On Transitive Points in a Generalized Shift Dynamical System

Abstract

Considering point transitive generalized shift dynamical system for discrete with at least two elements and infinite , we prove that is countable and has at most elements. Then, we find a transitive point of the dynamical system for with and show that point transitive , for infinite countable , is a factor of .

1. Introduction

By a topological dynamical system we mean where is a topological space and is a continuous map. In the dynamical system for , we call the orbit of and denote it by . Also we call the dynamical system point transitive if there exists with ; moreover, we call a transitive point. Moreover, we call topologically transitive if for every nonempty open subset of there exists with .

For arbitrary map and nonvoid sets , define by for all . If is a topological space and is equipped with the product (pointwise convergence) topology, then is continuous; we call a generalized shift dynamical system. Generalized shifts in this sense have been introduced for the first time in [1]. So many dynamical and nondynamical properties of generalized shifts like algebraic entropy, topological entropy, and Devaney chaos [2] of a generalized shift have been studied recently.

One of the famous dynamical systems is the one-sided shift dynamical system where and for all ; we recall once more the one-sided shift in our final example.

It is well known that, in dynamical system with Hausdorff , if does not have any isolated point and it is point transitive, then is topological transitive [3, Proposition 1]; however, as it has been mentioned in [3], two concepts, topological transitivity and point transitivity, are not equivalent in general (one may find notes regarding these two concepts (and their historical points of view) in [4]; however, the reader will find [5, 6] interesting too). Moreover, for discrete with at least two elements, infinite countable , and , if the generalized shift dynamical system is topological transitive, then is one to one without periodic points [2, Theorem 2.11]; the main aim of the following text is to construct a transitive point of in the above case. We recall that is a periodic point of if there exists such that .

In the following text, suppose is discrete with at least two elements, is an infinite set, and is an arbitrary map. Also suppose is the set of all natural numbers, is the set of all integers, and , and we use the symbol for .

2. Two Cardinal Bounds in the Point Transitive Generalized Shift Dynamical Systems

In this section, we prove that, for discrete with at least two elements, infinite , and arbitrary map , if the generalized shift dynamical system is point transitive, then(i)the map is one to one without periodic point;(ii)the set is countable, and .

Theorem 1. If is point transitive, then is countable.

Proof. Assume that is uncountable, and let . Choose and let . There exists since is uncountable and is countable. For , is an open subset of such that therefore, and ; hence, is not point transitive.

Definition 2. Define relation on letting for . It is easy to verify that is an equivalence relation on .

As a matter of fact, for injective and , we have (where ) so exactly one of the following cases occurs:(I)there exists with ( is a periodic point of ),(II)the sequence is one to one and there exists with ,(III)the sequence is one to one and for all we have .Now one can verify that the above cases are equivalent to the following statements in the corresponding case: (I) is finite (and equal to ),(II) is infinite and there exists with ,(III) is infinite and there exists a (one to one) bisequence such that and for all we have .In particular, in all of the above cases, is countable.

Theorem 3. For , if   is point transitive, then .

Proof. Let be point transitive; then, as it has been mentioned in the Introduction, by [2, Theorem 2.11], is one to one without periodic points. Now suppose is a transitive point of . Using Theorem 1, is countable; also is one to one without periodic point. Consider as in Definition 2. Let ; using the Axiom of Choice there exists a choice function ; thus, for all , we have ; we denote simply by . Since for distinct we have . Define with (for ). The map is one to one; otherwise, there exist distinct with , so for all we have and which is a contradiction. Therefore, is one to one, and (note the fact that every is countable, by injection of )

3. Towards the Main Construction

In the first part of this section, we construct a point transitive generalized shift dynamical system which plays a key role in the final part and our main construction.

3.1. Construction of a Point Transitive Generalized Shift Dynamical System

We denote the collection of matrices with the arrays in an arbitrary set , by .

Definition 4. For set with elements, suppose is the following matrix up to rearrangement of : However, is well defined up to the rearrangement of ; that is, if is a permutation (on ), one may consider the following matrix as too, where for (see Example 5): Also, note that the columns of are pairwise distinct.

