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\title{New Types of Transitive Maps and Minimal Mappings}
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\begin{document}

             \author[1]{Mohammed Nokhas Murad  Kaki}

             \affil[1]{  University of Sulaimani}

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\date{\small \em Received: 16 December 2018 Accepted: 2 January 2019 Published: 15 January 2019}

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\begin{abstract}
        


In this paper, we have introduced the relationship between two different concepts of maps, namely topological   transitive and  transitive maps and investigate some of their properties in two topological  spaces   and  ,   denotes the  topology and   denotes the  topology of a given topological space  The two concepts are defined by using the concepts of  irresolute and   irresolute maps respectively  Also, we studied  the relationship between  two types of minimal systems, namely,  - minimal and  minimal systems, The main results are the following propositions: XThe two concepts are defined by using the concepts of ? ? irresolute and ? ? irresolute maps respectively Also, we studied the relationship between two types of minimal systems, namely, ? -minimal and ? ? minimal systems, The main results are the following propositions.

\end{abstract}


\keywords{opologically   transitive, - irresolute, - transitive, dense.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
d exists an integer N, such that, for all N n > , one has ? ? ?V U f n ) ( ,topologically ? -mixing if for any nonempty ? -open set U, there exists such that? N n n U f ? )\par
( is ? -dense in X. With the above concepts, some new theorems have been introduced and studied. Furthermore, we have the following results:\par
? Every topologically ? ? transitive map implies topologically ? ? transitive map, but the converse not necessarily true.\par
? Every ? ? minimal system implies ? ? minimal system, but the converse not necessarily true.? ) ( ) ( ? ? ET E ? ; ? ) ( ) ( ) ( ? ? ? TT WM TM ? ? ; II.\par
Preliminaries and TheoremsDefinition 3.1 [2] A map Y X f ? : is called ?-irresolute if for every ?-open set H of Y, ) ( 1 H f ? is ?- open in X.\par
Proposition 2.2 The product of two topologically ?mixing systems must be topologically ? -mixing.\par
Proof: Suppose that Y V V ? ' , so that W V U ? × and '. ' ' W V U ? × By definition of topological ? -mixing of ), , ( f X there exists N such that for any , N n > . ) ( ? ? ?V U f n By definition of topological ? -mixing[3] of\par
), , ( g Y there exists N? such that for any, ' N n > . ' ) ' ( ? ? ?V U g n Then, for any ), ' , max( N N n > both V U f n ? ) ( and ' ) ' ( V U g n ? ?\par
are nonempty, and therefore) ' ( ) ' ( ) ( V V U U g f n × ? × × is nonempty as well. But this implies that ? ? ? × ' ) ( ) ( W W g f n\par
, since W and W' were arbitrary, this implies that be ? -irresolute map, then the set Keywords:) , ( g f Y X × × is topologically ? -mixing.X A ? is called topologically ? -mixing set[1] if, given any nonempty ? -open subsets X V U ? , with ? ? ?U A and ? ? ?V A then 0 > ? N such that ? ? ?V U f n ) ( for all N n > , weakly ? -mixing set[4] of ) , ( f X if for any choice of nonempty ? -open subsets 2 1 ,V V of A and nonempty ? -opensubsets 2 1 , U U of X with ? ? ? 1 U A and ? ? ? 2 U A there exists n ?N such that ? ? ? 1 1 ) ( U V f n and ? ? ? 2 1 ) ( U V f n , strongly ? - mixing if for any pair of open sets V and U with , A V and A U ? ? ? ? ? ? there exist some N ? n such that ? ? ? V U f k ) ( for any n k ? . A point x which has? -dense orbit ) (x O d in X.\par
Theorem 2.3 The product of two ? -transitive maps is not necessarily ? -transitive map [4]. Corollary 2.4 The product of two topologically ?transitive systems is not necessarily topologically ?transitive. 
\section[{III. New Types of Chaos of Topological Spaces}]{III. New Types of Chaos of Topological Spaces}\par
In this section, I introduced and defined ? -type transitive maps[3] and ? -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. Definition 3.1 Let X is a separable and second category space with no isolated point, if forX 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\par
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 X f ? : is called ? r- homeomorphism if f is ? -irresolute bijective and X X f ? ? : 1 is ? -irresolute. Definition 3.3 Two topological systems X X f ? : , ) ( 1 n n x f x = + and Y Y g ? : , ) ( 1 n n y g y = + are topologically r ? -conjugate if there is ? r- homeomorphism Y X h ? : such that h g f h ? ? = x)). x) 
\section[{) = g(h( (i.e. h(f(}]{) = g(h( (i.e. h(f(}\par
We call h a topological r ? -Conjugacy. Then I have proved some of the following statements:\par
1. The maps f and g have the same kind of dynamics.\par
