Relative Invariance for Monoid Actions

Let S be a topological monoid acting on the topological space M. Let J be a subset of M. Our purpose here is to study the subsets of M which correspond, under the action of S, to the relative (with respect to J) invariant control sets for control systems (see [4] section 3.3). The relation x ∼ y if y ∈ cl(Sx) and x ∈ cl(Sy) is an equivalence relation and the classes with respect to this relation with nonempty interior in M are the control sets for the action of S. It is given conditions for the existence and uniqueness of relative invariant classes. As it was done for the control sets, we define an order in the classes and relate it to the relative invariant classes. We also show under certain condition that the relative invariant classes are relatively closed in J.


Introduction
One of the principal dynamical concepts in control theory is the study of the controllability of the control systems.Many questions about the control system, especially those related to its controllability depend, in fact, only on the action of the semigroup of the system, so that it can be abstracted to arbitrary semigroup actions and solved in a more general setting.The regions of the state space where the controllability occurs are called control sets.The control sets for control systems were studied by Colonius and Kliemann in [1], [2], [3] and [4].In particular, Colonius and Kliemann introduced the concept of a control set which is relatively invariant with respect to a subset of the phase space of the control system (see Definition 3.1.9,pg.50, in [4]).From a more general point of view, the theory of control sets for semigroup actions was developed by San Martin and Tonelli (see [6], [7] and [8]).Let S be a topological monoid and suppose that S acts on a topological space M .Since the control sets are the regions where S is approximate transitive it is natural to define an equivalence relation by saying that two points are equivalent if they are approximate attainable by the action of S. We consider equivalence classes in M with respect to this relation.We show that a class with nonempty interior in M is a control set for S. The purpose of this paper is to study the relative invariant classes in M .We define relatively invariant classes.In case S is the system semigroup of a control system these relatively invariant classes, with nonempty interior in M , are the relative invariant control sets defined by Colonius and Kliemann.We develop the theory of relative invariant classes.As it was done for the control sets, we define an order in the classes and relate it to the relative invariant classes.We give conditions for the existence and uniqueness of relative invariant classes.Under the hypothesis of accessibility, we show that a relative invariant class is relatively closed.

Relative invariance
First, we define the concept of a topological semigroup.We refer to [9] for the theory of topological semigroups.Throughout this paper we always assume that all topological spaces involved are Hausdorff.

Definition 1. Let S be a non-void topological space which is provided with an associative multiplication
Then S is called a topological semigroup if µ is continuous as a map between the product space S × S and the space S. If S has an identity 1 then S is called a topological monoid.
Definition 2. Let M be a topological space and S a topological monoid.We say that S acts on M as a transformation monoid if there is a map satisfying the following conditions: 1. π is continuous as a map between the product space S × M and the space M .

g.(h.x) = (gh).
x for all g, h ∈ S and for all x ∈ M .
We suppose, in this paper, that S acts on a topological space M as a transformation monoid.We use the notation Sx = {gx : g ∈ S} and call it the orbit of x under S. Assume that x ∈ cl(Sy).Then Sx ⊂ Scl(Sy) ⊂ cl(SSy) = cl(Sy) because the action is continuous.Therefore cl(Sx) ⊂ cl(Sy) if x ∈ cl(Sy).We will use this fact frequently on the text.
The control sets for semigroup actions were defined by San Martin and Tonelli in [6].Here, we recall the definition.Let S be a topological monoid acting on a topological space M .We define a maximal class.
We will show later that a maximal class with int M ([x]) = ∅ is an invariant control set for the monoid S.
We observe that [x] ⊂ cl(Sx) for every x ∈ M .Definition 6.We say that a subset A ⊂ M is S-invariant, or invariant for the monoid S, if for every y ∈ A we have cl(Sy) ⊂ A.
A S-invariant class is always maximal.In fact, suppose [y] we have x y and y ∈ cl(Sx) ⊂ [x].Therefore x ∼ y and [x] = [y], showing the maximality.
We define O = {cl(Sx) ⊂ M : x ∈ M }.Now, we relate maximality and S-invariance.Lemma 1.For x ∈ M the following statements are equivalent:

[x] is closed and S-invariant
Proof: Let's assume that cl(Sy) ⊂ cl(Sx) for some y ∈ M .Take z ∈ cl(Sy).By the continuity of the action of S on M we have cl(Sz) ⊂ cl(Sy).Since z ∈ cl(Sx) the maximality of [x] implies that cl(Sx) ⊂ cl(Sz).Therefore cl(Sx) ⊂ cl(Sy) showing that cl(Sx) is minimal.Suppose that cl(Sx) is minimal in O. Then cl(Sy) ⊂ cl(Sx) for all y ∈ cl(Sx).By minimality cl(Sy) = cl(Sx) for all y ∈ cl(Sx).This implies that cl(Sx) is entirely contained in an equivalence class so that cl The maximal classes with nonempty interior in M are the invariant control sets, more specifically, we have.On the existence of maximal classes we have the following proposition.
Proposition 2. Let J be a S-invariant compact subset of M .Then for every x ∈ J there exists a maximal equivalence class [w] ⊂ cl(Sx).
Proof: Fix x ∈ J and consider the family of subsets This family is not empty because it contains cl(Sx).Let us order O x by inclusion and show with the aid of Hausdorff 's maximality principle that it contains minimal elements: Take a chain {cl(Sy)} y∈I of subsets in O x , where I is an index set.Since J is S-invariant cl(Sx) ⊂ J. Therefore we have a chain of closed subsets of J. Hence they are compact which implies that the intersection y∈I cl(Sy) is not empty.Take z ∈ y∈I cl(Sy).Then cl(Sz) belongs to O x and is contained in cl(Sy) for all y ∈ I.These means that cl(Sz) is a lower bound of the chain.Applying the maximality principle we conclude that O x contains a minimal element, say cl(Sw).Any element of Remark: We comment that the Proposition 2 has already been proved in the context of topological dynamics.Since an invariant class [x] is a minimal set the Proposition 2 is the classical result from topological dynamics which says that every compact invariant set has a minimal set.(see [5] Theorem 2.22).We define a maximal class relatively to a subset J contained in M .
For subsets J of M which are compact and S-invariant we have.
Corollary 2. Suppose that J ⊂ M is compact and S-invariant.Then there are J-maximal classes and they are the maximal classes contained in J.
Proof: Take x ∈ J. Since J is compact and invariant under the action of S we have by the Lemma 2 that there exist a maximal class, say, [w] ⊂ cl(Sx) ⊂ J.We also have that [w] is J-maximal.Conversely, a maximal class contained in J is J-maximal.2 Now, we define a class which is invariant with respect to a subset Therefore, if a class [x] ⊂ J is SJ-invariant it cannot leave by the closure of an orbit without leaving J. Remark: The relative invariant control sets for control systems were studied by Colonius and Kliemann in [4], pg.70, section 3.3.The definition of a SJ-invariant class was motivated by the definition of a relative invariant control set (see [4], pg.50, Definition 3.1.9).Roughly speaking, a control set D for a control system contained in a subset J of the phase space is said to be a relative invariant control set, if x ∈ D and a trajectory of the system starting at x leaves D, for some time and for some control, then the trajectory also leaves J.
It follows immediately that a SJ-invariant class is J-maximal.As a consequence a SM -invariant class is M -maximal and therefore it is a maximal class.Conversely, by the Lemma 1, a maximal class is On the existence of SJ-invariant classes we have.
Corollary 3. Suppose that J ⊂ M is compact and S-invariant.Then [x] is a SJ-invariant class if and only if [x] is J-maximal.In this case there exist SJ-invariant classes.
The no-return condition defined bellow was introduced, in the context of control systems, by Colonius and Kliemann (see [4], pg.50, Theorem 3.1.10)in the study of relatively invariant control sets.Definition 9. We say that a subset J ⊂ M satisfy the no-return condition if z ∈ cl(Sx) for some x ∈ J and cl(Sz) ∩ J = ∅, then z ∈ J.
This condition says that if we leave J we cannot go back to J again thorough the closure of an orbit of S. Now, we translate the no-return condition in terms of an union of equivalence classes.Proposition 3. Suppose that J ⊂ M satisfy the no-return condition.Then J is exhaustive for the equivalence relation ∼, i.e., any class [x] is entirely contained in J or in J c .Proof: Let [x] be a class such that [x] ∩ J = ∅ and [x] ∩ J c = ∅ and we will obtain a contradiction.Take z ∈ [x] ∩ J and y ∈ [x] ∩ J c .Then z ∼ y, i.e., z ∈ cl(Sy) and y ∈ cl(Sz).By the no-return condition we have that y ∈ J which is a contradiction. 2 Corollary 4. Suppose J satisfies the no-return condition.Then Proof: Take x ∈ J. Then by the Proposition 3 we have The converse of the last corollary is not always true.In fact, suppose that J = The next theorem gives conditions for the existence of SJ-invariant classes.It also generalizes Proposition 3.3.3,pg.72, in [4].
Theorem 1.Let J be a subset of M satisfying the no-return condition.Suppose there is an x ∈ J and a compact set Then there exists a SJ-invariant class [w] ⊂ cl(Sx).
Proof: For y ∈ cl(Sx) ∩ J we define the compact K y = cl(Sy) ∩ K. Since 1 ∈ S we have that K x is defined.Now, consider the family define the following order on T We will show that D ⊂ cl(Sx) is a SJ-invariant class.We know that r ∈ Sr ∩ J ⊂ D. By its only definition, every z ∈ D is approximately reachable from r and r z.Conversely, take z ∈ D and we will show that z r.Since z ∈ D we have z ∈ cl(Sr) and by the definition of K r we have r ∈ cl(Sx).Thus z ∈ cl(Sx) ∩ J.It follows from the hypothesis that K z = cl(Sz) ∩ K = ∅.Pick an element z 0 ∈ K z .It is obvious that z z 0 .Then K z 0 ∈ T and by the maximality of K r one obtains z 0 r.The transitivity of implies that z r.Therefore we have approximate transitivity in D. We also have that D is a class.Otherwise, there exists a class The theorem above allows us to show that the J-maximal and JS-invariant classes coincide.
Corollary 5. Let J ⊂ M be a subset satisfying the no-return condition.Take x ∈ J and assume that there exists a compact set K ⊂ J such that for all y ∈ cl(Sx) ∩ J We will see next that a closed subset is a maximal class if and only if it is the closure of the orbit of its elements.Now, we give conditions for the existence of a unique SJ-invariant class.
Proposition 5. Let J ⊂ M be a subset satisfying the no-return condition.Take x ∈ J and assume that there exists a compact set K x ⊂ J which is S-invariant and such that for all y ∈ cl(Sx) Then C is a unique SJ-invariant class contained in J.
Proof: First, we show that C is a SJ-invariant class.It is easy to see that C is closed and it is contained in J.By the Proposition As the invariant control sets are closed in M under the hypothesis of accessibility of S, we show that the SJ-invariant classes are relatively closed in J.
Definition 10.We say that a subset J ⊂ M satisfies the J-accessibility condition for a monoid S if for all y ∈ J, int M (Sy ∩ J) = ∅.
Remark: The hypothesis of J-accessibility was also considered in the context of control systems (see [4], pg.50, Theorem 3.1.10).

