# On theS-Invariance Property for S-Flows

Authors: Amin Saif; Adem Kılıçman
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We define an equivalence relation on a topological space which is acted by topological monoidS as a transformation semigroup. Then, we give some results about the S-invariant classes for this relation. We also provide a condition for the existence of relative S-invariant classes.


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## Body

## 1. Introduction

The invariance theory is one of the principal concepts in the topological dynamics system, see [1, 2]. In [3], 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. From a more general point of view, the theory of control sets for semigroup actions was developed by San Martin and Tonelli in [4].In this paper, we define an equivalence relation on a topological space which is acted by topological monoidS as a transformation semigroup. Then, we provide the necessary and sufficient conditions for the equivalence classes to be S-invariant classes which correspond with the control sets for control systems. Then, we study the S-invariant classes for this relation in X, in particular, and we provide the conditions for the existence and uniqueness of S-invariant classes.Throughout this paper,cl(A) will denote the closure set of a set A, and int(A) will denote the interior set of A and all topological spaces involved Hausdorff.Definition 1.1 (see [2]).
LetS be a monoid with the identity element e and also a topological space. Then, S will be called a topological monoid if the multiplication operation of: (s,t)→st is continuous mapping from S×S to S.Definition 1.2 (see [4]).
LetS be a topological monoid and X a topological space. We say that Sacts onXas a transformation semigroup if there is a continuous map a:S×X→X between the product space S×X and X satisfying
(1.1)a(st,x)=a(s,a(t,x)),∀s,t∈S,x∈X,
we further require that a(e,x)=x for all x∈X. The triple (S,X,a) is called an S-flow; sa̅x will denote a(s,x). In particular, an S-flow (S,X,a) is called S-phase flow if S is a compact space.Theorbit of x∈X under S is the set Oa(x)={sa̅x:s∈S}. For a subset M of X, S(M) denotes the set {sa̅m:s∈S,m∈M}. And a subset M is called an S-invariant set if M≠∅ and S(M)⊂M. A control set for S on X is a subset C of X which satisfies(1)
int(C)≠∅,(2)
for allx∈C,C⊂cl(Oa(x)),(3)
C is a maximal with these properties.Then, we say that a subsetM⊂X, satisfies the no-return conditionif y∈cl(Oa(x)) for some x∈M and cl(Oa(y))∩M≠∅, then y∈M.Lemma 1.3 (see [5, Zorn's Lemma]).
If each chain in a partially ordered set has an upper bound, then there is a maximal element of the set.


## 2.S-Invariant Classes

Let(S,X,a) be an S-flow. From the action on X, we can define the relation ~ on X by (2.1)x~yifx∈Oa(y),y∈Oa(x),x,y∈X.
It is clear that the relation ~ is an equivalence relation, and [X] will denote the set of all equivalence classes induced by ~ on X. We observe that [x]⊂Oa(x) for all x∈X, and if y∈Oa(x), then Oa(y)⊂Oa(x) for all x,y∈X.The following theorem shows that an equivalence class with nonempty interior set is a control set forS on X.Theorem 2.1.
Let(S,X,a) be an  S-phase flow. A class  [x]∈[X] with  intX([x])≠∅ is a control set for  S on  X.Proof.
It is clear that[x]⊂Oa(x)⊂Oa(y)⊂cl(Oa(y)) for all y∈[x]. Let C be a subset of X satisfying the property
(2.2)C⊂cl⁡(Oa(z)),∀z∈C,[x]⊂C.
Now if ω∈C then ω∈cl(Oa(z)) for all z∈C. Since S is a compact space, X is a Hausdorff space and by the continuity of the action a, then the orbit Oa(x) is a closed subset of X for all x∈X (i.e., cl(Oa(x))=Oa(x) for all x∈X). Then, ω∈Oa(z) for all z∈C. Since x∈C, then ω∈Oa(x). On the other hand, since x∈[x]⊂C⊂Oa(ω), then ω∈[x]. Hence, C=[x].In the following lemma, we give necessary and sufficient conditions for the equivalence classes to beS-invariant classes.Lemma 2.2.
Let(S,X,a) be an  S-flow. A class  [x]∈[X] is an  S-invariant class if and only if  [x]=Oa(x).Proof.
Suppose that[x]∈[X] is an S-invariant and let y∈Oa(x), then y=sa̅x for some s∈S. Since x∈[x], then y∈S([x])⊂[x]. Hence, Oa(x)⊂[x], and we have [x]⊂Oa(x). Therefore, [x]=Oa(x).
Conversely, let[x]=Oa(x) and y∈S([x]), then y=sa̅z for some s∈S, z∈[x]. Hence, z∈Oa(x). Take z=s′a̅x for some s′∈S. Hence
(2.3)y=sa̅z=sa̅(s′a̅x)=ss′a̅x∈Oa(x)=[x].
Therefore, [x] is an S-invariant class.Theorem 2.3.
Let(S,X,a) be an  S-phase flow. Then, for all  x∈X, there exists an  S-invariant class  [y]⊂Oa(x).Proof.
Forx∈X, consider the family of subsets
(2.4)Ex={z:Oa(z)⊂Oa(x)}.
We can define the relation ⪯ on Ex by
(2.5)x1⪯x2,ifOa(x2)⊂Oa(x1)forx1,x2∈Ex.
Then, it is clear that the family Ex with ⪯ is a partially order set. Let {zi:i∈∧} be a linearly ordered subset of Ex, where ∧ is an index set. Since S is a compact space, X is a Hausdorff space and by the continuity of the action a, then the orbit Oa(x) is a compact closed subset of X for all x∈X. Hence we have a chain {Oa(zi):i∈∧} of closed subsets of a compact Oa(x). Hence the intersection
(2.6)⋂i∈∧Oa(zi)≠∅.
Take r∈Oa(zi) for all i∈∧. Then, Oa(r)⊂Oa(zi) for all i∈∧, implies that Oa(r) is a lower bound of the chain {Oa(zi):i∈∧} (i.e., r is an upper bound of the linearly order subset {zi:i∈∧} of Ex). Hence, Zorn's lemma implies that the family Ex has a maximal element, say y. Then, [y]⊂Oa(y)⊂Oa(x).
Now, we show that[y] is an S-invariant. Let z∈Oa(y), then z∈Oa(z)⊂Oa(x) and y⪯z, but by the maximality of y, we get that z⪯y, this implies y∈Oa(z). Hence, z∈[y] (i.e., Oa(y)⊂[y]) and we have that [y]⊂Oa(y). Then, by Lemma 2.2, [y] is an S-invariant class.Now, we propose an open problem that whetherS-invariant class is unique?Theorem 2.4.
Let(S,X,a) be an  S-phase flow. Every  [x]∈[X] satisfies the no-return condition for all  x∈X.Proof.
SinceS is a compact space, X is a Hausdorff space and by the continuity of the action a, then the orbit Oa(x) is a compact closed subset of X for all x∈X (i.e., cl(Oa(x))=Oa(x) for all x∈X). Let z∈Oa(y) for some y∈[x] and Oa(z)∩[x]≠∅. Take ω∈Oa(z) and ω∈[x]. Hence,
(2.7)x∈Oa(x)⊂Oa(ω)⊂Oa(z).
On the other hand, z∈Oa(y) for some y∈[x], we have
(2.8)z∈Oa(z)⊂Oa(y)⊂Oa(x).
Hence, z∈[x].The next theorem states that ifM has the no-return condition, then any class [x] is entirely contained in M or Mc. Further M is also an S-invariant if [x] an S-invariant class for all x∈M.Theorem 2.5.
Let(S,X,a) be S-phase flow and M be a subset of X has no-return condition. Then, M is an S-invariant set if [x] is an S-invariant class for all x∈M.Proof.
It is clear thatM⊂⋃x∈M[x] because x∈[x]. Since S is a compact space, X is a Hausdorff space and by the continuity of the action a, then the orbit Oa(x) is a compact closed subset of X for all x∈X (i.e., cl(Oa(x))=Oa(x) for all x∈X). Let y∈⋃x∈M[x], then y∈[x] for some x∈M. Hence, [x]=[y] (i.e., x∈Oa(y) and y∈Oa(x)). Since x∈M, then Oa(y)∩M≠∅. By the no-return condition, we have that y∈M. Hence,
(2.9)M=⋃x∈M[x].
Now, we show that M is an S-invariant set. Let y∈S(M). Then, y=sa̅x for some x∈M. Hence, y∈Oa(x). Since [x] is an S-invariant class then by Lemma 2.2, [x]=Oa(x) and by (2.9), we get that y∈[x]⊂M. Hence, M is an S-invariant.



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