New Types of Lambda-Transitive and Weakly Lambda-Mixing Sets

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**Subject Areas:** **Dynamical System**

1. Introduction

We will consider a system given by a locally compact Hausdorff space (phase space) X and λ-irresolute map. A point “moves,” its trajectory being the sequence, where is the nth iteration of f. The point is the position of x after n units of time. The set of points of the trajectory of x under f is called the orbit of x, denoted by. As a motivation for the notion of topological transitivity of one may think of a real physical system, where a state is never given or measured exactly, but always up to a certain error. So instead of points one should study λ-open subsets of the phase space and describe how they move in that space. If for instance the λ-type minimality of is defined by requiring that every point visits every λ-open set V in X (i.e., for some) then, instead, one may wish to study the following concept: every nonempty λ-open subset U of X visits every nonempty λ-open subset V of X in the following sense: for some. If the topological system has this property; then it is called topologically λ-type transitive. We also say that f itself is topologically λ-transitive.

In this paper, new types of topologically λ-type transitive sets are introduced and studied. This is intended as a survey article on transitive sets in a system given by a λ-irresolute self-map on a topological space. On one hand it introduces postgraduate students to the study of new types of topological transitive sets and gives an overview of results on the topic; but, on the other hand, it covers some of the recent developments of mathematical science, technology, electronic and computer science. I introduced and defined a new type of transitive sets called λ-type transitive set and some of its properties are investigated. Relationships with some other types of transitive sets are given. I list some relevant properties of the λ-type transitive set. I have proved that every λ-type transitive set is transitive set but the converse not necessarily true. A topologically λ-type transitive set does not partition into nonempty λ-closed subsets. If A is a topologically λ-type transitive set, then there does not exist nonempty disjoint λ-closed subsets B and C of A such that. Every topologically transitive set is nonempty λ-closed and invariant. The set of all λ-cluster points is called the λ-closure of A and is denoted by A point x ∈ X is said to be a λ-interior point of a subset A ⊂ X if there exists a λ-open set U containing x such that U ⊂ A. The set of all λ-interior points of A is said to be the λ-interior of A and is denoted by, [1] - [3] .

2. Preliminaries and Definitions

To study the dynamics of a self-map means to study the qualitative behavior of the sequences as n goes to infinity when x varies in X, is the position of x after n unit of time, where denotes the composition of f with itself n times.

Definition 2.1. [4] By a topological system I mean a pair where X is a locally compact Hausdorff space (the phase space), and is a continuous function. The dynamics of the system is given by and the solution passing through x is the sequence.

Definition 2.2. Let, then the set is called an orbit of x under f, so is the set of points which occur on the orbit of x at some positive time, and the sequence is called the trajectory of x.

The set of limit points of the orbit is called the -limit set of x, and is denoted by A subset D of X is f-invariant if A non-empty closed invariant subset D of X is minimal, if for every A point is minimal if it is contained in some minimal subset of X.

For a point we say that f is λ-type open at x if for every open set U containing x, for a subset we say that f is λ-type open on A, if f is λ-type open at x for every Note that if f is open at x then it is λ-type open at x but not conversely.

Definition 2.3. [5] A function is called λ-irresolute if the inverse image of each λ-open set is a λ-open set in X.

Proposition 2.4. Let be a map, where X, Y are λ-compact, second countable, Hausdorff spaces. If for each there exists a λ-open set U containing p such that is λ-irresolute, then f is λ-irresolute.

Proof: Suppose that for every there exists a λ-open set U such that is λ-irresolute. So there is a cover with this property. Consider a λ-open set. Note that so which is λ-open since f is λ-irresolute.

Definition 2.5. (1) Let be a topological space, be λ-irresolute map, then the set is called λ-type transitive set if for every pair of non-empty λ-open sets U and V in X with and there is a positive integer n such that_{.}

(2) A topological system is λ-type chaotic if for every λ-open pair of not empty subsets there are a periodic point and such that

Note that:

(1) Every λ-type transitive set is transitive set but not conversely.

(2) Every λ-type transitive map is transitive map but not conversely.

(3) The reason of the foregoing statements is that the map defined on the λ-transitive set is λ-transitive map. For more knowledge see [6] .

Definition 2.6 (1). Let be a topological space, be λ-irresolute map, then the set is called topologically λ-mixing set if given any nonempty λ-open subsets with and then such that for all

(2) A λ-closed set is called a weakly λ-mixing set of if for any choice of nonempty λ-open subsets of A and nonempty λ-open subsets of X with and there exists

n ∈ N such that and

Proposition 2.7. If A is a weakly λ-mixing set of, then A is a λ-type transitive set of.

3. λ-Type Transitive Sets and Topological λr-Conjugacy

In the present section, I will introduce and define λ-type transitive sets. I will study some of their properties and prove some results associated with these new definitions. Some properties and characterizations of such sets are investigated.

A homeomorphism is a bijective continuous map with continuous inverse. More explicitly, to say that “a bijective mapping f of X onto Y is a homeomorphism” means that “is open if and only if is open”.

Definition 3.1. A function is called λr-homeomorphism if f is λ-irresolute bijective and is λ-irresolute. More explicitly, to say that “a bijective mapping f of X onto Y is λr-homeomor- phism” means that “is λ-open if and only if is λ-open”.

Definition 3.2. Two topological systems, and, are said to be topologically λr-conjugate if there is λr-homeomorphism such that I will call h a topological λr-conjugacy.

Then I have proved some of the following statements:

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

2) is a topological λr-conjugacy.

3)

4) A set B is λ-mixing set if and only if is λ-mixing set.

5) A set B is weakly λ-mixing set if and only if is weakly λ-mixing set.

