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

).0 ) ( ( > ≠∩ n some for B Ag show to n ϕ ) ( ) ( 1 1 Bh V and A hU − − = = are α –open subsets of X since h is an α -irresolute. Then there exists some n>0 such that ϕ ≠∩ V Uf n ) ( since the set T is α –type transitive subset of X, with φ ≠∩ TU and φ ≠∩ T V .Thus ). ( 1 1 1 1 n n g h h f implies g h h f as     − − − − = = ). ( )) ( ( ) ( )) ( ( 1 1 1 1 Bh Ag h Bh A h f n n − − − − ∩ = ∩ ≠ φ Therefore, . h since ) ( ) ) ( ( -1 1 invertible is B Ag implies B Ag h n n φ φ ≠∩ ≠ ∩ − So h(T) is α -type transitive subset of Y. Proof (2) We only prove that if T is topologically α - mixing subset of Y then )( 1 Th − is also topologically α -mixing subset of X. Let U,V be two α -open subsets of X with φ ≠ ∩ − )( 1 Th U and φ ≠ ∩ − )( 1 Th V . We have to show that there is N>0 such that for any n>N, . ) ( φ ≠∩ V Uf n ) ( 1 Uh − and ) ( 1 Vh − are two α -open sets since h is α -irresolute with φ ≠∩ − T Vh ) ( 1 and φ ≠∩ − T Uh ) ( 1 . If the set Tis topologically α -mixing then there is N >0 such that for any n>M, . ) ( )) ( ( 1 1 φ ≠ ∩ − − Vh Uhg n S ∃ ) ( )) ( ( 1 1 Vh Uhg x n − − ∩ ∈ . That is and Uhg x n )) ( ( 1 − ∈ ) ( 1 Vh x − ∈ ⇔ ) ( )( 1 Uh y for y g x n − ∈ = .h(x) ε V. Thus, since h f gh n n   = , so that )( ( )( y gh xh n = ) ( )) (( U f yh f n n ∈ = and we have V xh ∈ )( that is . ) ( φ ≠∩ V Uf n So, h -1 (T) is α -mixing set. Proposition 3.7 Let ) , ( fX be a topological system and A be a nonempty α -closed set of X. Then the following conditions are equivalent. 1. A is a α -transitive set of ) , ( fX . 2. Let V be a nonempty α -open subset of A and U be a nonempty α -open subset of X with φ ≠∩ A U . Then ther exists N ∈ n such that φ ≠ ∩ − ) ( U f V n . 3. Let U be a nonempty α -open set of X with φ ≠∩ A U . Then  N ∈ − n n U f ) ( is α -dense in A. Theorem 3.8 Let ) , ( fX be topological dynamical system and A be a nonempty α -closed invariant set of X. Then A is a α - transitive set of ) , ( fX if and only if ) , ( fA is α -type transitive system. Proof: ) ⇒ Let 1 1 Uand V be two nonempty α -open subsets of A. For a nonempty α -open subset 1 U of A, there exists a α - open set U of X such that A UU ∩= 1 Since A is a α -type transitive set of ) , ( fX , there exists n ∈ N such that . ) ( 1 φ ≠∩ U Vf Moreover, A is invariant, i.e., A Af ⊂ ) ( , which implies that A Af ⊂ ) ( Therefore, φ ≠∩∩ UA Vf ) ( 1 , i.e. φ ≠ ∩ 1 1 ) ( U Vf . These shows that ) , ( fA is α - type transitive. ) ⇐ Let 1 V be a nonempty α -open set of A and U be a nonempty α -open set of X with , φ ≠∩ UA Since U is an α -open set of X and , φ ≠∩ UA , it follows that U ∩ A is a nonempty α -open set of A. Since ) , ( fA is topologically α -type transitive, there existsn ∈ N such that , ) ( ) ( 1 φ ≠ ∩∩ UA Vf which implies that . ) ( 1 φ ≠∩ U Vf . This shows that A is a α -type transitive set of ). , ( fX IV. N ew T ypes of C haos in P roduct S paces We will give a new definition of chaos for δ - irresolute self map X X f → : of a compact Hausdorff topological space X, so called δ -type chaos. This new definition induces from John Tylar definition which coincides with Devaney's definition for chaos when the topological space happens to be a metric space. Definition 4.1 [4] Let ) , ( fX be a topological dynamical system; the dynamics is obtained by iterating the map. Then, f is said to be δ -type chaotic on X provided that for any nonempty δ -open sets U and V in X, there is a periodic point X p ∈ such that φ ≠ ∩ ) ( pOU f and φ ≠ ∩ ) ( pOV f . Proposition 4.2 Let ) , ( fX be a topological dynamical system. The map f is δ -type chaotic on X if and only if f © 2019 Global Journals Global Journal of Researches in Engineering ersion I 45 Year 2019 New Types of Transitive Maps and Minimal Mappings ( ) Volume XIxX Issue II V I J is δ -type transitive and the periodic points of the map are δ -dense in X . Proof: ) ⇒ If f is δ -type chaotic on X, then for every pair of nonempty δ -open sets U and V, there is a