# On Transitive Points in a Generalized Shift Dynamical System

Authors: Bahman Taherkhani; Fatemah Ayatollah Zadeh Shirazi
Journal: Chinese Journal of Mathematics
 (2015)

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## Abstract

Considering point transitive generalized shift dynamical system(XΓ,σφ) for discrete X with at least two elements and infinite Γ, we prove that X is countable and Γ has at most 2ℵ0 elements. Then, we find a transitive point of the dynamical system (NN×Z,στ) for τ:N×Z→N×Z with τ(n,m)=(n,m+1) and show that point transitive (XΓ,σφ), for infinite countable Γ, is a factor of (NN×Z,στ).


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## Body

## 1. Introduction

By a topological dynamical system we mean(Z,h) where Z is a topological space and h:Z→Z is a continuous map. In the dynamical system (Z,h) for z0∈Z, we call {hi(z0):i≥0} theorbit of z0∈Z and denote it by Oh(z0). Also we call the dynamical system (Z,h)point transitive if there exists z0∈Z with Oh(z0)¯=Z; moreover, we call z0 atransitive point. Moreover, we call (Z,h)topologically transitive if for every nonempty open subset U,V of Z there exists n≥1 with hn(U)∩V≠⌀.For arbitrary mapφ:Γ→Γ and nonvoid sets X,Γ, define σφ:XΓ→XΓ by σφ((xγ)γ∈Γ)=(xφ(γ))γ∈Γ for all (xγ)γ∈Γ∈XΓ. If X is a topological space and XΓ is equipped with the product (pointwise convergence) topology, then σφ:XΓ→XΓ is continuous; we call (XΓ,σφ) ageneralized 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({1,…,k}N,σ) where k≥2 and σ((xn)n∈N)=(xn+1)n∈N for all (xn)n∈N∈{1,…,k}N; we recall once more the one-sided shift in our final example.It is well known that, in dynamical system(Z,h) with Hausdorff Z, if Z does not have any isolated point and it is point transitive, then (Z,h) 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 X with at least two elements, infinite countable Γ, and φ:Γ→Γ, if the generalized shift dynamical system (XΓ,σφ) is topological transitive, then φ:Γ→Γ 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 (XΓ,σφ) in the above case. We recall that α∈Γ is a periodic point of φ:Γ→Γ if there exists n≥1 such that φn(α)=α.In the following text, suppose
X
is discrete with at least two elements,
Γ
is an infinite set, and
φ
:
Γ
→
Γ
is an arbitrary map. Also suppose N={1,2,…} is the set of all natural numbers, Z={0,±1,±2,…} is the set of all integers, and ℵ0=card(N), and we use the symbol c for 2ℵ0.


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

In this section, we prove that, for discreteX with at least two elements, infinite Γ, and arbitrary map φ:Γ→Γ, if the generalized shift dynamical system (XΓ,σφ) is point transitive, then(i)
the mapφ:Γ→Γ is one to one without periodic point;(ii)
the setX is countable, and card(Γ)≤c.Theorem 1.
If(XΓ,σφ) is point transitive, then X is countable.Proof.
Assume thatX is uncountable, and let b=(bα)α∈Γ∈XΓ. Choose β∈Γ and let Y={bφn(β):n≥0}. There exists c∈X∖Y since X is uncountable and Y is countable. For V={(xα)α∈Γ:xβ=c}, V is an open subset of XΓ such that
(1)V∩σφnbαα∈Γ:n≥0=⌀;
therefore, V∩Oσφ(b)=⌀ and XΓ≠Oσφ(b)¯; hence, (XΓ,σφ) is not point transitive.Definition 2.
Define~φ relation on Γ letting
(2)α~φβ⟺∃n,m∈N(φn(α)=φm(β))
for α,β∈Γ. It is easy to verify that ~φ is an equivalence relation on Γ.As a matter of fact, for injectiveφ:Γ→Γ and α∈Γ, we have α/~φ=⋃‍{φn(α):n∈Z} (where α/~φ={β∈Γ:α~φβ}) so exactly one of the following cases occurs:(I)
there existsn≥1 with φn(α)=α (α is a periodic point of φ),(II)
the sequence{φn(α)}n≥1 is one to one and there exists k≥1 with φ-k(α)=⌀,(III)
the sequence{φn(α)}n≥1 is one to one and for all k≥1 we have φ-k(α)≠⌀.Now one can verify that the above cases are equivalent to the following statements in the corresponding case: (I)
α/~φ is finite (and equal to {φi(α):i≥0}),(II)
α/~φ is infinite and there exists β∈Γ with α/~φ={φi(β):i≥0},(III)
α/~φ is infinite and there exists a (one to one) bisequence {bn}n∈Z such that α/~φ={bn:n∈Z} and for all n∈Z we have bn+1=φ(bn).In particular, in all of the above cases, α/~φ is countable.Theorem 3.
Forφ:Γ→Γ, if  (XΓ,σφ) is point transitive, then card(Γ)≤c.Proof.
