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\title{Relationship between New Types of Transitive and Chaotic Maps}
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             \author[1]{Mohammed Nokhas Murad  Kaki}

             \affil[1]{  University of Sulaimani Faculty of Science and Science Education, School of Science, Math Department}

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\date{\small \em Received: 16 December 2013 Accepted: 4 January 2014 Published: 15 January 2014}

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\begin{abstract}
        


The concepts of topological ?-transitive maps, ?-transitive maps, ?-type chaotic and ?-type chaotic maps were introduced by M. Nokhas Murad Kaki. In this paper, I study the relationship between two different notions of transitive maps, namely topological ?-transitive maps, topological ?-transitive maps and investigate some of their properties in two topological spaces (X, ??) and (X, ??), ?? denotes the ?-topology(resp. ?? denotes the ?-topology) of a given topological space (X, ? ).. The two notions are defined by using the concepts of ?-irresolute map and ?-irresolute map respectively Also, we study the relationship between two new types of chaotic maps, namely, ?-type chaotic maps and ?-chaotic maps, and I will prove that the properties of ?-transitive, ?-type chaotic are preserved under ?r-conjugacy and ?-transitive, ?-chaotic maps are preserved under ?r-conjugacy The main results are the following propositions:1) Every topologically ?-type transitive map is a topologically ?-type transitive map which implies topologically transitive map, but the converse not necessarily true..2) Every ?-type chaotic map is ?-chaotic map which implies chaotic map in topological spaces, but the converse not necessarily true..

\end{abstract}


\keywords{topologically ?- transitive, - type chaotic, ?-type chaotic, ?-dense.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
ecently there has been some interest in the notion of a locally closed subset of a topological space. According to Bourbaki \hyperref[b15]{[16]} a subset S of a space (X, ) is called locally closed if it is the intersection of an open set and a closed set. Ganster and Reilly used locally closed sets in \hyperref[b13]{[13]} and \hyperref[b14]{[14]} to define the concept of LC-continuity, i.e. a function f : (X, ) (X, ) is LC-continuous if the inverse with respect to f of any open set in Y is closed in X. The study of semi open sets and semi continuity in topological spaces was initiated by Levine \hyperref[b5]{[6]}. Bhattacharya and Lahiri \hyperref[b7]{[8]} introduced the concept of semi generalized closed sets in topological spaces analogous to generalized closed sets which was introduced by Levine \hyperref[b4]{[5]}. Throughout this paper, the word "space " will mean topological space The collections of semi-open, semi-closed sets and ?-sets in ( X, ) will be denoted by SO (X, ) , SC (X, ) and respectively. Njastad \hyperref[b6]{[7]} has shown that is a topology on X with the following properties: and if and only if N where and N is nowhere dense in (X, ). Hence if and only if every nowhere dense (nwd) set in (X, ) is closed, therefore every transitive map implies ?-transitive. Also if every ?-open set is locally closed then every transitive map implies ?transitive; and this structure also occurs if ( X, ) is locally compact and Hausdorff  {\ref [36,} p. 140, Ex. B]] and every ?-open set is locally compact, then every ?-open set is locally closed.\par
In 1943, Fomin \hyperref[b27]{[27]} introduced the notion of ?continuous maps. The notions of ?-open sets, ?-closed sets and ?-closure where introduced by Veli?cko \hyperref[b18]{[19]} for the purpose of studying the important class of H-closed spaces in terms of arbitrary fiber-bases. Dickman and Porter \hyperref[b19]{[20]}, \hyperref[b20]{[21]}, Joseph \hyperref[b22]{[22]} and Long and Herrington \hyperref[b31]{[31]} continued the work of Velic ?ko. We introduce the notions of ?-type transitive maps, ?-minimal maps and show that some of their properties are analogous to those for topologically transitive maps. Also, we give some additional properties of ?-irresolute maps. We denote the interior and the closure of a subset A of X by Int(A) and Cl(A), respectively. By a space X, we mean a topological space (X, ) A point x ? X is called a ?-adherent point of A \hyperref[b18]{[19]}, if for every open set V containing x. The set of all ?-adherent points of a subset A of X is called the ?-closure of A and is denoted by . A subset A of X is called -closed if . Dontchev and Maki \hyperref[b22]{[22]} have shown that if A and B are subsets of a space (X, ), then also Note also that the ?-closure of a given set need not be a -closed set. But it is always closed. Dickman and Porter \hyperref[b19]{[20]} proved that a compact subspace of a Hausdorff space is ?-closed. Moreover, they showed that a ?-closed subspace of a Hausdorff space is closed. Jankovi´ \hyperref[b25]{[25]} proved that a space (X, ) is Hausdorff if and only if every compact set is ?-closed. The complement of a ?-closed set is called a ?-open set. The family of all ?-open sets forms a topology on X and is denoted by or topology.\par