2. If x is a periodic point of the map f with stable set ) (x W f , then the stable set of h(x) is )). ( ( x W h f 3. The map f is ? -exact if and only if g is ? -exact 4. The map f is ? -mixing if) ( 1 n n x f x = + then ), h(x = y \{ 0 0 ), h(x = y 1 1 \} . . ), h(x = y 2 2\par
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.\par
I introduced and defined the new type of transitive in such a way that it is preserved under topologically ? r-conjugation. Proposition 3.5 Let X and Y are ? -separable and ?second category spaces. IfX X f ? : Y Y g and ? : are r ? -conjugated by the ? r-homeomorphism X Y h ? : then, for each ? -hyper-cyclic point y in Y if and only if h(y) is ? -hyper-cyclic point in X Proof: Suppose that X X f ? : Y Y g and ? : are maps ? r ? conjugate via X Y h ? : such that h f g h ? ? = , then if y?Y is ? -hyper-cyclic in Y i.e. the orbit ),.......\} ( ), ( , \{ ) ( 2 y g y g y y O g = is ? -dense in Y, let X V ? be a nonempty ? -open set. Then since h is a? r-homeomorphism, ) ( 1 V h ? is ? -open in Y, so there exists N ? n with ) ( ) ( 1 V h y g n ? ? . From h f g h n n ? ? = it follows that V y h f y g h n n ? = )) ( ( )) ( ( , So that \} )),....... ( ( )), ( ( ), ( \{ )) ( ( 2 y h f y h f y h y h O f = is ? -dense in X so h(y) is hyper-cyclic in X . Similarly, if ) ( y h is ? -hyper-cyclic in X, then y is? -hyper-cyclic in Y. Proposition 3.6 if Y Y g and X X f ? ? : : are r ? -conjugate via Y X h ? : . Then (1) T is ? -type transitive subset of X â??" ) (T h is ? - type transitive subset of Y; (2) X T ? is? -mixing set â??" ) (T h is ? -mixing subset of Y. Proof (1) Assume that Y Y g and X X f ? ? : : are topological systemswhich are topologically ? r- conjugated by Y X h ? :\par
. Thus, h is ? rhomeomorphism (that is, h is bijective and thus invertible and both h and ? ? ) (T h B and ? ? ? ) (T h A ) )) = g(y )) = g(h(x ) = h(f(x = h(x y n n n n+ n+ 1 1 , i.e. f ). 0 ) ( ( > ? ? n some for B A g show to n ? ) ( ) ( 1 1 B h V and A h U ? ? = =\par
are? -open subsets of X since h is an ? -irresolute. Then there exists some n>0 such that? ? ?V U f n ) ( since the set T is ? -type transitive subset of X, with ? ? ? T U and ? ? ? T V .Thus ). ( 1 1 1 1 n n g h h f implies g h h f as ? ? ? ? ? ? ? ? = = ). ( )) ( ( ) ( )) ( ( 1 1 1 1 B h A g h B h A h f n n ? ? ? ? ? = ? ? ? Therefore, . h since ) ( ) ) ( ( -1 1 invertible is B A g implies B A g h n n ? ? ? ? ? ? ? So h(T) is ? -type transitive subset of Y. Proof (2) We only prove that if T is topologically ? - mixing subset of Y then ) ( 1 T h ? is also topologically ? -mixing subset of X. Let U,V be two ? -open subsets of X with ? ? ? ? ) ( 1 T h U and ? ? ? ? ) ( 1 T h V\par
. We have to show that there is N>0 such that for any n>N,. ) ( ? ? ?V U f n ) ( 1 U h ? and ) ( 1 V h ? are two ? -open sets since h is ? -irresolute with ? ? ? ? T V h ) ( 1 and ? ? ? ? T U h ) ( 1 .\par
If the set Tis topologically ? -mixing then there is N >0 such that for any n>M, .? ? ? ? V h U h g n S ? ) ( )) ( ( 1 1 V h U h g x n ? ? ? ? . That is and U h g x n )) ( ( 1 ? ? ) ( 1 V h x ? ? â??" ) ( ) ( 1 U h y for y g x n ? ? = .h(x) ? V. Thus, since h f g h n n ? ? = , so that ) ( ( ) ( y g h x h n = ) ( )) ( ( U f y h f n n ? = and we have V x h ? ) ( that is . ) ( ? ? ?V U f n So, h -1 (T) is ? -mixing set.\textbf{) ( )) ( ( 1 1 ?}\par
Proposition 3.7 Let ) , ( f X be a topological system and A be a nonempty ? -closed set of X. Then the following conditions are equivalent.\par
1. A is a ? -transitive set of ) , ( f X .\par
2. Let V be a nonempty ? -open subset of A and U be a nonempty ? -open subset of X with ? ? ? A U . Then ther existsN ? n such that ? ? ? ? ) (U f V n . 3. Let U be a nonempty ? -open set of X with ? ? ? A U . Then ? N ? ? n n U f ) Year 2019\par
New Types of Transitive Maps and Minimal Mappings ( ) Volume XIx X Issue II Version I J is ? -type transitive and the periodic points of the map are? -dense in X .\par
Proof:\par
) ? If f is ? -type chaotic on X, then for every pair of nonempty ? -open sets U and V, there is a periodic orbit intersects them; in particular, the periodic points are ? -dense in X. Then there is a periodic point y x f n = ) ( , so that ) ( ) ( U f x f y n n ? = therefore ? ? ?V U f n ) ( . :)\par
? The ? -type transitivity [5]of f on X implies, for any nonempty ? -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 ? -open) ( ? TT if for every non-empty ? -open set X D ? , ? ? =1 ) ( n n D f is ? -dense, 2. Weak ? -Mixing ) ( ? WM if f f × is topologically ? -transitive .     \begin{figure}[htbp]
\noindent\textbf{} \par 
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\tabcellsep \multicolumn{12}{l}{isweakl y? -mixing.}\\
\multicolumn{7}{l}{Remark 3.4}\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
If\tabcellsep \{x\tabcellsep 0,\tabcellsep x\tabcellsep 1\tabcellsep ,\tabcellsep x\tabcellsep 2\tabcellsep ,\tabcellsep .\tabcellsep .\tabcellsep .\}\tabcellsep denotes an orbit of\end{longtable} \par
 