Let(XΓ,σφ) be point transitive; then, as it has been mentioned in the Introduction, by [2, Theorem 2.11], φ:Γ→Γ is one to one without periodic points. Now suppose w=(xα)α∈Γ∈XΓ is a transitive point of XΓ. Using Theorem 1, X is countable; also φ:Γ→Γ is one to one without periodic point. Consider ~φ as in Definition 2. Let Λ:=Γ/~φ; using the Axiom of Choice there exists a choice function ϖ:Λ→⋃‍Λ=Γ; thus, for all D∈Λ, we have ϖ(D)∈D; we denote ϖ(D) simply by αD. Since
(3)Oσφ(w)¯=xφnαα∈Γ:n≥0¯=XΓ,
for distinct D,G∈Λ we have (xφn(αD))n∈N≠(xφn(αG))n∈N. Define ξ:Λ→XN with ξ(D)=(xφn(αD))n∈N (for D∈Λ). The map ξ:Λ→XN is one to one; otherwise, there exist distinct D,G∈Λ with ξ(D)=ξ(G), so for all n≥1 we have xφn(αD)=xφn(αG) and
(4)Oσφw¯=σφnxαα∈Γ:n≥0¯=xφnαα∈Γ:n≥0¯⊆yαα∈Γ∈XΓ:yαD=yαG¯=yαα∈Γ∈XΓ:yαD=yαG≠XΓ
which is a contradiction. Therefore, ξ:Λ→XN is one to one, and (note the fact that every D∈Λ is countable, by injection of φ:Γ→Γ)
(5)cardΓ=card⋃‍Λ≤ℵ0card(Λ)≤ℵ0card(XN)≤ℵ0(ℵ0ℵ0)=2ℵ0=c.


## 3. Towards the Main Construction

In the first part of this section, we construct a point transitive generalized shift dynamical system(NN×Z,στ) 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 ofn×m matrices with the arrays in an arbitrary set A, by Matn×m(A).Definition 4.
For setA={x1,…,xn} with n elements, suppose
(6)M(A)∈Matn×nn(A)
is the following matrix up to rearrangement of x1,…,xn:(7)x1···x1︸nn-1times⋯xn···xn︸nn-1timesx1⋯x1︸nn-2timesx2⋯x2︸nn-2times⋯xn⋯xn︸nn-2times⋯x1⋯x1︸nn-2timesx2⋯x2︸nn-2times⋯xn⋯xn︸nn-2times⋮⋮⋮x1x2⋯xn⋯x1x2⋯xn⋯x1x2⋯xn⋯x1x2⋯xnn×nn.However, M(A) is well defined up to the rearrangement of x1,…,xn; that is, if ϱ:{1,…,n}→{1,…,n} is a permutation (on {1,…,n}), one may consider the following matrix as M(A) too, where yi=xϱ(i) for i=1,…,n (see Example 5):(8)y1···y1︸nn-1times⋯yn···yn︸nn-1timesy1⋯y1︸nn-2timesy2⋯y2︸nn-2times⋯yn⋯yn︸nn-2times⋯y1⋯y1︸nn-2timesy2⋯y2︸nn-2times⋯yn⋯yn︸nn-2times⋮⋮⋮y1y2⋯yn⋯y1y2⋯yn⋯y1y2⋯yn⋯y1y2⋯ynn×nn.Also, note that the columns of M(A) are pairwise distinct.Example 5.
ForA={1,2}, M(A) is any of the following matrices:
(9)11221212,22112121.Note 1.
Under the same notations as in Definition4, if M(A)=[a(i,j)]n×nn and b1,…,bn∈A, then there exists (a unique) j∈{1,…,nn} such that
(10)a1ja2j⋮anj=b1b2⋮bn.Definition 6.
Forp∈N, A:={1,…,p} and B:=Ap; then, M(B)∈Mat(pp)×(pp)(pp)(B); in other words, M(B) is a matrix whose arrays belong to Mat1×p(A). By removing partitions of the matrix M(B), we have a new matrix Kp∈Mat(pp)×p(pp)(pp)(A).
For example, letp=2; then, A={1,2} and B={(1,1),(1,2),(2,1),(2,2)} and we have the following matrix as M(B):(11)(1,1)···(1,1)︸43times⋯(2,2)···(2,2)︸43times1,1⋯(1,1)︸16times⋯(2,2)⋯(2,2)︸16times⋯(1,1)⋯(1,1)︸16times⋯(2,2)⋯(2,2)︸16times(1,1)⋯(1,1)︸4times⋯⋯(1,1)⋯(1,1)︸16times⋯(2,2)⋯(2,2)︸4times(1,1)(1,2)(2,1)(2,2)⋯⋯(1,1)⋯(1,1)︸16times⋯(1,1)(1,2)(2,1)(2,2)4×44so K2 is the following matrix:(12)1···1︸43×2times⋯2···2︸43×2times1···1︸32times⋯⋯⋯2···2︸32times11111111121212122121212122222222⋯⋯⋯1111111112121212212121212222222211122122111221221112212211122122⋯⋯⋯111221221112212211122122111221224×29.Note 2.