Example 5. For , is any of the following matrices:

Note 1. Under the same notations as in Definition 4, if and , then there exists (a unique) such that

Definition 6. For , and ; then, ; in other words, is a matrix whose arrays belong to . By removing partitions of the matrix , we have a new matrix .
For example, let ; then, and and we have the following matrix as : so is the following matrix:

Note 2. Using Note 1 and Definition 6, if is an matrix with arrays in and (introduced in Definition 6), then there exists such that However there exists a unique column in equal to

Lemma 7. If is a nonempty open subset of , then there exist and with for all , such that is a subset of .

Proof. Suppose is a nonempty open subset of ; here, we search for a suitable open subset of , namely, . Choose . There exist such that Let
Since for all we have , , and , we have . Moreover, using the definition of and s, we have .

Convention 1. Consider the matrix such that where is introduced in Definition 6. In other words, for , if (as in Definition 6), then

Lemma 8. Let Then, is a transitive point of the generalized shift dynamical system for with . In particular, is point transitive.

Proof. We just should prove that for all nonempty open subsets of there exists with . Suppose is a nonempty open subset of . By Lemma 7, there exist and such that is a subset of , and . Let and let . Using Note 2 for there exists such that Now according to Convention 1, we have which leads to since and using the definition of s.
So for and . Therefore, for and , we have hence, for , we have So we have . Thus, , and is point transitive.

Definition 9. The dynamical systems and are topologically semiconjugate, if there exists an onto and continuous map, with . Such a map is called a topological semiconjugacy. We call a factor of if there exists a topological semiconjugacy .

Lemma 10. In the dynamical systems and , if is a topological semiconjugacy and is a transitive point of , then is a transitive point of .
In particular, the factor of a point transitive dynamical system is point transitive too.

Proof. Suppose is a topological semiconjugacy and is a point transitive of ; we have therefore, , is a transitive point of , and is point transitive.

Lemma 11. For with and countable , consider as in Lemma 8; then, is a transitive point of ; in particular, is point transitive.

Proof. We use the same notation for and .
The map with is a semiconjugacy. By Lemma 8 and Definition 9 we have the desired result.

3.2. Main Construction

Now we are ready to have our main construction; suppose is infinite countable, is countable, and is point transitive; hence, is topologically transitive and, by [2, Theorem 2.11], is one to one without any periodic point.

Using the same notations as in Theorem 3 and Definition 2 for , we have and is infinite countable. Since is countable, is countable too and there exists a one to one map .

Now define with for all and . Regarding , we have the following facts:(i) is well defined, since for every there exist unique with and unique with ;(ii) is one to one, since is one to one.Now consider as in Lemma 11. For , choose and with ; therefore, Consider defined by ; for , let ; we have Consider and ; let (This definition is well defined, since is one to one.)

Then, we have ; thus, is a semiconjugacy; hence, by Lemma 11 and Definition 9 and using the same notations, is point transitive, and is a transitive point of .

We recall that the dynamical system with metric space is called Devaney chaotic if is sensitive dependence to initial conditions, the collection of periodic points of is dense in , and is topological transitive [7]. Now note the fact that, for metric space with at least two elements, is metrizable if and only if is countable [8].

Remark 12. It has been proved in [2, Theorem 2.13] that for infinite countable the generalized shift dynamical system is Devaney chaotic, if and only if it is topological transitive.

Using Remark 12, Theorem 1, and the above construction, we have the following corollary, which shows an interaction between Devaney chaos and point transitivity in .

Corollary 13. For infinite countable in the dynamical system , the following statements are equivalent: (i)the system is point transitive,(ii)the set is countable and is one to one without any periodic point,(iii)the set is countable and the system is Devaney chaotic.

Example 14. Let us return to our well known one-sided shift in the Introduction, that is, for with (), where . Since is one to one without any periodic point, is point transitive by Corollary 13 (note that in the Introduction). However, it is easy to see that is a transitive point of . However, using our method of construction and notations, for we have ; thus, is the unique element of ; we make the following choices (for fix ): (a),(b) (),(c), with . is , so is a transitive point of ; however, we may find different transitive points by choosing another , , or s in (a), (b), and (c).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.