# Introduction

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 ? )
( is ? -dense in X. With the above concepts, some new theorems have been introduced and studied. Furthermore, we have the following results:

? Every topologically ? ? transitive map implies topologically ? ? transitive map, but the converse not necessarily true.

? Every ? ? minimal system implies ? ? minimal system, but the converse not necessarily true.
? ) ( ) ( ? ? ET E ? ; ? ) ( ) ( ) ( ? ? ? TT WM TM ? ? ; II.
Preliminaries and Theorems
Definition 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.
Proposition 2.2 The product of two topologically ?mixing systems must be topologically ? -mixing.

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
), , ( 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 ? ?
are nonempty, and therefore
) ' ( ) ' ( ) ( V V U U g f n × ? × × is nonempty as well. But this implies that ? ? ? × ' ) ( ) ( W W g f n
, 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.
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.


# III. New Types of Chaos of Topological Spaces

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 for
X 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
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)

# ) = g(h( (i.e. h(f(

We call h a topological r ? -Conjugacy. Then I have proved some of the following statements:

1. The maps f and g have the same kind of dynamics.

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
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.

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. If
X 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 ? :
. 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 ? ? = =
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
. 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 .
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.) ( )) ( ( 1 1 ?
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.

1. A is a ? -transitive set of ) , ( f X .

2. Let V be a nonempty ? -open subset of A and U be a nonempty ? -open subset of X with ? ? ? A U . Then ther exists
N ? 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
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 .

Proof:

) ? 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 ) ( . :)
? 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 .




isweakl y? -mixing.Remark 3.4If{x0,x1,x2,...}denotes an orbit of
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( 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

. These shows that ) , ( f A is ? -type transitive.

. This shows that A is a ? -type transitive set of ). , ( f X IV. New Types of Chaos in Product Spaces

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

) , ( f X be a topological dynamical system. The map f is ? -type chaotic on X if and only if f
			
			

				


* 
	
		
			Mohammed Nokhas MuradKaki
		
		
			IntroductionTo
		
		
			Dynamical Systems I
		
		Publisher SciencePG
				Book; New York, USA
		
			2015