Using Note1 and Definition 6, if Δ=[d(i,j)]n×n is an n×n matrix with arrays in {1,…,n} and Kn=[kijn]nn×n(nn)(nn) (introduced in Definition 6), then there exists t∈{0,1,…,n(nn)(nn)-1} such that
(13)k1(t+1)nk1(t+2)n⋯k1(t+n)nk2(t+1)nk2(t+2)n⋯k2(t+n)n⋮⋮⋮kn(t+1)nkn(t+2)n⋯kn(t+n)n=d(1,1)d(1,2)⋯d(1,n)d(2,1)d(2,2)⋯d(2,n)⋮⋮⋮d(n,1)d(n,2)⋯d(n,n).
However there exists a unique column in M({1,…,n}n) equal to
(14)d(1,1)d(1,2)⋯d(1,n)d(2,1)d(2,2)⋯d(2,n)⋮d(n,1)d(n,2)⋯d(n,n)nn-ntimes11⋯1⋮11⋯1.Lemma 7.
IfV is a nonempty open subset of NN×Z, then there exist N∈N and Δ=[d(i,j)]1≤i≤2N+1,-N≤j≤N with 1≤d(i,j)≤N for all i,j, such that
(15)xi,j(i,j)∈N×Z∈NN×Z:xi,j(i,j)∈N×Z∈NN×Z∀i∈1,…,2N+1∀j∈-N,…,N-1,Nx(i,j)=d(i,j)
is a subset of V.Proof.
SupposeV is a nonempty open subset of NN×Z; here, we search for a suitable open subset of V, namely, W. Choose (a(i,j))(i,j)∈N×Z∈V. There exist (i1,j1),…,(ip,jp)∈N×Z such that
(16)V1:=xi,j(i,j)∈N×Z∈NN×Z:xi,j(i,j)∈N×Z∈NN×Z∀k∈{1,…,p}x(ik,jk)=a(ik,jk)⊆V.
Let
(17)N=max⁡{ik:k=1,…,p}∪{jk:k=1,…,p}aik,jk:k=1,…,p∪aik,jk:k=1,…,pd(i,j)=a(i,j)∃k∈{1,…,p}(i,j)=(ik,jk)1otherwiseW=xi,j(i,j)∈N×Z∈NN×Z:xi,ji,j∈N×Z∈NN×Z∀i∈1,…,2N+1∀j∈-N,…,Nxi,j=di,j.
Since for allk∈{1,…,p} we have ik≤N<2N+1, |jk|≤N, and a(ik,jk)=d(ik,jk), we have W⊆V1(⊆V). Moreover, using the definition of N and d(i,j)s, we have max⁡{d(i,j):i∈N,j∈Z}≤N.Convention 1.
Consider the matrix(18)Ψ=ψi,j(i,j)∈N×N=ψ(1,1)ψ(1,2)⋯ψ(2,1)ψ(2,2)⋯ψ(3,1)ψ(3,2)⋯⋮⋮⋮
such that
(19)Ψ=K1K2K3K4⋯1111⋮⋮⋮⋮⋱1111⋮⋮⋮⋮,
where Kn is introduced in Definition 6. In other words, for n∈N, if Kn=[kijn]nn×n(nn)(nn) (as in Definition 6), then
(20)ψ(i,j)=kisn1≤i≤nn,j=s+∑1≤p≤(n-1)‍ppp(pp),1≤i≤nn,j=s+1≤1≤s≤nnn(nn),1otherwise.Lemma 8.
Let(21)a(i,j)=ψ(i,j)i≥1,j≥1,1i≥1,j≤0.
Then, (a(i,j))(i,j)∈N×Z is a transitive point of the generalized shift dynamical system (NN×Z,στ) for τ:N×Z→N×Z with τ(n,m)=(n,m+1). In particular, (NN×Z,στ) is point transitive.Proof.
We just should prove that for all nonempty open subsetsV of NN×Z there exists n≥0 with στn((a(i,j))(i,j)∈N×Z)=(a(i,j+n))(i,j)∈N×Z∈V. Suppose V is a nonempty open subset of NN×Z. By Lemma 7, there exist N∈N and Δ=[d(i,j)]0≤i≤2N+1,-N≤j≤N such that
(22)xi,j(i,j)∈N×Z∈NN×Z:xi,j(i,j)∈N×Z∈NN×Z∀i∈1,…,2N+1∀j∈-N,…,N-1,Nxi,j=di,j
is a subset of V, and max⁡{d(i,j):1≤i≤2N+1,-N≤j≤N}≤N<2N+1. Let M:=(2N+1)((2N+1)(2N+1))((2N+1)(2N+1))-1 and let L:=∑1≤p≤2N‍p(pp)(pp). Using Note 2 for
(23)Q:=d(1,-N)d(1,-N+1)⋯d(1,N)d(2,-N)d(2,-N+1)⋯d(2,N)⋮⋮⋮d(2N+1,-N)d(2N+1,-N+1)⋯d(2N+1,N),
there exists t∈{0,…,M} such that
(24)Q=k1(t+1)2N+1k1(t+2)2N+1⋯k1(t+2N+1)2N+1k2(t+1)2N+1k2(t+2)2N+1⋯k2(t+2N+1)2N+1⋮⋮⋮k(2N+1)(t+1)2N+1k(2N+1)(t+2)2N+1⋯k(2N+1)(t+2N+1)2N+1.