This topology is coarser than and that a space (X, ) is regular if and only if \hyperref[b26]{[26]}. Then we observe that every theta-type transitive maps is transitive if (X, ) is regular. In general, will not be the closure of A with respect to (X , ). It is easily seen that one always has ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? , ? ? ? ? ? ? ) ( ? ? ? S U S \textbackslash  ? ? ? U ) )) ( ( . . ( ? ? N Cl Int e i ? ? ? ? ? ? ? ? ? ? ) (V Cl A ) (A Cl ? ) (A Cl A ? ? ? ? (B) Cl (A) Cl = B) (A Cl ? ? ? ? ? (B) Cl (A) Cl = B) (A Cl ? ? ? ? ? . ? ? ? ? ? ? ? ? ? ? ? ? ) (A Cl ? ? ? ? ? ? A A Cl A Cl A Cl A ? ? ? ? ) ( ) ( ) (\par
where denotes the closure of A with respect to (X, ).\par
It is also obvious that a set A is ?-closed in (X, ) if and only if it is closed in (X, ). The space (X, ) is called sometimes the semi regularization of (X, ). A function is closure continuous \hyperref[b29]{[29]} (? continuous) at x ? X if given any open set V in Y containing f(x), there exists an open set U in X containing x such that . \hyperref[b29]{[29]} In this paper, we will study the relationship between new classes of topological transitive maps called type transitive and -type transitive, also, new classes of -type chaotic maps and -type chaotic maps. We have shown that every ?-type transitive map is a ?-type transitive map, but the converse not necessarily true and that every ?-type chaotic map is ?-type chaotic map, but the converse not necessarily true we will also study some of their properties. 
\section[{II.}]{II.} 
\section[{Preliminaries and Definitions}]{Preliminaries and Definitions}\par
In this section, we recall some of the basic definitions. Let X be a space and A X . The intersection (resp. closure) of A is denoted by Int(A) (resp. Cl(A).\par
? Definition 2.1 \hyperref[b5]{[6]} A subset A of a topological space X will be termed semi-open (written S.O.) if and only if there exists an open set U such that .\par
? Definition 2.2 \hyperref[b7]{[8]} Let A be a subset of a space X then semi closure of A defined as the intersection of all semi-closed sets containing A is denoted by sClA.\par
? Definition 2.3 \hyperref[b8]{[9]} Let (X, ) be a topological space and ? an operator from to ?(X) i.e ?: ? ? ?(X), where ?(X) is a power set of X. We say that ? is an operator associated with if for all\par
? Definition 2.4 \hyperref[b9]{[10]} Let (X, ) be a topological space and ? an operator associated with ?. A subset A of X is said to be ?-open if for each x Ñ?" X there exists an open set U containing x such that . Let us denote the collection of all ?-open, semi-open sets in the topological space ( ) by , SO( ), respectively. We then haveSO . A subset B of X is said to be ?-closed [7] if its complement is ?-open.\par
? Definition 2.5 \hyperref[b8]{[9]} Let (X, ) be a space. An operator ? is said to be regular if, for every open neighborhoods U and V of each x Ñ?" X, there exists a neighborhood W of x such that Note that the family of ? -open sets in (X, ) always forms a topology on X, when ? is considered to be regular finer than .\par
? Theorem 2.6 \hyperref[b30]{[30]} For subsets A, B of a space X, the following statements hold: \hyperref[b40]{(1)} , where D(A) is the derived set of A (2) If , then\par
Note that the family of ? -open sets in (X, ) always forms a topology on X denoted ?-topology and that ?-topology coarser than .\par
? Definition 2.7 \hyperref[b3]{[4]}: Let A be a subset of a space X. A point x is said to be an -limit point of A if for each -open U containing x, . The set of all -limit points of A is called the -derived set of A and is denoted by \hyperref[b3]{[4]} For subsets A and B of a space X, the following statements hold true: 1) where is the derived set of A 2) if then 3) 4)? Definition 2.8\par
? Definition 2.9 \hyperref[b9]{[10]}: The point x Ñ?" X is in the -closure of a set A X if ?(U) ? A??, for each open set U containing x. The -closure of a set A is the intersection of all -closed sets containing A and is denoted by Cl (A) . ? Remark 2.10: For any subset A of the space X,\par
? Definition 2.11 \hyperref[b9]{[10]} Let (X, ) be a topological space. We say that a subset A of X is -compact if for every -open covering ? of A there exists a finite subcollection of ? such that I Properties of ? -compact spaces have been investigated by Rosa, E etc. and Kasahara, S \hyperref[b8]{[9,}\hyperref[b9]{10]}.The following results were given by Rosas, E etc. \hyperref[b8]{[9]}. 
