the component of the identity of G. This implies that 1 ∈ T_{q} so that q ∈ int (S_{Q}q). Since q ∈ π^{−1} (C_{0}) was arbitrary we have that S_{Q}p ∩ π^{−1} (C_{0}) is open for every p ∈ π^{−1} (C_{0}). Therefore if we take into account that π^{−1} (C_{0}) is connected a standard argument shows that S_{Q}p ∩ π^{−1} (C_{0}) = π^{−1} (C_{0}) for all p ∈ π^{−1} (C_{0}). Hence S_{Q} is transitive over C_{0} and the result follows as above. □

4 Invariant control sets

For the invariant control sets the results of the previous section can be highly improved so we can get a clear picture of the construction of the invariant control sets on E from those on the base space and on the fibers.

We consider as above a semigroup S_{Q} acting on the principal bundle Q, which induces semigroups S_{E} acting on an associated bundle and S_{M} acting on the base space.

Any invariant control set for S_{E} projects down onto an invariant control set on M. So in order to get the invariant control sets on E it remains to analyze the structure of the invariant control sets over a given control set C ⊂ M. We shall do this with the further assumption that the typical fiber is compact.

Given such a set, we assume that S_{Q} is accessible over C. At this regard, it is convenient to mention that accessibility over C is equivalent to the accessibility of S_{Q} from the points above C, because of the invariance of this set. Also, this accessibility implies that C is closed.

The semigroup S_{q} ⊂ G given by the intersection of S_{Q}q with the fiber through q ∈ Q has nonempty interior in G. Let us assume that there are a finite number of invariant control sets for S_{q} on the typical fiber F. Denote this number by ic (S_{q}) and enumerate the invariant control sets by C_{q}^{j}, j = 1, …, ic (S_{q}). These subsets are contained in F. However by applying q we can map them into the fiber E_{x} over x = π (q) and obtain the subsets

q · C_{q}^{j} ⊂ E_{x}, j = 1, …, ic (S_{q}) .

These subsets will be the building blocks for the construction of the invariant control sets over C. We first point out that q · C_{q}^{j} is actually independent of the specific q ∈ Q in the fiber over x. Indeed, if q′ = qa then S_{q′} = a^{−1}S_{q}a. Hence the invariant control sets for S_{q′} on F are the subsets C_{q′}^{j} = a^{−1}C_{q}^{j}. So that

q′ · C_{q′}^{j} = q′ · a^{−1}C_{q}^{j} = q · C_{q}^{j}.

Therefore we are allowed to put

C_{x}^{j} = q · C_{q}^{j}, j = 1, …, ic (S_{q}) .

We have now the following lemmas, which we state with the assumption that the fiber F is compact, although this condition is not always required.

Lemma 4.1 For x, y ∈ C_{0} take ϕ ∈ S_{Q} such that ϕ (Q_{x}) = Q_{y}, and given q_{x} ∈ Q_{x} put q_{y} = ϕ (q_{x}). Let C_{q_{x}}^{j} be an invariant control set for S_{q_{x}}. Then for every v ∈ C_{q_{x}}^{j}, we have that C_{q_{x}}^{j} ⊂ cl (S_{q_{y}} (v)).

Proof: Pick an arbitrary w ∈ C_{q_{x}}. We wish to find a sequence (b_{k})_{k≥1} in S_{q_{y}} with b_{k}v → w as k → ∞. Since S_{M} is transitive on C_{0} there exists ψ ∈ S_{Q} such that ψ ∘ ϕ(q_{x}) ∈ Q_{x}. Let a ∈ G defined by ψ ∘ ϕ(q_{x}) = q_{x}a. Then a ∈ S_{q_{x}} so that av ∈ C_{q_{x}}^{j}. Since w ∈ C_{q_{x}}^{j}, there exists a sequence (a_{k})_{k≥1} in S_{q_{x}} with a_{k}av → w. Let b_{k} = a_{k}a and let us show that b_{k} ∈ S_{q_{y}}. We have that a_{k} is defined by ψ_{k}(q_{x}) = q_{x}a_{k} with ψ_{k} ∈ S_{Q}. Now

\begin{aligned}ϕ ∘ ψ_{k} ∘ ψ(q_{y}) & = & ϕ ∘ ψ_{k} ∘ ψ ∘ ϕ(q_{x}) = ϕ ∘ ψ_{k}(q_{x}a) \\ & = & ϕ(q_{x}a_{k}a) = q_{y}a_{k}a \end{aligned}

so that a_{k}a ∈ S_{q_{y}} proving the lemma. □

Lemma 4.2 With the notations as in the lemma above, C_{q_{x}}^{j} is contained in an invariant control set for S_{q_{y}}.

Proof: Let v ∈ C_{q_{x}}^{j}. The assumption that F is compact ensures the existence of an invariant control set for S_{q_{y}} contained in cl(S_{q_{y}}(v)) (see Lemma 3.1 in [2]). Denote it by C_{q_{y}}. Since S_{q_{y}} has interior points in G, C_{q_{y}} has nonempty interior, hence C_{q_{y}} ∩ S_{q_{y}}(v) ≠ ∅ and there exists b ∈ S_{q_{y}} with bv ∈ C_{q_{y}}. We claim that there exists b′ ∈ S_{q_{y}} with b′bv ∈ C_{q_{x}}^{j}. In fact, let ψ′ ∈ S_{Q} be such that ψ′(q_{y}) = q_{y}b and take ψ ∈ S_{Q} that maps Q_{y} into Q_{x}. Then ϕ ∘ ψ(q_{y}) = q_{y}b′ with b′ ∈ S_{q_{y}}. Also,

ϕ ∘ ψ ∘ ψ′(q_{y}) = q_{y}b′b = ϕ(q_{x})b′b = ϕ(q_{x}b′b),

so that ψ ∘ ψ′ ∘ ϕ(q_{x}) = q_{x}b′b and b′b ∈ S_{q_{x}}. Hence b′bv ∈ C_{q_{x}}^{j} because v ∈ C_{q_{x}}^{j}. This proves the claim. From it and the choice of b it follows that b′b · v ∈ C_{q_{x}}^{j} ∩ C_{q_{y}}. Putting w = b′b · v the previous lemma implies that

C_{q_{x}}^{j} ⊂ cl (S_{q_{y}}w) = C_{q_{y}},

proving the existence of the invariant control set. The uniqueness follows from the fact that the intersection of different invariant control sets for S_{q_{y}} is empty. □

Recall that we are using the same notation for the Q-bundle map and the induced E-bundle map. They are related by ψ (q · v) = ψ (q) · v for q ∈ Q, v ∈ F and q · v ∈ E. With this notation in mind we translate the statement of the above lemma, in terms of the subsets C_{x}^{j}, as follows:

Put C_{x}^{j} = q_{x} · C_{q_{x}}^{j} and C_{y} = q_{y} · C_{q_{y}} with C_{q_{y}} the S_{q_{y}}-invariant control set ensured by the lemma. If ϕ is as in the proof then

ϕ(C_{x}^{j}) = ϕ(q_{x} · C_{q_{x}}) = ϕ(q_{x}) · C_{q_{x}} ⊂ q_{y} · C_{q_{y}}.

Therefore ϕ maps the subset C_{x}^{j} inside one of the subsets C_{y}^{i} = q_{y} · C_{q_{y}}^{i} for i = 1, ..., ic(S_{q_{y}}). We show now that this C_{y}^{i} ⊂ E_{y} is the same for every ϕ′ that maps E_{x} into E_{y}.

Lemma 4.3 Let ϕ be as before with q_{y} = ϕ (q_{x}), take C_{x}^{j} and suppose that C_{y}^{j} is such that ϕ(C_{x}^{j}) ⊂ C_{y}^{j}. Then

i) If ψ ∈ S_{E} maps E_{y} into E_{x} then ψ(C_{y}^{j}) ⊂ C_{x}^{j}.

ii) If ϕ′ ∈ S_{E} maps E_{x} into E_{y} then ϕ′(C_{x}^{j}) ⊂ C_{y}^{j}.