Now according to Convention 1, we have
(25)Q=ψ(1,t+1+L)ψ(1,t+2+L)⋯ψ(1,t+2N+1+L)ψ(2,t+1+L)ψ(2,t+2+L)⋯ψ(2,t+2N+1+L)⋮⋮⋮ψ(2N+1,t+1+L)ψ(2N+1,t+2+L)⋯ψ(2N+1,t+2N+1+L)
which leads to
(26)Q=a(1,t+1+L)a(1,t+2+L)⋯a(1,t+2N+1+L)a(2,t+1+L)a(2,t+2+L)⋯a(2,t+2N+1+L)⋮⋮⋮a(2N+1,t+1+L)a(2N+1,t+2+L)⋯a(2N+1,t+2N+1+L)
since 0≤t≤M and using the definition of a(i,j)s.
Soa(p,q+N+t+L+1)=d(p,q) for p≤2N+1 and |q|≤N. Therefore, for p≤2N+1 and |q|≤N, we have
(27)στN+t+L+1ai,j(i,j)∈N×Z=ai,j+N+t+L+1(i,j)∈N×Z;
hence, for (b(i,j))(i,j)∈N×Z:=στN+t+L+1(a(i,j))(i,j)∈N×Z, we have
(28)b(p,q)=a(p,q+N+t+L+1)=d(p,q).
So we have στN+t+L+1((a(i,j))(i,j)∈N×Z)∈V. Thus, {στn((a(i,j))(i,j)∈N×Z):n≥0}¯=NN×Z, and (NN×Z,στ) is point transitive.Definition 9.
The dynamical systems(Z,f) and (W,g) aretopologically semiconjugate, if there exists an onto and continuous map, μ:(Z,f)→(W,g) with μ∘f=g∘μ. Such a map μ:(Z,f)→(W,g) is called atopological semiconjugacy. We call (W,g) afactor of (Z,f) if there exists a topological semiconjugacy μ:(Z,f)→(W,g).Lemma 10.
In the dynamical systems(Z,f) and (W,g), if μ:(Z,f)→(W,g) is a topological semiconjugacy and z0∈Z is a transitive point of (Z,f), then μ(z0) is a transitive point of (W,g).
In particular, the factor of a point transitive dynamical system is point transitive too.Proof.
Supposeμ:(Z,f)→(W,g) is a topological semiconjugacy and z0∈Z is a point transitive of (Z,f); we have
(29)W=μZ=μOfz0¯⊆μOfz0¯=Ogμz0¯⊆W;
therefore, Og(μ(z0))¯=W, μ(z0) is a transitive point of (W,g), and (W,g) is point transitive.Lemma 11.
Forτ:N×Z→N×Z with τ(n,m)=(n,m+1) and countable X={θ1,θ2,…}, consider (a(i,j))(i,j)∈N×Z as in Lemma 8; then, (θa(n,m))(n,m)∈N×Z is a transitive point of (XN×Z,στ); in particular, (XN×Z,στ) is point transitive.Proof.
We use the same notation forστ:XN×Z→XN×Z and στ:NN×Z→NN×Z.
The mapg:(NN×Z,στ)→(XN×Z,στ) with
(30)gxn,mn,m∈N×Z=θxn,m(n,m)∈N×Z
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; supposeΓ is infinite countable, X={θ1,θ2,…} is countable, and (XΓ,σφ) is point transitive; hence, (XΓ,σφ) is topologically transitive and, by [2, Theorem 2.11], φ:Γ→Γ is one to one without any periodic point.Using the same notations as in Theorem3 and Definition 2 for D∈Λ(=Γ/~φ), we have D=αD/~φ=⋃‍{φn(αD):n∈Z} and D is infinite countable. Since Γ is countable, Λ is countable too and there exists a one to one map ϑ:Λ→N.Now defineζ:Γ→N×Z with ζ(φn(αD))=(ϑ(D),n) for all D∈Λ and n∈Z. Regarding ζ:Γ→N×Z, we have the following facts:(i)
ζ:Γ→N×Z is well defined, since for every γ∈Γ there exist unique D∈Λ with γ∈D and unique n∈Z with γ=φn(αD);(ii)
ζ:Γ→N×Z is one to one, since ϑ is one to one.Now consider (XN×Z,στ) as in Lemma 11. For γ∈Γ, choose n∈Z and D∈Λ with γ=φn(αD); therefore,
(31)ζφγ=ζφn+1αD=ϑD,n+1=τϑD,n=τζγ.
