Global Journal of Researches in Engineering, J: General Engineering, Volume 1 9 Issue 2

Theorem 2.3 The product of two α - transitive maps is not necessarily α - transitive map [4]. Corollary 2.4 The product of two topologically α - transitive systems is not necessarily topologically α - transitive. III. N ew T ypes of C haos of T opological S paces In this section, I introduced and defined α -type transitive maps[3] and α –type minimal maps[3], and study some of their properties and prove some results associated with these new definitions. I investigate some properties and characterizations of such maps. Definiti on 3.1 Let X is a separable and second category space with no isolated point, if for X x ∈ the set } :)( { N ∈ n x f n is dense in X thenx is called hyper-cyclic point. If there exists such an X x ∈ , then f is called hyper-cyclic function or f is said to have a hyper-cyclic point. Here, we have an important theorem that is: f is a hyper-cyclic function if and only if f is transitive. Definition 3.2 A function X Xf → : is called α r- homeomorphism if f is α -irresolute bijective and X X f → − : 1 is α -irresolute. Definition 3.3 Two topological systems X Xf → : , ) ( 1 n n xf x = + and Y Yg → : , ) ( 1 n n yg y = + are topologically r α -conjugate if there is α r- homeomorphism Y Xh → : such that hg f h   = x)). x)) = g(h( (i.e. h(f( We call h a topological r α -Conjugacy. Then I have proved some of the following statements: 1. The maps f and g have the same kind of dynamics. 2. If x is a periodic point of the map f with stable set )( xW f , then the stable set of h(x) is )). ( ( xWh f 3. The map f is α -exact if and only if g is α -exact 4. The map f is α -mixing if and only if g is α - mixing 5. The map f is α -type chaotic if and only if g is α -type chaotic 6. The map f is weakly α -mixing if and only if g isweakl y α - mixing. Remark 3.4 If .} . . , x, x {x 2 1 0, denotes an orbit of ) ( 1 n n xf x = + then ), h(x =y{ 0 0 ), h(x = y 1 1 }. . ), h(x = y 2 2 yields an. In particular, h maps periodic orbits of f onto periodic orbits of g . orbit of g since and g have the same kind of dynamics. I introduced and defined the new type of transitive in such a way that it is preserved under topologically α r- conjugation. Proposition 3.5 Let X and Y are α - separable and α - second category spaces. If X Xf → : Y Yg and → : are r α -conjugated by the α r-homeomorphism X Yh → : then, for each α -hyper-cyclic point y in Y if and only if h(y) is α -hyper-cyclic point in X Proof: Suppose that X Xf → : Y Yg and → : are maps − r α conjugate via X Yh → : such that h f gh   = , then if y ∈ Y is α -hyper-cyclic in Y i.e. the orbit ),.......} ( ), ( ,{ )( 2 y g ygy yO g = is α -dense in Y, let X V ⊂ be a nonempty α - open set. Then since h is a α r-homeomorphism, ) ( 1 V h − is α - open in Y , so there exists N ∈ n with ) ( )( 1 V h y g n − ∈ . From h f gh n n   = it follows that V yh f y gh n n ∈ = )) (( )) ( ( , So that } )),....... (( )), (( ), ({ )) (( 2 yh f yhf yh yhO f = is α - dense in X so h(y) is hyper-cyclic in X . Similarly, if )( yh is α - hyper-cyclic in X , then y is α -hyper-cyclic in Y. Proposition 3.6 if Y Yg and X Xf → → : : are r α -conjugate via Y Xh → : . Then (1) T is α -type transitive subset of X ⇔ )( Th is α - type transitive subset of Y; (2) X T ⊂ is α -mixing set ⇔ )( Th is α -mixing subset of Y. Proof (1) Assume that Y Yg and X Xf → → : : are topological systemswhich are topologically α r- conjugated by Y Xh → : . Thus, h is α r- homeomorphism (that is, h is bijective and thus invertible and both h and 1 − h are α -irresolute) and hg f h   = Suppose T is α -type transitive subset Global Journal of Researches in Engineering 44 Year 2019 © 2019 Global Journals New Types of Transitive Maps and Minimal Mappings ( ) Volume XIxX Issue II Version I J of X. Let A, B be α -open subsets of Y with φ ≠ ∩ )( Th B and φ ≠ ∩ )( Th A ) )) = g(y )) = g(h(x ) = h(f(x = h(x y n n n n+ n+ 1 1 , i.e. f