\section[{?}]{?}\par
Let (X, ) be a topological space and ? an operator associated with ?. A X and K A. If A is ?-compact and K is ? -closed then K is ? -compact.\par
? Theorem2.13 Let (X, ) be a topological space and ? be a regular operator on ?. If X is ? -2 T (see Rosa, E etc. and Kasahara, S) \hyperref[b8]{[9,}\hyperref[b9]{10]} and K X is ? -compact then K is ? -closed.\par
? Definition 2.14 \hyperref[b9]{[10]} The intersection of all ? -closed sets containing A is called ? -closure of A, denoted by\par
? Remark 2.15 For any subset A of the space X, A ? Lemma2. \hyperref[b15]{16} For subsets A and of a space (X, ) , the following hold:1) A 2) closed; 3) If A B then 4) 
\section[{5)}]{5)}\par
? Lemma 2.17 The collection of ? -compact subsets of X is closed under finite unions. If ? is a regular operator and X is an ?-2 T space then it is closed under arbitrary intersection.\par
Global Journal of Researches in Engineering ( )F Volume XIV Issue V Version I 26 Year 2014 © 2014 Global Journals Inc. (US) ? A ? ? ? ? ? ? ? ? f : Y X? ) ( )) ( ( V Cl U Cl f ? ? ? ? ? ? ) (U Cl A U ? ? ? ? ? U ?a (U) U Ñ?" ? ? ? a(U) A ,? X ? ? ? ) (? ? ? ? ? ? ? ? ?(U) a(W) a(V).. ? ? ? ? ) ( ) ( A D A D ? ? B A ? ) ( ) ( B D A D ? ? ? . ) ( ) ( ) ( B A D B D A D ? ? ? ? ? ? ) ( ) ( ) ( B D A D B A D ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ) \textbackslash  ( x A U ? ? ) (A D ? . ) ( ) ( A D A D ? ? D(A) B A ? ) ( ) ( B D A D ? ? ? ) ( ) ( ) ( B A D B D A D ? ? ? ? ? ? ) ( )) ( ( A D A A D A D ? ? ? ? ? ? ? ? ? ? ? ) ( ) ( A Cl A Cl A ? ? ? ? ? ? \} ..., , 2 , 1 \{ n C C C i n i C A 1 ? ? ? . ? ? ? ? ? ) (A Cl ? . ? Cl(A) ? ) (A Cl ? . i A (i Ñ?" I) ? ? ) (A Cl ? ) (A Cl ? ) ( )) ( ( A Cl A Cl Cl ? ? ? ? ? ) (A Cl ? ? ) (B Cl ? ) : ) ( ( )) : ( ( I i A Cl I i A Cl i ? ? ? ? ? ? ? ) : ) ( ( )) : ( ( I i A Cl I i A Cl i ? ? ? ? ? ? ?\par
Relationship between New Types of Transitive and Chaotic Maps Theorem2.12 ? Theorem 2.21 \hyperref[b3]{[4]}: For any subset A of a space X,\par
? Theorem 2.22. \hyperref[b3]{[4]}: For subsets A, B of a space X, the following statements are true:1) int (A ) is the largest ? -open contained in A 2) 3) If 4) 5)\par
? Lemma 2.23 \hyperref[b6]{[7]} For any ? -open set A and any ?-closed set C, we have 1) 
\section[{2)}]{2)}\par
3) ? Theorem 2.26 Let (X, f )and (Y, g) be two topological systems, if and are topologically ?r-conjugate. Then (1) f is topologically ?-transitive map if and only if g is topologically ?-transitive map;? Remark 2.\par
(2) f is ?-type chaotic map if and only if g is ?-type chaotic map;\par
(3) f is ?-type chaotic map if and only if g is ?-type chaotic map. 