Consider υ:(XN×Z,στ)→(XΓ,σφ) defined by υ((x(n,m))(n,m)∈N×Z)=(xζ(γ))γ∈Γ; for (x(n,m))(n,m)∈N×Z, let (y(n,m))(n,m)∈N×Z=(x(n,m+1))(n,m)∈N×Z; we have
(32)υστxn,mn,m∈N×Z=υxτn,mn,m∈N×Z=υxn,m+1n,m∈N×Z=υyn,mn,m∈N×Z=yζγγ∈Γ=xτζγγ∈Γ=xζφγγ∈Γ=σφxζγγ∈Γ=σφυxαα∈Γ.
Consider (yγ)γ∈Γ∈XΓ and p∈X; let
(33)x(m,n)=yγm,n=ζγ,potherwise.
(This definition is well defined, since ζ:Γ→N×Z is one to one.)Then, we haveυ((x(m,n))(m,n)∈N×Z)=(yγ)γ∈Γ; thus, υ:(XN×Z,στ)→(XΓ,σφ) is a semiconjugacy; hence, by Lemma 11 and Definition 9 and using the same notations, (XΓ,σφ) is point transitive, and
(34)υθan,mn,m∈N×Z=θaζγγ∈Γ
is a transitive point of (XΓ,σφ).We recall that the dynamical system(Z,h) with metric space Z is calledDevaney chaotic if h:Z→Z is sensitive dependence to initial conditions, the collection of periodic points of h is dense in Z, and (Z,h) is topological transitive [7]. Now note the fact that, for metric space W with at least two elements, WΓ is metrizable if and only if Γ is countable [8].Remark 12.
It has been proved in [2, Theorem 2.13] that for infinite countable Γ the generalized shift dynamical system (XΓ,σφ) is Devaney chaotic, if and only if it is topological transitive.Using Remark12, Theorem 1, and the above construction, we have the following corollary, which shows an interaction between Devaney chaos and point transitivity in (XΓ,σφ).Corollary 13.
For infinite countableΓ in the dynamical system (XΓ,σφ), the following statements are equivalent: (i)
the system(XΓ,σφ) is point transitive,(ii)
the setX is countable and φ:Γ→Γ is one to one without any periodic point,(iii)
the setX is countable and the system (XΓ,σφ) is Devaney chaotic.Example 14.
Let us return to our well known one-sided shift in the Introduction, that is,({1,…,k}N,σκ) for κ:N→N with κ(n)=n+1 (n∈N), where k≥2. Since κ is one to one without any periodic point, ({1,…,k}N,σκ) is point transitive by Corollary 13 (note that σκ=σ in the Introduction). However, it is easy to see that(35)(1,2︸,1,1,1,2,2,1,2,2︸allpossiblepairsin{1,2},1,1,1,1,1,2,1,2,1,1,2,2,2,1,1,2,1,2,2,2,1,2,2,2︸allpossibletriplesin{1,2},…)is a transitive point of ({1,2}N,σκ). However, using our method of construction and notations, for φ=κ we have Λ={N}; thus, D=N is the unique element of Λ; we make the following choices (for fix m): (a)
αD=1,(b)
θi=i (i≥1),(c)
ϑ:Λ→N, with ϑ(D)=m.ζ:N→N×Z is ζ(n)=(m,n), so (θaζ(n))n∈N=(θa(m,n))n∈N is a transitive point of ({1,…,k}N,σκ); however, we may find different transitive points by choosing another m, αD, or θis in (a), (b), and (c).


## 3.1. Construction of a Point Transitive Generalized Shift Dynamical System

We denote the collection ofn×m matrices with the arrays in an arbitrary set A, by Matn×m(A).Definition 4.
For setA={x1,…,xn} with n elements, suppose
(6)M(A)∈Matn×nn(A)
is the following matrix up to rearrangement of x1,…,xn:(7)x1···x1︸nn-1times⋯xn···xn︸nn-1timesx1⋯x1︸nn-2timesx2⋯x2︸nn-2times⋯xn⋯xn︸nn-2times⋯x1⋯x1︸nn-2timesx2⋯x2︸nn-2times⋯xn⋯xn︸nn-2times⋮⋮⋮x1x2⋯xn⋯x1x2⋯xn⋯x1x2⋯xn⋯x1x2⋯xnn×nn.However, M(A) is well defined up to the rearrangement of x1,…,xn; that is, if ϱ:{1,…,n}→{1,…,n} is a permutation (on {1,…,n}), one may consider the following matrix as M(A) too, where yi=xϱ(i) for i=1,…,n (see Example 5):(8)y1···y1︸nn-1times⋯yn···yn︸nn-1timesy1⋯y1︸nn-2timesy2⋯y2︸nn-2times⋯yn⋯yn︸nn-2times⋯y1⋯y1︸nn-2timesy2⋯y2︸nn-2times⋯yn⋯yn︸nn-2times⋮⋮⋮y1y2⋯yn⋯y1y2⋯yn⋯y1y2⋯yn⋯y1y2⋯ynn×nn.Also, note that the columns of M(A) are pairwise distinct.Example 5.