Definition 3 .Definition 4 .
A control set for S on M is a subset D ⊂ M which satisfies 1. int(D) = ∅, 2. For every x ∈ D, D ⊂ cl(Sx) and 3. D is maximal with these properties.The action on M induces the pre-order relation defined byx y if y ∈ cl(Sx), x, y ∈ M. We define x ∼ y if x y and y x.Therefore [x] = [y] if and only if y ∈ cl(Sx) and x ∈ cl(Sy).Thus ∼ is the equivalence relation associated with .The pre-order in M induces a partial order in the quotient space M/ ∼.This order in the quotient space is also denoted by .We denote by [x] ∈ M/ ∼ the equivalence class of x ∈ M .From the control theoretic point of view a class [x] with int M ([x]) = ∅ is a control set for the action of the monoid S. Proposition 1.Let D = [x] be a class with respect to the equivalence relation ∼.Suppose that int M (D) = ∅.Then D is a control set for S on M .Proof: We first show that D = [x] ⊂ cl(Sy), for every y ∈ [x].Take y ∈ [x] then x ∼ y.For z ∈ [x], we have z ∼ x.By the transitivity of the relation ∼ we have z ∼ y and y z.This implies that z ∈ cl(Sy).Now, suppose D ⊆ D with D satisfying the condition D ⊂ cl(Sz) for every z ∈ D .Take y ∈ D .Then y ∈ cl(Sz), for every z ∈ D , in particular for x ∈ D. Therefore y ∈ cl(Sx) and x y.On the other hand, since x ∈ D ⊆ D , we have x ∈ cl(Sy) once y ∈ D and y x.Hence x ∼ y and y ∈ [x] = D, showing the maximality of D. 2

