periodic orbit intersects them; in particular, the periodic points are δ -dense in X. Then there is a periodic point p and ) ( , pOyx f ∈ with x ∈ U and y ∈ V and some positive integer n such that y x f n = )( , so that ) ( )( Uf x f y n n ∈ = therefore φ ≠∩ V Uf n ) ( . :) ⇐ The δ -type transitivity [5]of f on X implies, for any nonempty δ -open subsets U, V ⊂ X, there is n such that for some x ∈ U, V x f n ∈ )( . Now,define U V f W n ∩ = − ) ( . Then W is δ -open and nonempty with the property that V Wf n ⊂ ) ( . We will define some concepts as follows: R eferences R éférences R eferencias 1. Mohammed Nokhas Murad Kaki, INTRODUCTION TO TOPOLOGICAL DYNAMICAL SYSTEMS I , Book with ISBN 978-1-840366-52-4, (2015), Publisher SciencePG, New York, USA. Global Journal of Researches in Engineering 46 Year 2019 © 2019 Global Journals New Types of Transitive Maps and Minimal Mappings ( ) Volume XIxX Issue II Version I J But since the periodic points of f are δ -dense in X, there is a p ∈ W such that V p f n ∈ ) ( . Therefore, φ ≠ ∩ ) ( pOU f and φ ≠ ∩ ) ( pOV f . So, the map f is δ -type chaotic. 2. Maheshwari N. S., and Thakur S. S., On α -irresolute mappings, Tamkang J. Math. Vol. 11,(1980), pp. 209- 214. 3. Mohammed Nokhas Murad Kaki ,Topologically α - Transitive Maps and Minimal Systems Gen. Math. Notes, Vol. 10, No. 2, (2012), pp. 43-53 1. ) ( δ TT if for every non-empty δ -open set XD ⊂ , ∞ = 1 ) ( n n D f is δ -dense, 2. Weak δ -Mixing ) ( δ WM if f f × is topologically δ -transitive . 3. Exact δ -Transitive ) ( δ ET if for every pair of non- empty δ -open set XWD ⊂ , , X in dense isWf D f n n n − ∩ ∞ = δ 1 ) ( ) ( ( , 4. Topologically δ - Mixing ) ( δ TM if for every pair of non-empty δ -open set XWD ⊂ , , there exits an N ∈ N such that φ ≠∩ WD f n ) ( for all . Nn ≥ 5. δ - Exact ) ( δ E if for every non-empty δ -open set XD ⊂ , there exists N ∈ N such that X D f N = ) ( 6. Then the following implications hold: • ) ( ) ( α α ET E ⇒ ; • ) ( ) ( ) ( α α α TT WM TM ⇒ ⇒ ; 4. Mohammed Nokhas Murad Kaki, INTRODUCTION TO TOPOLOGICAL DYNAMICAL SYSTEMS II;Book with ISBN: 978-3-659-80680-3. Publisher: Lambert academic publisher / Germany 5. Mohammed Nokhas Murad Kaki, Chaos, Mixing, Weakly Mixing and Exactness, Oalib Journal, China (2018) , Vol. 5, No. 6, pp.1-6 . 6. Mohammed Nokhas Murad Kaki, New Types of δ - Transitive Maps, International Journal of Engineering & Technology IJET-IJENS No.06 (2013), pp. 134- 136

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