Proposition 3.3. If are λr-conjugated by the λr-homeomorphism then for all y ∈ Y the orbit is λ-dense in Y if and only if the orbit of h(y) is λ-dense in X.

Proof: Suppose that are maps λr-conjugate via such that , then if for all y ∈ Y the orbit is λ-dense in Y, let be a nonempty λ-open set. Then since h is a λr-homeomorphism, is λ-open in Y, so there exists with. From it follows that so that is λ-dense in X. Similarly, if is λ-dense in X, then is λ-dense in Y.

If h is not λr-homeomorphism but only λ-irresolute surjection (a semi-λr-conjugacy), then the orbit is λ-dense in Y implies the orbit of h(y) is λ-dense in X, but not conversely.

Proposition 3.4. if are λr-conjugate. Then,

(1) T is λ-type transitive subset of X is λ-type transitive subset of Y;

(2) T is λ-mixing subset of X is λ-mixing subset of Y.

Proof (1).

Assume that are topological systems which are topologically λr-conjugated by.Thus, h is λr-homeomorphism (that is, h is bijective and thus invertible and both h and are λ-irresolute) and

Suppose T is λ-type transitive subset of X. Let A, B be λ-open subsets of Y with and (to show for some). are λ-open subsets of X since h is an λ-irresolute.

Then there exists some n > 0 such *that * *since* T is λ-type transitive subset of X, with and. Thus (as implies).

Therefore, implies since is invertible. So h(T) is λ-type transitive subset of Y.

Proof (2).

We only prove that if T is topologically λ-mixing subset of Y then h^{−1}(T) is also topologically λ-mixing subset of X. Let U, V be two λ-open subsets of X with and. We have to show that there is N > 0 such that for any n > N, and are two λ-open sets since h is λ-irresolute with and. If the set T is topologically λ-mixing then there is N > 0 such that for any n > M, So. That is and for,. Thus, since, so that, , and we have that is So, h^{−1}(T) is λ-mixing set.

Proposition 3.5. Let be a topological system and A be a nonempty λ-closed set of X. Then the following conditions are equivalent.

1) A is a λ-transitive set of.

2) Let V be a nonempty λ-open subset of A and U be a nonempty λ-open subset of X with. Then there exists such that.

3) Let U be a nonempty λ-open set of X with. Then is λ-dense in A.

Note that for any,; the n^{th} inverse image of A.

Theorem 3.6. Let be topological system and A be a nonempty λ-closed invariant set of X. Then A is a λ-type transitive set of if and only if is λ-type transitive system.

Proof:

Þ) Let be two nonempty λ-open subsets of A. For a nonempty λ-open subset of A, there exists a λ-open set U of X such that. Since A is a λ-type transitive set of, there exists n ∈

N such that Moreover, A is invariant, i.e., , which implies that Therefore, , i.e.. This shows that is λ-type transitive system.

Ü) Let be a nonempty λ-open set of A and U be a nonempty λ-open set of X with Since U is a λ-open set of X and, it follows that U ∩ A is a nonempty λ-open set of A. Since is topologically λ-type transitive system, there exists n ∈ N such that which implies that. This shows that A is a λ-type transitive set of.

4. Conclusions

There are the following results:

Proposition 4.1. Every topologically λ-transitive set is transitive set but not conversely.

Proposition 4.2. If are -conjugate. Then,

(1) T is λ-type transitive set in X if and only if is λ-type transitive set in Y;

(2) T is λ-type mixing set in X if and only if is topologically λ-mixing set in Y.

Definition 4.3. Let be a topological space and be λ-irresolute map, then the set is called topologically λ-mixing set if, given any nonempty λ-open subsets with and then such that for all

Definition 4.4. A λ-closed set is called a weakly λ-mixing set of if for any choice of nonempty λ-open subsets of A and nonempty λ-open subsets of X with and

there exists n ∈ N such that and

Proposition 4.5. If A is a weakly λ-mixing set of, then A is a λ-type transitive set of.

Proposition 4.6. Let be a topological system and A be a nonempty λ-closed set of X. Then the following conditions are equivalent.

1) A is a λ-transitive set of.

2) Let V be a nonempty λ-open subset of A and U be a nonempty λ-open subset of X with. Then there exists such that.

3) Let U be a nonempty λ-open set of X with. Then is λ-dense in A.

Theorem 4.7. Let be topological system and A be a nonempty λ-closed invariant set of X. Then A is a λ-type transitive set of if and only if is λ-type transitive system.

References

[1] Maki, H. (1986) Generalized Λ-Sets and the Associated Closure Operator. The Special Issue in Commemoration of Prof. Kazusada IKED’s Retirement, 1 October, 139-146.

[2] Arenas, F.G., Dontchev, J. and Ganster, M. (1997) On λ-Sets and the Dual of Generalized Contiuity. Questions and Answers in General Topology, 15, 3-13.

[3] Caldas, M. and Jafari, S. (2005) On Some Low Separation Axioms via λ-Open and λ-Closure Operator. Rendiconti del Circolo Matematico di Palermo, 54, 195-208.

http://dx.doi.org/10.1007/BF02874634

[4] Murad, M.N. (2013) New Types of Λ-Transitive Maps. International Journal of Research in Electrical and Electronics Engineering, 2, 19-24.

[5] Caldas, M., Jafari, S. and Navalagi, G. (2007) More on λ-Closed Sets in Topological Spaces. Revista Colombiana, 41, 355-369.

[6] Murad, M.N. (2014) New Types of Chaotic Maps on Topological Spaces. International Journal of Electrical and Electronic Science, American Association for Science and Technology, 1, 1-5.