Corollary 1 .
Let D = [x] be a maximal class with respect to the relation ∼.Assume int M (D) = ∅.Then D is an invariant control set for S on M .Proof: First we show that cl(D) = cl(Sy) for every y ∈ D. Since [x] is maximal we have by the Lemma 1 that [x] = cl(Sx).Hence D = cl(D).For y, z ∈ [x] we have z ∈ cl(Sy).Conversely, take z ∈ cl(Sy), since x ∼ y we have y ∈ cl(Sx).Therefore z ∈ cl(Sx) = [x].Now we will show the maximality of D. Suppose [x] = D ⊆ D and D satisfies the equality cl(D ) = cl(Sy) for every y ∈ D .Take z ∈ D .Then cl(D ) = cl(Sz).Since x ∈ D ⊆ D ⊆ cl(D ) we have x ∈ cl(Sz) and z x.On the other hand, since x ∈ D ⊆ D we have cl(D ) = cl(Sx).But, z ∈ D ⊆ cl(D ) = cl(Sx) and therefore x z.It follows that x ∼ z and z ∈ [x] = D, showing the maximality of D. 2 . Take x ∈ J and z ∈ cl(Sx) with cl(Sz) ∩ J = ∅.Pick w ∈ cl(Sz) ∩ J. Then x z w.It follows that [x] [z] [w].Now, It is not always possible to conclude that z ∈ [z] ⊂ J.
[w] ⊃ D containing a point w / ∈ D = cl(Sr) ∩ J.The no-return condition and Proposition 3 implies that D ⊂ [w] ⊂ J.It follows that w ∼ r and w ∈ cl(Sr) ∩ J contradicting the choice of w.It remains to show the SJ-invariance of D. Take z ∈ D = cl(Sr) ∩ J and suppose that there exists k ∈ cl(Sz) ∩ J. Then k ∈ cl(Sz) ⊂ cl(Sr) and k ∈ cl(Sr) ∩ J, showing the SJ-invariance of D. 2 cl(Sy) ∩ K = ∅ Then for a class [w] ⊂ cl(Sx) ∩ J we have that [w] is J-maximal if and only if [w] is SJ-invariant.Proof: Suppose [w] is J-maximal.Assume that there exists y ∈ [w] ⊂ cl(Sx) and z ∈ cl(Sy) ∩ J with z / ∈ [w].By the last proposition there exist a SJ-invariant class [w 1 ] ⊂ cl(Sy) ⊂ cl(Sx).Thus [w] [w 1 ] and [w] = [w 1 ].This contradicts the J-maximality of [w].Hence [w] is SJ-invariant.2

Proposition 4 .
Let C be a closed subset of M .Then C is maximal class if and only if C = cl(Sx) for every x ∈ C. Proof: Suppose C = cl(Sx) for every x ∈ C. Then for x, y ∈ C we have y ∈ cl(Sx) and x ∈ cl(Sy), i.e., x ∼ y and therefore C is contained in a equivalence class, say, [x] ⊃ C. Furthermore, for y ∈ [x] we have y ∈ cl(Sx), and therefore [x] ⊂ cl(Sx) = C, showing that [x] = C. Now, suppose that C = [x] [y].Then x y and y ∈ cl(Sx) = [x].Therefore C = [x] = [y] is a maximal class.Conversely, assume that C = [x] is a maximal class.Thus by the Lemma 1 we have C = [x] = cl(Sx) for every x ∈ C. 2 4 and Corollary 5 it is enough to show that C = cl(Sx) for every x ∈ C. Take x ∈ J and y ∈ cl(Sx) ∩ J. Then cl(Sy) ⊂ cl(Sx) and cl(Sx) ∩ K x ⊂ cl(Sx).Therefore C ⊂ x∈J (cl(Sx)) ⊂ cl(Sx) for every x ∈ C. Now take w ∈ cl(Sz) with z ∈ C. Then for every x ∈ J and every y ∈ cl(Sx) ∩ J we have z ∈ cl(Sy) ∩K x .Since K x is S-invariant we have w ∈ cl(Sz) ⊂ cl(Sy) ∩ K x .Thus w ∈ C. It remains to show the uniqueness of C. Suppose that C 1 ⊂ J is a maximal class.By the Proposition 4, cl(Sx) = C 1 for every x ∈ C 1 .Thus C 1 = x∈C 1 cl(Sx).Therefore C = x∈J y∈cl(Sx)∩J (cl(Sy) ∩ K x ) ⊂ x∈J (cl(Sx)) ⊂ x∈C 1 cl(Sx) = C 1 .By the maximality of C it follows that C = C 1 .2