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

New Types of Transitive Maps and Minimal Mappings Mohammed Nokhas Murad Kaki Abstract- In this paper, we have introduced the relationship between two different concepts of maps, namely topological − α transitive and − δ transitive maps and investigate some of their properties in two topological spaces ) , ( α τ X and ) , ( δ τ X , α τ denotes the − α topology and δ τ denotes the − δ topology of a given topological space ). , ( τ X The two concepts are defined by using the concepts of − α irresolute and − δ irresolute maps respectively Also, we studied the relationship between two types of minimal systems, namely, α - minimal and − δ minimal systems, The main results are the following propositions: I. I ntroduction d exists an integer N , such that, for all Nn > , one has φ ≠∩ V Uf n ) ( , topologically α -mixing if for any non- empty α - open set U, there exists such that  Nn n U f ≥ ) ( is α -dense in X. With the above concepts, some new theorems have been introduced and studied. Furthermore, we have the following results: • Every topologically − α transitive map implies topologically − δ transitive map, but the converse not necessarily true. • Every − α minimal system implies − δ minimal system, but the converse not necessarily true. • ) ( ) ( α α ET E ⇒ ; • ) ( ) ( ) ( α α α TT WM TM ⇒ ⇒ ; II. P reliminaries and T heorems Definition 3.1 [2] A map Y X f → : is called α -irresolute if for every α -open set H of Y, ) ( 1 H f − is α - open in X. Proposition 2.2 The product of two topologically α - mixing systems must be topologically α - mixing. Proof: Suppose that ) , ( ) , ( gY and fX are two α - mixing systems, and consider any α -open sets ' , WW in YX × . By definition of the product topology, there exist α -open sets X UU ⊂ ' , ′ and Y VV ⊂ ' , so that WVU ⊂× and '. ' ' WVU ⊂× By definition of topological α -mixing of ), , ( fX there exists N such that for any , Nn > . ) ( φ ≠∩ V Uf n By definition of topological α -mixing[3] of ), , ( gY there exists N ′ such that for any ,' Nn > . ' )' ( φ ≠∩ V Ug n Then, for any ), ' , max( NN n > both V Uf n ∩ ) ( and ' )' ( V Ug n ∩ ′ are nonempty, and therefore )' ( )' () ( VV UUg f n ×∩ × × is nonempty as well. But this implies that φ ≠ ∩ × ' ) () ( WWg f n , since W and W’ were arbitrary, this implies that ) , ( g fYX × × is topologically α - mixing. Ν∈ N © 2019 Global Journals Global Journal of Researches in Engineering ersion I 43 Year 2019 ( ) Volume XIxX Issue II V J Author: College of Science University of Sulaimani. e-mail: Mohammed.murad@univsul.edu.iq et ) , ( τ X be a topological space, X X f → : be α -irresolute map, then the set X A ⊆ is called topologically α -mixing set[1] if, given any nonempty α -open subsets X VU ⊆ , with φ ≠∩ UA and φ ≠∩ VA then 0 > ∃ N such that φ ≠∩ V Uf n ) ( for all Nn > , weakly α - mixing set[4] of ) , ( fX if for any choice of nonempty α -open subsets 2 1 , VV of A and nonempty α –opensubsets 2 1 , UU of X with φ ≠ ∩ 1 UA and φ ≠ ∩ 2 UA there exists n ∈ N such that φ ≠ ∩ 1 1 ) ( U V f n and φ ≠ ∩ 2 1 ) ( U V f n , strongly α - mixing if for any pair of open sets V andU with , AV and AU φ φ ≠∩ ≠∩ there exist some N ∈ n such that φ ≠∩ V Uf k ) ( for any n k ≥ . A point x which has α -dense orbit )( xO d in X . is called type − α hyper-cyclic point.A system is α - mixing[1] if, given α -open sets U and V in X, there L 1. Every topologically − α transitive map implies topologically − δ transitive map, but the converse not necessarily true. 2. Every − α minimal system implies − δ minimal system, but the converse not necessarily true . topologically − δ transitive, α - irres olute, δ - transitive, − δ dense. Keywords: