Abstract
We define an equivalence relation on a topological space which is acted by topological monoid as a transformation semigroup. Then, we give some results about the -invariant classes for this relation. We also provide a condition for the existence of relative -invariant classes.
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 monoid as a transformation semigroup. Then, we provide the necessary and sufficient conditions for the equivalence classes to be -invariant classes which correspond with the control sets for control systems. Then, we study the -invariant classes for this relation in , in particular, and we provide the conditions for the existence and uniqueness of invariant classes.
Throughout this paper, will denote the closure set of a set , and will denote the interior set of and all topological spaces involved Hausdorff.
Definition 1.1 (see [2]). Let be a monoid with the identity element and also a topological space. Then, will be called a topological monoid if the multiplication operation of: is continuous mapping from to .
Definition 1.2 (see [4]). Let be a topological monoid and a topological space. We say that acts on as a transformation semigroup if there is a continuous map between the product space and satisfying we further require that for all . The triple is called an flow; will denote . In particular, an flow is called phase flow if is a compact space.
The orbit of under is the set . For a subset of , denotes the set . And a subset is called an invariant set if and . A control set for on is a subset of which satisfies(1),(2)for all ,(3) is a maximal with these properties.
Then, we say that a subset , satisfies the no-return condition if for some and , then .
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. Invariant Classes
Let be an flow. From the action on , we can define the relation ~ on by It is clear that the relation ~ is an equivalence relation, and will denote the set of all equivalence classes induced by ~ on . We observe that for all , and if , then for all .
The following theorem shows that an equivalence class with nonempty interior set is a control set for on .
Theorem 2.1. Let be an phase flow. A class with is a control set for on .
Proof. It is clear that for all . Let be a subset of satisfying the property Now if then for all . Since is a compact space, is a Hausdorff space and by the continuity of the action , then the orbit is a closed subset of for all (i.e., for all ). Then, for all . Since , then . On the other hand, since , then . Hence, .
In the following lemma, we give necessary and sufficient conditions for the equivalence classes to be invariant classes.
Lemma 2.2. Let be an flow. A class is an invariant class if and only if .
Proof. Suppose that is an invariant and let , then for some . Since , then . Hence, , and we have . Therefore, .
Conversely, let and , then for some , . Hence, . Take for some . Hence
Therefore, is an invariant class.
Theorem 2.3. Let be an phase flow. Then, for all , there exists an invariant class .
Proof. For , consider the family of subsets
We can define the relation on by
Then, it is clear that the family with is a partially order set. Let be a linearly ordered subset of , where is an index set. Since is a compact space, is a Hausdorff space and by the continuity of the action , then the orbit is a compact closed subset of for all . Hence we have a chain of closed subsets of a compact . Hence the intersection
Take for all . Then, for all , implies that is a lower bound of the chain (i.e., is an upper bound of the linearly order subset of ). Hence, Zorn's lemma implies that the family has a maximal element, say . Then, .
Now, we show that is an invariant. Let , then and , but by the maximality of , we get that , this implies . Hence, (i.e., ) and we have that . Then, by Lemma 2.2, is an invariant class.
Now, we propose an open problem that whether invariant class is unique?
Theorem 2.4. Let be an phase flow. Every satisfies the no-return condition for all .
Proof. Since is a compact space, is a Hausdorff space and by the continuity of the action , then the orbit is a compact closed subset of for all (i.e., for all ). Let for some and . Take and . Hence, On the other hand, for some , we have Hence, .
The next theorem states that if has the no-return condition, then any class is entirely contained in or . Further is also an invariant if an invariant class for all .
Theorem 2.5. Let be phase flow and be a subset of has no-return condition. Then, is an invariant set if is an invariant class for all .
Proof. It is clear that because . Since is a compact space, is a Hausdorff space and by the continuity of the action , then the orbit is a compact closed subset of for all (i.e., for all ). Let , then for some . Hence, (i.e., and ). Since , then . By the no-return condition, we have that . Hence, Now, we show that is an invariant set. Let . Then, for some . Hence, . Since is an invariant class then by Lemma 2.2, and by (2.9), we get that . Hence, is an invariant.
Acknowledgments
The authors wish to express their sincere gratitude to anonyms referees for their very helpful comments and suggestions which improved the paper. The authors would also acknowledge that this research was partially supported by the University Putra Malaysia under the Research University Grant Scheme (RUGS) 05-01-09-0720RU and Fundamental Research Grant Scheme 01-11-09-723FR.