ForA={1,2}, M(A) is any of the following matrices:
(9)11221212,22112121.Note 1.
Under the same notations as in Definition4, if M(A)=[a(i,j)]n×nn and b1,…,bn∈A, then there exists (a unique) j∈{1,…,nn} such that
(10)a1ja2j⋮anj=b1b2⋮bn.Definition 6.
Forp∈N, A:={1,…,p} and B:=Ap; then, M(B)∈Mat(pp)×(pp)(pp)(B); in other words, M(B) is a matrix whose arrays belong to Mat1×p(A). By removing partitions of the matrix M(B), we have a new matrix Kp∈Mat(pp)×p(pp)(pp)(A).
For example, letp=2; then, A={1,2} and B={(1,1),(1,2),(2,1),(2,2)} and we have the following matrix as M(B):(11)(1,1)···(1,1)︸43times⋯(2,2)···(2,2)︸43times1,1⋯(1,1)︸16times⋯(2,2)⋯(2,2)︸16times⋯(1,1)⋯(1,1)︸16times⋯(2,2)⋯(2,2)︸16times(1,1)⋯(1,1)︸4times⋯⋯(1,1)⋯(1,1)︸16times⋯(2,2)⋯(2,2)︸4times(1,1)(1,2)(2,1)(2,2)⋯⋯(1,1)⋯(1,1)︸16times⋯(1,1)(1,2)(2,1)(2,2)4×44so K2 is the following matrix:(12)1···1︸43×2times⋯2···2︸43×2times1···1︸32times⋯⋯⋯2···2︸32times11111111121212122121212122222222⋯⋯⋯1111111112121212212121212222222211122122111221221112212211122122⋯⋯⋯111221221112212211122122111221224×29.Note 2.
Using Note1 and Definition 6, if Δ=[d(i,j)]n×n is an n×n matrix with arrays in {1,…,n} and Kn=[kijn]nn×n(nn)(nn) (introduced in Definition 6), then there exists t∈{0,1,…,n(nn)(nn)-1} such that
(13)k1(t+1)nk1(t+2)n⋯k1(t+n)nk2(t+1)nk2(t+2)n⋯k2(t+n)n⋮⋮⋮kn(t+1)nkn(t+2)n⋯kn(t+n)n=d(1,1)d(1,2)⋯d(1,n)d(2,1)d(2,2)⋯d(2,n)⋮⋮⋮d(n,1)d(n,2)⋯d(n,n).
However there exists a unique column in M({1,…,n}n) equal to
(14)d(1,1)d(1,2)⋯d(1,n)d(2,1)d(2,2)⋯d(2,n)⋮d(n,1)d(n,2)⋯d(n,n)nn-ntimes11⋯1⋮11⋯1.Lemma 7.
IfV is a nonempty open subset of NN×Z, then there exist N∈N and Δ=[d(i,j)]1≤i≤2N+1,-N≤j≤N with 1≤d(i,j)≤N for all i,j, such that
(15)xi,j(i,j)∈N×Z∈NN×Z:xi,j(i,j)∈N×Z∈NN×Z∀i∈1,…,2N+1∀j∈-N,…,N-1,Nx(i,j)=d(i,j)
is a subset of V.Proof.
SupposeV is a nonempty open subset of NN×Z; here, we search for a suitable open subset of V, namely, W. Choose (a(i,j))(i,j)∈N×Z∈V. There exist (i1,j1),…,(ip,jp)∈N×Z such that
(16)V1:=xi,j(i,j)∈N×Z∈NN×Z:xi,j(i,j)∈N×Z∈NN×Z∀k∈{1,…,p}x(ik,jk)=a(ik,jk)⊆V.
Let
(17)N=max⁡{ik:k=1,…,p}∪{jk:k=1,…,p}aik,jk:k=1,…,p∪aik,jk:k=1,…,pd(i,j)=a(i,j)∃k∈{1,…,p}(i,j)=(ik,jk)1otherwiseW=xi,j(i,j)∈N×Z∈NN×Z:xi,ji,j∈N×Z∈NN×Z∀i∈1,…,2N+1∀j∈-N,…,Nxi,j=di,j.
Since for allk∈{1,…,p} we have ik≤N<2N+1, |jk|≤N, and a(ik,jk)=d(ik,jk), we have W⊆V1(⊆V). Moreover, using the definition of N and d(i,j)s, we have max⁡{d(i,j):i∈N,j∈Z}≤N.Convention 1.
Consider the matrix(18)Ψ=ψi,j(i,j)∈N×N=ψ(1,1)ψ(1,2)⋯ψ(2,1)ψ(2,2)⋯ψ(3,1)ψ(3,2)⋯⋮⋮⋮
such that
(19)Ψ=K1K2K3K4⋯1111⋮⋮⋮⋮⋱1111⋮⋮⋮⋮,
where Kn is introduced in Definition 6. In other words, for n∈N, if Kn=[kijn]nn×n(nn)(nn) (as in Definition 6), then
(20)ψ(i,j)=kisn1≤i≤nn,j=s+∑1≤p≤(n-1)‍ppp(pp),1≤i≤nn,j=s+1≤1≤s≤nnn(nn),1otherwise.Lemma 8.
