On the S-Invariance Property for S-Flows

and Applied Analysis 3 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 be S-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 sax 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 saz for some s ∈ S, z ∈ x . Hence, z ∈ Oa x . Take z s′ax for some s′ ∈ S. Hence y saz sa ( s′ax ) ss′ax ∈ Oa x x . 2.3 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. For x ∈ X, consider the family of subsets Ex {z : Oa z ⊂ Oa x }. 2.4 We can define the relation on Ex by x1 x2, if Oa x2 ⊂ Oa x1 for x1, x2 ∈ Ex. 2.5 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 orbitOa 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 ⋂ i∈∧ Oa zi / ∅. 2.6 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. 4 Abstract and Applied Analysis Now, we propose an open problem that whether S-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. 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 z ∈ Oa y for some y ∈ x and Oa z ∩ x / ∅. Take ω ∈ Oa z and ω ∈ x . Hence, x ∈ Oa x ⊂ Oa ω ⊂ Oa z . 2.7 On the other hand, z ∈ Oa y for some y ∈ x , we have z ∈ Oa z ⊂ Oa ( y ) ⊂ Oa x . 2.8 Hence, z ∈ x . The next theorem states that if M has the no-return condition, then any class x is entirely contained inM orM. FurtherM 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 that M ⊂ 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,