Abstract

In this paper, we have introduced some concepts about topological dynamical systems and proved some new corollary and theorems of transitivity of a theta irresolute function defined on topological space.

Keywords

θ-Irresolute Function, Dynamics in Topological Spaces, Transitive Functions, θ-Adherent Points

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1. Introduction

In this paper, we have investigated and introduced some new definitions of transitivity in topological space. To study the dynamics of a self-map $f:X\to X$ means to study the qualitative behavior of the sequences $\left\{f^n\left(x\right)\right\}$ as n goes to infinity when x varies in X, where $f^n$ denotes the composition of f with itself n times:

By a topological system I mean a pair $\left(X,f\right)$ , where X is a locally compact Hausdorff topological space (the phase space), and $f:X\to X$ is a continuous function. The dynamics of the system is given by $x_{n+1}=f\left(x_n\right),x_0\in X,n\in N$ and the solution passing through x is the sequence $\left\{f\left(x_n\right)\right\}$ where $n\in N$ .

Let $x\in X$ , then the set $\left\{x,f\left(x\right),f^2\left(x\right),\cdots \right\}$ is called an orbit of x under f and is denoted by $O_f\left(x\right)$ , so $O_f\left(x\right)$ is the set of points which occur on the orbit of x at some positive time, and the sequence $x,f\left(x\right),f^2\left(x\right),\cdots$ is called the trajectory of x. Any point with dense orbit is called a transitive point. A point which is not transitive is called intransitive.

Topological dynamics is concerned with the behavior of iterations of a continuous map f from a space X into itself. Suppose for some $x\in X$ , sequence $x,f\left(x\right),f^2\left(x\right),\cdots$ converges to some point say $x_0\in X$ , then we must have $f\left(x_0\right)=x_0$ , because f is continuous. Such points we call as fixed points. We say that the point x is attracted by the fixed point $x_0$ . The set of all points in X attracted by $x_0$ is called the stable set or the basin of attraction of the fixed point $x_0$ and is denoted by $W_f\left(x_0\right)$ . A fixed point $x_0$ is said to be attracting if its stable set is a neighborhood of it.

A point $x\in X$ is said to be periodic if there exists a positive integer $n\in N$ such that $f^n\left(x\right)=x$ . The set of all periodic points of the map f is denoted by per(f).

A point $x\in X$ is called a θ-adherent point of A [1] , if $A\cap Cl\left(U\right)\ne \varphi$ for every open set U containing x. The set of all θ-adherent points of a subset A of X is called the θ-closure of A and is denoted by $C{l}_{\theta }\left(A\right)$ . A subset A of X is called θ-closed if $A=Cl{}_{\theta }(A)$ . Dontchev and Maki [2] have shown that if A and B are subsets of a space X, then $C{l}_{\theta }\left(A\cup B\right)=C{l}_{\theta }\left(A\right)\cup C{l}_{\theta }\left(B\right)$ and that $C{l}_{\theta }\left(A\cap B\right)=C{l}_{\theta }\left(A\right)\cap C{l}_{\theta }\left(B\right)$ . Recall that a space (X, τ) is Hausdorff if and only if every compact set is θ-closed. The complement of a θ-closed set is called a θ-open set. The family of all θ-open sets forms a topology on X and is denoted by ${\tau }^{\theta }$ . This topology is coarser than τ and that a space (X, τ) is regular if and only if $\tau ={\tau }^{\theta }$ [3] .

2. Basic Definition and Theorems

Definition 2.1 [4] By a topological system I mean a pair $\left(X,f\right)$ , where X is a locally compact Hausdorff topological space (the phase space), and $f:X\to X$ is a continuous function. The dynamics of the system is given by $x_{n+1}=f\left(x_n\right),x_0\in X,n\in N$ and the solution passing through $x_0$ is the sequence $\left\{f\left(x_n\right)\right\}$ where $n\in N$ .

Definition 2.2. 1) Let $x\in X$ , then the set $\left\{x,f\left(x\right),f^2\left(x\right),\cdots \right\}$ is called an orbit of x under f and is denoted by $O_f\left(x\right)$ , so $O_f\left(x\right)$ is the set of points which occur on the orbit of x at some positive time, and the sequence $x,f\left(x\right),f^2\left(x\right),\cdots$ is called the trajectory of x.

2) Let X be a topological space, $f:X\to X$ , ${\left\{f^n\left(x_0\right)\right\}}_{n=0}^{\infty }$ be a sequence in X, and let $x\in X$ . Then $\left\{f^n\left(x_0\right)\right\}$ converges to x if for all open sets U containing x, there exists an integer N such that $f^n\left(x_0\right)\in U$ for all n >N, Note that if this sequence is convergence then it converges to a fixed point, say y, i.e. $f\left(y\right)=y$ .

Any point with dense orbit is called a transitive point. A point which is not transitive is called intransitive.

Definition 2.3. 1) (Transitivity) Let X be a topological space with no isolated point. Then the function $f:X\to X$ is said to be transitive if for any two open sets U and V in X, there is a point $x\in U$ and an n > 0 such that $f^n\left(x\right)\in V$ . It is easily to show that if f is transitive then for every pair U, V of non-empty open sets, there exist a positive integer n such that $f^n\left(U\right)\cap V\ne \varphi$ .

2) Let X be a topological space, the function $f:X\to X$ , is said to be topologically mixing if for every pair U, V of non-empty open sets, if there exist N such that $f^n\left(U\right)\cap V\ne \varphi$ for all $n>N$ .