Let(21)a(i,j)=ψ(i,j)i≥1,j≥1,1i≥1,j≤0.
Then, (a(i,j))(i,j)∈N×Z is a transitive point of the generalized shift dynamical system (NN×Z,στ) for τ:N×Z→N×Z with τ(n,m)=(n,m+1). In particular, (NN×Z,στ) is point transitive.Proof.
We just should prove that for all nonempty open subsetsV of NN×Z there exists n≥0 with στn((a(i,j))(i,j)∈N×Z)=(a(i,j+n))(i,j)∈N×Z∈V. Suppose V is a nonempty open subset of NN×Z. By Lemma 7, there exist N∈N and Δ=[d(i,j)]0≤i≤2N+1,-N≤j≤N such that
(22)xi,j(i,j)∈N×Z∈NN×Z:xi,j(i,j)∈N×Z∈NN×Z∀i∈1,…,2N+1∀j∈-N,…,N-1,Nxi,j=di,j
is a subset of V, and max⁡{d(i,j):1≤i≤2N+1,-N≤j≤N}≤N<2N+1. Let M:=(2N+1)((2N+1)(2N+1))((2N+1)(2N+1))-1 and let L:=∑1≤p≤2N‍p(pp)(pp). Using Note 2 for
(23)Q:=d(1,-N)d(1,-N+1)⋯d(1,N)d(2,-N)d(2,-N+1)⋯d(2,N)⋮⋮⋮d(2N+1,-N)d(2N+1,-N+1)⋯d(2N+1,N),
there exists t∈{0,…,M} such that
(24)Q=k1(t+1)2N+1k1(t+2)2N+1⋯k1(t+2N+1)2N+1k2(t+1)2N+1k2(t+2)2N+1⋯k2(t+2N+1)2N+1⋮⋮⋮k(2N+1)(t+1)2N+1k(2N+1)(t+2)2N+1⋯k(2N+1)(t+2N+1)2N+1.
Now according to Convention 1, we have
(25)Q=ψ(1,t+1+L)ψ(1,t+2+L)⋯ψ(1,t+2N+1+L)ψ(2,t+1+L)ψ(2,t+2+L)⋯ψ(2,t+2N+1+L)⋮⋮⋮ψ(2N+1,t+1+L)ψ(2N+1,t+2+L)⋯ψ(2N+1,t+2N+1+L)
which leads to
(26)Q=a(1,t+1+L)a(1,t+2+L)⋯a(1,t+2N+1+L)a(2,t+1+L)a(2,t+2+L)⋯a(2,t+2N+1+L)⋮⋮⋮a(2N+1,t+1+L)a(2N+1,t+2+L)⋯a(2N+1,t+2N+1+L)
since 0≤t≤M and using the definition of a(i,j)s.
Soa(p,q+N+t+L+1)=d(p,q) for p≤2N+1 and |q|≤N. Therefore, for p≤2N+1 and |q|≤N, we have
(27)στN+t+L+1ai,j(i,j)∈N×Z=ai,j+N+t+L+1(i,j)∈N×Z;
hence, for (b(i,j))(i,j)∈N×Z:=στN+t+L+1(a(i,j))(i,j)∈N×Z, we have
(28)b(p,q)=a(p,q+N+t+L+1)=d(p,q).
So we have στN+t+L+1((a(i,j))(i,j)∈N×Z)∈V. Thus, {στn((a(i,j))(i,j)∈N×Z):n≥0}¯=NN×Z, and (NN×Z,στ) is point transitive.Definition 9.
The dynamical systems(Z,f) and (W,g) aretopologically semiconjugate, if there exists an onto and continuous map, μ:(Z,f)→(W,g) with μ∘f=g∘μ. Such a map μ:(Z,f)→(W,g) is called atopological semiconjugacy. We call (W,g) afactor of (Z,f) if there exists a topological semiconjugacy μ:(Z,f)→(W,g).Lemma 10.
In the dynamical systems(Z,f) and (W,g), if μ:(Z,f)→(W,g) is a topological semiconjugacy and z0∈Z is a transitive point of (Z,f), then μ(z0) is a transitive point of (W,g).
In particular, the factor of a point transitive dynamical system is point transitive too.Proof.
Supposeμ:(Z,f)→(W,g) is a topological semiconjugacy and z0∈Z is a point transitive of (Z,f); we have
(29)W=μZ=μOfz0¯⊆μOfz0¯=Ogμz0¯⊆W;
therefore, Og(μ(z0))¯=W, μ(z0) is a transitive point of (W,g), and (W,g) is point transitive.Lemma 11.
Forτ:N×Z→N×Z with τ(n,m)=(n,m+1) and countable X={θ1,θ2,…}, consider (a(i,j))(i,j)∈N×Z as in Lemma 8; then, (θa(n,m))(n,m)∈N×Z is a transitive point of (XN×Z,στ); in particular, (XN×Z,στ) is point transitive.Proof.
