Int. Journal of Math. Analysis, Vol. 6, 2012, no. 14, 651 - 657

n-Tuples and Chaoticity

Mezban Habibi

Department of MathematicsDehdasht Branch, Islamic Azad University, Deahdasht, Iran

P.O. Box 181 40, Lidingo, Stockholm, Swedenhabibi.m@iaudehdasht.ac.ir

Abstract

In this paper we characterize the Condition for Chaoticity of Tuplesof operators on a Frechet space.

Mathematics Subject Classification: 47A16, 47B37

Keywords: Hypercyclic vector, Hypercyclicity Criterion, Periodic vector,Chaotic n-tuple, n-Tuple of operators

1 Introduction

Let T1, T2, ..., Tn be commutative bounded linear operators on a Banach spaceX . For n-Tuple T = (T1, T2, ..., Tn), put

Γ = {Tm11 Tm2

2 ...Tmnn : m1, m2, ...,mn ≥ 0}

the semigroup generated by T . For x ∈ X , the orbit of x under T is the setOrb(T , x) = {S(x) : S ∈ Γ}, that is

Orb(T , x) = {Tm11 Tm2

2 ...Tmnn (x) : m1, m2, ...,mn ≥ 0}

The vector x is called Hypercyclic vector for T and n-Tuple T is called Hy-percyclic n-Tuple, if the set Orb(T , x) is dense in X , that is

Orb(T , x) = {Tm11 Tm2

2 ...Tmn2 (x) : m1, m2, ...,mn ≥ 0} = X

The vector x in X is called a Periodic vector for the n-Tuple T = (T1, T2, ..., Tn),if there exist some numbers μ1, μ2, ..., μn ∈ N such that

T μ11 T μ2

2 ...T μnn (x) = x.

652 M. Habibi

Also the n-Tuple T = (T1, T2, ..., Tn), is called chaotic tuple, if we have treebelow conditions together,(1). It is topologically transitive, that is, if for any given open sets U and V,there exist positive integer numbers α1, α2, ..., αn ∈ N such that

T α11 T α2

2 ...Tαnn (U) ∩ V �= φ

(2). It has a dense set of periodic points, in other word, there is a set X suchthat for each x ∈ X , there exist some numbers β1, β2, ..., βn ∈ N such that

T β11 T β2

2 ...T βnn (x) = x

(3). It has a certain property called sensitive dependence on initial conditions.

Notice that, all operators in this paper are commutative. For some topics see[1–12].

2 Main Results

Theorem 2.1.[The Hypercyclicity Criterion] Let X be a separable Ba-nach space and T = (T1, T2, ..., Tn) is an n-tuple of continuous linear mappingson X . If there exist two dense subsets Y and Z in X , and n strictly increasingsequences {mj,1}, {mj,2}, ...,{mj,n} such that :1. T

mj,1

1 Tmj,2

2 ...Tmj,nn → 0 on Y as j → ∞,

2. There exist function {Sj : Z → X} such that for every z ∈ Z, Sjz → 0,and T

mj,1

1 Tmj,2

2 ...Tmj,nn Sjz → z,

then T is a Hypercyclic n-tuple.If the tuple T satisfying the hypothesis of previous theorem then we say thatT satisfying the Hypercyclicity criterion.

Theorem 2.2. Suppose X be an F-sequence space whit the unconditionalbasis {eκ}κ∈N . Let T1, T2, ..., Tn are unilateral weighted backward shifts withweight sequence {ai,1 : i ∈ N}, {ai,2 : i ∈ N},..., {ai,n : i ∈ N} and T =(T1, T2, ..., Tn) be an n-tuple of operators T1, T2, ..., Tn. Then the followingassertions are equivalent:(1). T is chaotic,(2). T is Hypercyclic and has a non-trivial periodic point,(3). T has a non-trivial periodic point,(4). the series Σ∞

m=1(∏m

k=1(ak,i)−1em) convergence in X for i = 1, 2, ..., n.

Proof. Proof of the cases (1) → (2) and (2) → (3) are trivial, so we justproof (3) → (4) and (4) → (1). First we proof (3) → (4), for this, Suppose

n-tuples and chaoticity 653

that T has a non-trivial periodic point, and x = {xn} ∈ X be a non-trivialperiodic point for T , that is there are μ1, μ2, ..., μn ∈ N such that,

T μ11 T μ2

2 ...T μnn (x) = x.

Comparing the entries at positions j+kN , and j+kN , k ∈ N ∪{0}, of x andTM

1 TN2 (x) we find that

xj+kM1 = (M∏

t=1

(aj+kN+t))xj+(k+1)

xj+kM2 = (N∏

t=1

(bj+kN+t))xj+(k+1)

...

xj+kMn = (N∏

t=1

(bj+kN+t))xj+(k+1)

so that we have,

xj+kM1 = (j+kM1∏

t=j+1

(at))−1xj = c1(

j+kM1∏

t=1

(at))−1, k ∈ N ∪ {0}

xj+kM2 = (j+kM2∏

t=j+1

(at))−1xj = c2(

j+kM2∏

t=1

(at))−1, k ∈ N ∪ {0}

...

xj+kMn = (j+kMn∏

t=j+1

(at))−1xj = cn(

j+kMn∏

t=1

(at))−1, k ∈ N ∪ {0}

with

c1 = (j∏

t=1

(mj,1))xj

c2 = (j∏

t=1

(mj,2))xj

...

cn = (j∏

t=1

(mj,n))xj .