Definition 2.4. (topological weak mixing) Let X has no isolated point. g is topologically weakly mixing, if the product of two functions $g\times g$ is topologically transitive.

Proposition 2.5. Every topological mixing function implies topological weak mixing. But the converse is no necessarily true.

Proof: It is easily to prove the foregoing theorem.

Definition 2.6. A map f is said to be transitive (resp., θ-transitive [5] ) if for any non-empty open (resp., θ-open) sets U and V in X, there exists $n\in N$ such that $f^n\left(U\right)\cap V\ne \varphi$ .

Theorem 2.7 [5] . Let X be a non-empty locally θ-compact Hausdorff space. Then the intersection of a countable collection of θ-open θ-dense subsets of X is θ-dense in X.

Corollary 2.8. A subset A of a space $\left(X,\tau \right)$ is θ-dense if and only if $A\cap U\ne \varphi$ for all $U\in {\tau }^{\alpha }$ other than $U=\varphi$ .

Two topological spaces $\left(X,\tau \right)$ and $\left(Y,{\tau }_1\right)$ are called homeomorphic if there exists a one-to-one onto function $f:\left(X,\tau \right)\to \left(Y,{\tau }_1\right)$ such that f and $f^{-1}$ are both continuous.

Note that any homeomorphic spaces have the same dynamics, if we have any notion about first space then we have the same notion about the other one.

A map $h:X\to Y$ is a homeomorphism if it is continuous, bijective and has a continuous inverse.

A function $f:X\to X$ is called θ-irresolute [6] if the inverse image of each θ-open set is a θ-open set in X.

A map $h:X\to Y$ is θr-homeomorphism if it is bijective and thus invertible and both h and $h^{-1}$ are θ-irresolute.

Theorem 2.9. Let $\left(X,f\right)$ be a topological system where X is a non-empty θ-compact topological space and $f:X\to X$ is θ-irresolute map and that X is separable. Suppose that f is topologically θ-transitive. Then there is an element $x\in X$ such that the orbit $O_f\left(x\right)=\left\{x,f\left(x\right),f^2\left(x\right),\cdots ,f^n\left(x\right),\cdots \right\}$ is θ-dense in X.

Proof: Let $B=\left\{U_i\right\},i=1,2,3,\cdots$ be a countable basis for the θ-topology of X. For each i, let $O_i=\left\{x\in X:f^n\left(x\right)\in U_i\text{forsome}n\ge 0\right\}$

Then, clearly $O_i$ is θ-open and θ-dense. It is θ-open since f is θ-irresolute, so, $O_i={\displaystyle \underset{i=1}{\overset{\infty }{\cup }}{f}^{-1}\left(U_i\right)}$ is θ-open and θ-dense since f is topological θ-transitive map. Further, for every θ-open set V, there is a positive integer n such that $f^n\left(V\right)\cap U_i\ne \varphi$ , since f is θ transitive.

Now, apply theorem 2.7 to the countable θ-dense set $\left\{O_i\right\}$ to say that ${\displaystyle \underset{i=0}{\overset{\infty }{\cap }}{O}_i}$ is θ-dense and so non-empty. Let $y\in {\displaystyle \underset{i=0}{\overset{\infty }{\cap }}{O}_i}$ . This means that, for each i, there is a positive integer n such that $f^n\left(y\right)\in U_i$ for every i. By Corollary 2.8 this implies that $O_f\left(x\right)$ is θ-dense in X.

Definition 2.10. The function $f:X\to X$ , is strongly transitive [7] if for any nonempty open set $U\subset X$ , $X={\displaystyle \underset{k=0}{\overset{s}{\cup }}{f}^k\left(U\right)}$ for some s > 0. It is easily seen that $X={\displaystyle \underset{k=0}{\overset{\infty }{\cup }}{f}^k\left(U\right)}$ for any nonempty open set $U\subset X$ if and only if ${\displaystyle \underset{k=0}{\overset{\infty }{\cup }}{f}^{-k}\left(x\right)}$ is dense in X for any $x\in X$ .

We may consider that, the last statement of the foregoing definition as lemma, because we can use this statement to prove the following corollary.

Lemma 2.11. $X={\displaystyle \underset{k=0}{\overset{\infty }{\cup }}{f}^k\left(U\right)}$ for any nonempty open set $U\subset X$ if and only if ${\displaystyle \underset{k=0}{\overset{\infty }{\cup }}{f}^{-k}\left(x\right)}$ is dense in X for any $x\in X$ .

According to the definition 2.10 and lemma 2.11, we have the following important corollary.

Corollary 2.12. If ${\displaystyle \underset{k=0}{\overset{\infty }{\cup }}{f}^{-k}\left(x\right)}$ is dense in X for any $x\in X$ , then the function $f:X\to X$ , is strongly transitive.

3. Conclusion:

There are the following results:

Proposition 3.1. Every topological mixing function implies topological weak mixing. But the converse is no necessarily true.

Theorem 3.2. Let $\left(X,f\right)$ be a topological system where X is a non-empty θ-compact topological space and $f:X\to X$ is θ-irresolute map and that X is separable. Suppose that f is topologically θ-transitive. Then there is an element $x\in X$ such that the orbit $O_f\left(x\right)=\left\{x,f\left(x\right),f^2\left(x\right),\cdots ,f^n\left(x\right),\cdots \right\}$ is θ-dense in X.

Conflicts of Interest

The author declares no conflicts of interest.

References