We use the same notation forστ:XN×Z→XN×Z and στ:NN×Z→NN×Z.
The mapg:(NN×Z,στ)→(XN×Z,στ) with
(30)gxn,mn,m∈N×Z=θxn,m(n,m)∈N×Z
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; supposeΓ is infinite countable, X={θ1,θ2,…} is countable, and (XΓ,σφ) is point transitive; hence, (XΓ,σφ) is topologically transitive and, by [2, Theorem 2.11], φ:Γ→Γ is one to one without any periodic point.Using the same notations as in Theorem3 and Definition 2 for D∈Λ(=Γ/~φ), we have D=αD/~φ=⋃‍{φn(αD):n∈Z} and D is infinite countable. Since Γ is countable, Λ is countable too and there exists a one to one map ϑ:Λ→N.Now defineζ:Γ→N×Z with ζ(φn(αD))=(ϑ(D),n) for all D∈Λ and n∈Z. Regarding ζ:Γ→N×Z, we have the following facts:(i)
ζ:Γ→N×Z is well defined, since for every γ∈Γ there exist unique D∈Λ with γ∈D and unique n∈Z with γ=φn(αD);(ii)
ζ:Γ→N×Z is one to one, since ϑ is one to one.Now consider (XN×Z,στ) as in Lemma 11. For γ∈Γ, choose n∈Z and D∈Λ with γ=φn(αD); therefore,
(31)ζφγ=ζφn+1αD=ϑD,n+1=τϑD,n=τζγ.
Consider υ:(XN×Z,στ)→(XΓ,σφ) defined by υ((x(n,m))(n,m)∈N×Z)=(xζ(γ))γ∈Γ; for (x(n,m))(n,m)∈N×Z, let (y(n,m))(n,m)∈N×Z=(x(n,m+1))(n,m)∈N×Z; we have
(32)υστxn,mn,m∈N×Z=υxτn,mn,m∈N×Z=υxn,m+1n,m∈N×Z=υyn,mn,m∈N×Z=yζγγ∈Γ=xτζγγ∈Γ=xζφγγ∈Γ=σφxζγγ∈Γ=σφυxαα∈Γ.
Consider (yγ)γ∈Γ∈XΓ and p∈X; let
(33)x(m,n)=yγm,n=ζγ,potherwise.
(This definition is well defined, since ζ:Γ→N×Z is one to one.)Then, we haveυ((x(m,n))(m,n)∈N×Z)=(yγ)γ∈Γ; thus, υ:(XN×Z,στ)→(XΓ,σφ) is a semiconjugacy; hence, by Lemma 11 and Definition 9 and using the same notations, (XΓ,σφ) is point transitive, and
(34)υθan,mn,m∈N×Z=θaζγγ∈Γ
is a transitive point of (XΓ,σφ).We recall that the dynamical system(Z,h) with metric space Z is calledDevaney chaotic if h:Z→Z is sensitive dependence to initial conditions, the collection of periodic points of h is dense in Z, and (Z,h) is topological transitive [7]. Now note the fact that, for metric space W with at least two elements, WΓ is metrizable if and only if Γ is countable [8].Remark 12.
It has been proved in [2, Theorem 2.13] that for infinite countable Γ the generalized shift dynamical system (XΓ,σφ) is Devaney chaotic, if and only if it is topological transitive.Using Remark12, Theorem 1, and the above construction, we have the following corollary, which shows an interaction between Devaney chaos and point transitivity in (XΓ,σφ).Corollary 13.
For infinite countableΓ in the dynamical system (XΓ,σφ), the following statements are equivalent: (i)
the system(XΓ,σφ) is point transitive,(ii)
the setX is countable and φ:Γ→Γ is one to one without any periodic point,(iii)
the setX is countable and the system (XΓ,σφ) is Devaney chaotic.Example 14.
Let us return to our well known one-sided shift in the Introduction, that is,({1,…,k}N,σκ) for κ:N→N with κ(n)=n+1 (n∈N), where k≥2. Since κ is one to one without any periodic point, ({1,…,k}N,σκ) is point transitive by Corollary 13 (note that σκ=σ in the Introduction). However, it is easy to see that(35)(1,2︸,1,1,1,2,2,1,2,2︸allpossiblepairsin{1,2},1,1,1,1,1,2,1,2,1,1,2,2,2,1,1,2,1,2,2,2,1,2,2,2︸allpossibletriplesin{1,2},…)is a transitive point of ({1,2}N,σκ). However, using our method of construction and notations, for φ=κ we have Λ={N}; thus, D=N is the unique element of Λ; we make the following choices (for fix m): (a)
αD=1,(b)
θi=i (i≥1),(c)
ϑ:Λ→N, with ϑ(D)=m.ζ:N→N×Z is ζ(n)=(m,n), so (θaζ(n))n∈N=(θa(m,n))n∈N is a transitive point of ({1,…,k}N,σκ); however, we may find different transitive points by choosing another m, αD, or θis in (a), (b), and (c).



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*Source: 519357-2015-01-05.xml*