\section[{III.}]{III.} 
\section[{Transitive and Minimal Systems}]{Transitive and Minimal Systems}\par
Topological transitivity is a global characteristic of dynamical systems. By a dynamical system ( , )  {\ref [15]} we mean a topological space X together with a continuous map . The space X is sometimes called the phase space of the system. A set is called inveriant if . A topological system (X, f ) is called minimal if X does not contain any nonempty, proper, closed inveriant subset. In such a case we also say that the map itself is minimal. Thus, one cannot simplify the study of the dynamics of a minimal system by finding its nontrivial closed subsystems and studying first the dynamics restricted to them.\par
Given a point x in denotes its orbit (by an orbit we mean a forward orbit even if is a homeomorphism) and denotes its ? -limit set, i.e. the set of limit points of the sequence .\par
The following conditions are equivalent:? is ?-minimal (resp. ?-minimal),\par
? every orbit is ?-dense (resp. ?-dense) in X ,\par
? for every A minimal map is necessarily surjective if X is assumed to be Hausdorff and compact.\par
Now, we will study the Existence of minimal sets. Given a dynamical system , a set is called a minimal set if it is non-empty, closed and invariant and if no proper subset of A has these three properties. So, is a mi nimal set if and only if is a minimal system. A system is minimal if and only if X is a minimal set in . The basic fact discovered by G. D. Birkhoff is that in any compact system there are minimal sets. This follows immediately from the Zorn's lemma.\par
Since any orbit closure is invariant, we get that any compact orbit closure contains a minimal set. This is how compact minimal sets may appear in non-compact spaces. Two minimal sets in either are disjoint or coincide. A minimal set A is strongly inveriant , i.e.\par
Provided it is compact Hausdorff. Let be a topological system, and ?r-homeomorphism of X onto itself. For A and B subsets of X, we let\par
We write for a singleton thus For a point we write for the orbit of x and for the ?-closure of . We say that the topological system is ?type point transitive if there is a point with ?dense. Such a point is called ?-type transitive. We say that the topological systems is topologically ?-type transitive (or just ?-type transitive) if the set is nonempty for every pair U and V of nonempty ?-open subsets of X.\par
? Theorem 2.8 [37] Let be a topological system where X is a non-empty locally ?-compact Hausdorff topological space and is ?-irresolute map and that X is ?-type separable. Suppose that f is topologically ?-type transitive. Then there is an element such that the orbit is ?-dense in X. 
\section[{a) Topologically ?-Transitive Maps}]{a) Topologically ?-Transitive Maps}\par
In \hyperref[b37]{[35]}, we introduced and defined a new class of transitive maps that are called topologically ?transitive maps on a topological space (X, ), and we studied some of their properties and proved some Global Journal of Researches in Engineering ( )F Volume XIV Issue V Version I 27 Year 2014 © 2014 Global Journals Inc. (US) ? U U A : \{ ) ( ? ? ? A U ? \}. A A ? ) ( ? . ? ? Y X f ? : ) ( 1 H f ? ) ( ) ( A Cl A A Cl ? ? ? ? ? ) ( int )) ( (int int A A ? ? ? ? ? B then ) ( int ) ( int B A ? ? ? A ) ( int ) ( int ) ( int B A B A ? ? ? ? ? ? ) ( int ) ( int ) ( int B A B A ? ? ? ? ? ? ) ( ) ( A Cl A Cl ? ? ) int( ) ( int C C ? ? int A Cl ? ? ? )) ( int( A Cl ? =\{ d\}, X\}.\par
? ,\{c, ? ,\{c, d\},\{b, c, d\},? \{a, c, d\},X\} X X f ? : Y Y g ? : f X X X f ? : X A ? f ? A A f ? ) ( f ? f X, ...\} ), ( ), ( , \{ ) ( 2 x f x f x x O f ? ) (x f ? f ... ), ( ), ( , 2 x f x f x ) , ( f X X x f ? ) ( ? x ? X. . f ) , ( f X X A ? X A ? ) , ( A f A ) , ( f X ) , ( f X ) , ( f X ) , ( f X f ? A A f ? ) ( . ) , ( f X X X f ? : \} ) ( : \{ ) , ( ? ? ? ? ? B A f n B A N n Z ) , ( ) , ( B x N B A N ? \} \{x A ? \} ) ( : \{ ) , ( B x f n B x N n ? ? ? Z X x? \} : ) ( \{ ) ( Z ? ? n x f x O n f )) ( ( x O Cl f ? ) (x O f ) , ( f X X x ? ) (x O f ) , ( f X ) , ( V U N ) , ( f X X X f ? : X x ? \} : ) ( \{ ) ( N ? ? n x f x O n f\par
? results associated with these new definitions. We also defined and introduced a new class of ?-minimal maps. In this paper we discuss the relationship between topologically ?-transitive maps and ?-transitive maps. On the other hand, we discuss the relationship between ?-minimal and ?-minimal in topological systems.\par
? Definition 3.1.1 Let (X, ) be a topological space. A subset A of X is called ?-dense in X if .\par
? Remark 3.1.2 Any ?-dense subset in X intersects any ?-open set in X.\par
Proof: Let A be an ?-dense subset in X, then by definition, , and let U be a non-empty ?-open set in X. Suppose that A?U=? . Therefore is ?-closed and i.e.\par
B, but , so B, this contradicts that U? Definition 3.1.3 [12] A map is called ?- irresolute if for every ?-open set H of Y, is ?-open in X.\par
? Example 3.1.4 \hyperref[b37]{[35]} Let (X, ?) be a topological space such that X=\{a, b, c, d\} and ? =\{?, X ,\{a, b\}, \{b\}\}. We have the set of all ?-open sets is ?(X, ?)=\{?, X, \{b\}, \{a, b\}, \{b, c\}, \{b, d\}, \{a, b, c\}, \{a, b, d\}\}and the set of all ?-closed sets is ?C(X, ?)=\{?, X, \{c, d, \{a, c, d\}, \{a, d\}, \{\}a, c\}, \{d\}, \{c\}\}. Then define the map f : X?X as follows f ? Definition 3.1.5 A subset A of a topological space (X, ) is said to be nowhere ?-dense, if its ?-closure has an empty ?-interior, that is, .(a)= a, f(b)= b, f(c)= d, f(d)= c, we have f is ?-irresolute because \{b\} is ? open and f-1(\{b\})=\{b\} is ?-open; \{a,\par
? Definition 3.1.6 \hyperref[b37]{[35]} Let (X, ) be a topological space, be ?-irresolute map then f is said to be topological ?-transitive if every pair of non-empty ?-open sets U and V in X there is a positive integer n such that . In the forgoing example 3.1.4: we have f is ?-transitive because b belongs to any non-empty ?-open set V and also belongs to f(U) for any ?-open set it means that so f is . ? -transitive.\par
? Example 3.1.7 Let (X, ?) be a topological space such that X =\{a, b, c\}and ?=\{?, \{a\}, X\}.Then the set of all ?-open sets is ??=\{?, \{a\}, \{a, b\}, \{a, c\}, X\}. Define f : X?X as follows f(a)=b, f(b)=b, f(c)=c. Clearly f is continuous because \{a\} is open and f(\{a\})=? is open. Note that f is transitive because f(\{a\})=\{b\} implies that f(\{a\})?\{b\}??. But f is not ?-transitive because for each n in N , fn(\{a\})?\{a, c\}=?; since fn(\{a\})=\{b\} for every n ? N, and \{b\}?\{a, c\}=?. So we have f is not ?-transitive, so we show that transitivity not implies ?-transitivity.\par
? Definition 3.1.8 Let (X, ?) be a topological space. A subset A of X is called ?-dense in X if ? Remark 3.1.9 [38] Any ?-dense subset in X intersects any ?-open set in X.\par
Proof: Let A be a ?-dense subset in X, then by definition, ? Definition 3.1.11 A subset A of a topological space (X, ) is said to be nowhere ?-dense, if its ?-closure has an empty ?-interior, that is,\par
? Definition 3.1.12 \hyperref[b36]{[34]} Let (X, ) be a topological space, and .-irresolute) map, then is said to be topologically ?-type transitive map if for every pair of ?-open sets U and V in X there is a positive integer n such that\par
We introduced a new definition on ?minimal \hyperref[b37]{[35]} (resp. ?-minimal \hyperref[b36]{[34]}) maps and we studied some new theorems associated with these definitions. Given a topological space X, we ask whether there exists ?-irresolute (resp. ?-irresolute) map on X such that the set , called the orbit of x and Global Journal of Researches in Engineering ( ) F Volume XIV Issue V Version I\begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_0}}\end{figure}
    			\footnote{Year 2014 © 2014 Global Journals Inc. (US)} 			\footnote{© 2014 Global Journals Inc. (US) denoted by O ( x ) f , i s ?dense(resp. ?dense) in X for each x Ñ?" X.. A partial answer will be given in this section. Let us begin with a new definition.} 			\footnote{© 2014 Global Journals Inc. (US)} 		 		\backmatter  			 \par
Relationship between New Types of Transitive and Chaotic Maps IV. 
\subsection[{Alpha-Minimal Functions}]{Alpha-Minimal Functions}\par
Associated with this new definition we can prove the following new theorem.\par
? Theorem 3.1.13 \hyperref[b37]{[35]}: Let (X, ) be a topological space and be ? -irresolute map. Then the following statements are equivalent: (1) f is topological ?-transitive map ? Theorem 3.1.14 : \hyperref[b36]{[34]} Let (X, ) be a topological space and be ? -irresolute map. Then the following statements are equivalent:\par
in X.\par
? Definition 4.1 (?-minimal) Let X be a topological space and f be ?-irresolute map on X with ?-regular operator associated with the topology on X. Then the dynamical system (X, f) is called ?-minimal system (or f is called ?-minimal map on X) if one of the three equivalent conditions \hyperref[b37]{[35]} hold:\par
1) The orbit of each point of X is ?-dense in X.\par
2) for each x Ñ?" X 3) Given x Ñ?" X and a nonempty ?-open U in X, there exists nÑ?" N such that ? Theorem 4.2 \hyperref[b37]{[35]} For (X, ) the following statements are equivalent: 
\subsection[{Topological Systems and Conjugacy}]{Topological Systems and Conjugacy}\par
In this section, I introduce and define ?r-conjugated topological systems (X, f ) and (Y, g), where X and Y are almost regular topological spaces. First I will define ?r-homeomorphism and then I will prove new theorem associated with these new definitions:\par
? Definition 5.1 A map.is said to be -homeomorphism if is bijective and thus invertible and both and are ?rirresolute\par
? Definition 5.2 Two topological systems (X, f ) and (Y, g) are said to be almost regular systems if X and Y are almost regular topological spaces.\par
? Definition [38] 5.3 Let (X, f ) and (Y, g) be two almost regular systems, then and are said to be topologically ?r-conjugate if there is ?rhomeomorphism such that . will call h a topological ?r-conjugacy. Thus, the two almost regular topological systems with their respective function acting on them share the same dynamics VI. 
\subsection[{New Types of Chaos of Topological Spaces}]{New Types of Chaos of Topological Spaces}\par
We will give a new definition of chaos for ?-irresolute (resp. ?-irresolute) self map of a locally compact Hausdorff topological space X, so called ?-type chaos (resp. ?-type chaos). These new definitions imply John Tylar definition which coincides with Devanney's definition for chaos when the topological space happens to be a metric space, but not conversely.\par
? Definition 4.1 Let (X , f ) be a topological system, the dynamics is obtained by iterating the map. Then, f is said to be ?-type chaotic (resp. ?-type chaotic) on X provided that for any nonempty ?-open (resp. ?open) sets U and V in X, there is a periodic point such that and .\par
? Proposition 4.2 Let (X, f ) be a topological system.\par
The map f is ?-type chaotic (resp.?-type chaotic) on X if and only if f is ?-type transitive (resp. ?-type transitive) and the set of periodic points of the map f is dense (resp. dense) in X.\par
Let us prove only for ?-type chaotic Proof: I f 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 p and with x ? U and y ? V and some positive integer n such that , so that therefore that is, f is ?-type transitive map.\par
The ?-type transitivity of f on X implies that for any nonempty ?-open subsets U, V ? X, there is n such that for some x ? U, Now define\par
. Then W is ?-open and nonempty with the property that . But since the periodic points of f are ?-dense in X, there is a p ? W such that . Therefore, and , so that f is ?-type chaotic map. 
\subsection[{VII.}]{VII.} 
\subsection[{Conclusion}]{Conclusion}\par
We have the following results\par
? Proposition 7.1. Every topologically ?-type transitive map is a topologically ?-type transitive map which implies topologically transitive map, but the converse not necessarily true.. ? proposition 7.2.Every ?-minimal map is ?-minimal map which implies minimal map, but the converse not necessarily true.. ? Theorem 7.3 Let (X, f )and (Y, g) be two topological systems, if and are topologically ?rconjugate. Then (1) f is topologically ?-transitive map if and only if g is topologically ?-transitive map;\par
(2) f is ?-type chaotic map if and only if g is ?-type chaotic map;\par
(3) f is ?-type chaotic map if and only if g is ?-type chaotic map.\par
? Proposition 7.4 Let (X, f ) be a topological system.\par
The map f is ?-type chaotic (resp.?-type chaotic) on X if and only if f is ?-type transitive (resp. ?-type transitive) and the set of periodic points of the map f is dense (resp.\par
dense) in X.\par
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