Proof: As above we take q_{x} and q_{y} in the fiber over x and y respectively such that ϕ (q_{x}) = q_{y}. If ψ is as in ii) then it maps the fiber over y into the fiber over x so that ψ ∘ ϕ maps the fiber over x into itself. Since both maps are in S_{Q} we have that there exists a ∈ S_{q_{x}} such that ψ ∘ϕ (q_{x}) = q_{x}.a. Now, take w ∈ C_{q_{x}}^{j}. Then

ψ(q_{y} · w) = ψ(q_{y}) · w = ψ ∘ ϕ(q_{x}) · w = q_{x} · aw.

Therefore ψ (q_{y} · w) ∈ C_{x}^{j}. However q_{y} · w ∈ C_{y}^{j} because q_{y} · w = ϕ (q_{x} · w) and by assumption ϕ (C_{x}^{j}) ⊂ C_{y}^{j}. Hence ψ (C_{y}^{j})∩C_{x}^{j} ≠ ∅ and i) follows from Lemma 4.2 (and the above comments) applied to ψ.

Concerning ii), we note that it is just a rephrasing of i) with ψ in place of ϕ and ϕ′ instead of ψ. □

As a consequence of the above lemmas we get the following description of the invariant control sets over C.

Theorem 4.4 Let C ⊂ M be an invariant control set for S_{M}, and assume that S_{Q} is accessible over C. Assume also that the fiber F is compact. Then

1. ic(S_{q}) is constant as a function of q ∈ π^{−1} (C_{0}).

2. There are invariant control sets for S_{E} over C and its number equals ic(S_{q}).

3. For every invariant control set D ⊂ π^{−1} (C) for S_{E} and q ∈ Q,

C ∩ E_{x} = q · B,

where x = π(q) and B is an invariant control set for S_{q} in F. Proof: Lemma 4.3 implies that ic(S_{q_{x}}) = ic(S_{q_{y}}) so 1) follows from the transitivity of S_{M} on C_{0} and the invariance of ic(S_{q}) with q varying in a fixed fiber.

Now, pick x ∈ C_{0} and some C_{x}^{j} ⊂ E_{x}. If q · v ∈ C_{x}^{j} then by Lemma 4.3 ii) there is defined for each y ∈ C_{0} a unique C_{y}^{j} with C_{y}^{j} ∩ S_{E}(q · v) ≠ ∅. Put

C^{j} = ⋃_{y∈C_{0}} C_{y}^{j}.

Then C^{j} is S_{E}-invariant as follows from Lemma 4.3 i). Hence cl C^{j} is also S_{E}-invariant and a fortiori cl(S_{E}(u)) = cl C^{j} for every u ∈ cl C^{j}. This shows that cl C^{j} is an invariant control set. By Theorem 3.5 its intersection with a fiber E_{x}, x ∈ C_{0} is the image under v → q · v of a control set for S_{q}. However, by construction, C_{x}^{j} is contained in cl C^{j} if x ∈ C_{0}. Hence

cl C^{j} ∩ E_{x} = C_{x}^{j}.

We have thus constructed ic (S_{q}) different control sets for S_{E}.

To see that these are the only possibilities, let D be an invariant control set for S_{E} such that π (D) ⊂ C. The invariance of D implies that π (D) = C. Take u ∈ D such that π (u) ∈ C_{0}, and put u = q · v, v ∈ F and q ∈ Q. Then the compactness of F ensures the existence of an invariant control set for S_{q} which meets S_{q}v. But qS_{q}v ⊂ S_{E}(q · v), which implies that for some C^{j} as above S_{E}(q · v) ∩ C^{j} ≠ ∅, i.e., C ∩ C^{j} ≠ ∅ so that C = C^{j}. This proves 2), 3) and the theorem. □

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