\caption{\label{tab_1}}\end{figure}
 			\footnote{© 2019 Global Journals} 		 		\backmatter  			 \par
( is ? -dense in A. Theorem 3.8 Let ) , ( f X be topological dynamical system and A be a nonempty ? -closed invariant set of X. Then A is a ? -transitive set of\par
. These shows that ) , ( f A is ? -type transitive.\par
. This shows that A is a ? -type transitive set of ). , ( f X IV. New Types of Chaos in Product Spaces\par
We will give a new definition of chaos for ?irresolute self map X X f ? : of a compact Hausdorff topological space X, so called ? -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 ) , ( f X be a topological dynamical system; the dynamics is obtained by iterating the map. Then, f is said to be ? -type chaotic on X provided that for any nonempty ? -open sets U and V in X, there is a periodic point\par
) , ( f X be a topological dynamical system. The map f is ? -type chaotic on X if and only if f			 			  				\begin{bibitemlist}{1}
\bibitem[Kaki et al. ()]{b0}\label{b0} 	 		 			Mohammed Nokhas Murad Kaki 		,  		 			Introduction To 		,  		 			Dynamical Systems I 		.  		\textit{Publisher SciencePG},  				 (Book; New York, USA)  		2015.  	 
\end{bibitemlist}
 			 		 	 
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