On the -Invariance Property for -Flows

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