Since {eκ} is an unconditional basis and x ∈ X it follows from [...] that

∞∑

k=0

(1

∏j+kM1t=1 (mj,1)

)ej+kM1 =1

c1

∞∑

k=0

xj+kM1.ej+kM1

654 M. Habibi

∞∑

k=0

(1

∏j+kM2t=1 (mj,2)

)ej+kM2 =1

c2

∞∑

k=0

xj+kM2.ej+kM2

...∞∑

k=0

(1

∏j+kMnt=1 (mj,n)

)ej+kMn =1

cn

∞∑

k=0

xj+kMn.ej+kMn

convergence in X . Without loss of generality we may assume that j ≥ N . Ap-plying the operators T , T 2, T 3,..., TR−1, with R = Min{Mi : i = 1, 2, ..., n},to this series and note that T1(en) = anen−1 and T2(en) = bnen−1 for n ≥ 2,we deduce that ∞∑

k=0

(1

∏j+kM1−υ1t=1 (mj,1)

)ej+kM1−υ1

∞∑

k=0

(1

∏j+kM2−υ2t=1 (mj,2)

)ej+kM2−υ2

...∞∑

k=0

(1

∏j+kMn−υ1t=1 (mj,n)

)ej+kMn−υn

convergence in X for γ = 0, 1, 2, ..., N − 1. By adding these series, we see thatcondition (4) holds.Proof of (4) ⇒ (1). It follows from theorem (2.1), that under condition (4)the operator T is Hypercyclic. Hence it remains to show that T has a denseset of periodic points. Since {eκ} is an unconditional basis, condition (4) withproposition 2.3 implies that for each j ∈ N and M,N ∈ N the series

ψ1(j,M1) =∞∑

k=0

(1

∏j+kM1t=1 (mk,1)

)ej+kM1 = (j∏

t=1

mk,1).(∞∑

k=0

1∏j+kM1

t=1 mk,1

ej+kM1)

ψ2(j,M2) =∞∑

k=0

(1

∏j+kM2t=1 (mk,2)

)ej+kM2 = (j∏

t=1

mk,2).(∞∑

k=0

1∏j+kM2

t=1 mk,2

ej+kM2)

...

ψn(j,Mn) =∞∑

k=0

(1

∏j+kMnt=1 (mk,n)

)ej+kMn = (j∏

t=1

mk,n).(∞∑

k=0

1∏j+kMn

t=1 mk,n

ej+kMn)

converges and define n elements in X . Moreover, if M ≥ i then

Tmj,1

1 Tmj,2

2 ...Tmj,nn = ψ1(j,M1)

Tmj,1

1 Tmj,2

2 ...Tmj,nn = ψ2(j,M2)

...

Tmj,1

1 Tmj,2

2 ...Tmj,nn = ψn(j,Mn)

Tmj,1

1 Tmj,2

2 ...Tmj,nn = ψ1(j,M1)T

M11 TM2

2 ...TMnn ψ1(j,M1) = ω(j,M1) (1)

n-tuples and chaoticity 655

Tmj,1

1 Tmj,2

2 ...Tmj,nn = ψ2(j,M2)T

M11 TM2

2 ...TMnn ψ2(j,M2) = ω(j,M2)

...

Tmj,1

1 Tmj,2

2 ...Tmj,nn = ψ1(j,M1)T

M11 TM2

2 ...TMnn ψn(j,Mn) = ω(j,Mn)

and, if N ≥ ji then

Tmj,1

1 Tmj,2

2 ...Tmj,nn ω((j, i), N) = ω((j, i), N) (2)

for mj,i ≥ N and i = 1, 2, ..., n. So that each ψ(j,N) for j ≤ N is a periodicpoint for T . We shall show that T has a dense set of periodic points. Since{eκ} is a basis, it suffices to show that for every element x ∈ span{eκ : κ ∈ N}there is a periodic point y arbitrarily close to it. For this, let x =

∑mj=1 xjej

and ε > 0. We can assume without lost of generality that

| xi

i∏

t=1

at,1 |≤ 1 , i = 1, 2, 3, ...,m1

| xj

i∏

t=1

at,2 |≤ 1 , i = 1, 2, 3, ...,m2

...

| xi

i∏

t=1

at,n |≤ 1 , i = 1, 2, 3, ...,mn

Since {en} is an unconditional basis, then condition (4) implies that there arean M,N ≥ m such that

‖∞∑

n=M1+1

εκ,11

∏κt=1 at,1

eκ‖ <ε

m1

‖∞∑

n=M2+1

εκ,21

∏κt=1 at,2

eκ‖ <ε

m2

...

‖∞∑

n=Mn+1

εκ,n1

∏κt=1 at,n

eκ‖ <ε

mn

for every sequences {εκ,i}, i = 1, 2, ..., n taking values 0 or 1. By (1) and (2)the elements

y1 =m1∑

i=1

xiψ(i,M1)

y2 =m2∑

i=1

xiψ(i,M2)

...

656 M. Habibi

yn =mn∑

i=1

xiψ(i,Mn)

of X is a periodic point for T , and we have‖yλ − x‖ = ‖∑mλ

i=1 xi(ψ(i,Mλ) − ei)‖= ‖∑mλ

i=1(xi∏i

t=1 dt,Mλ)(

∑∞k=1

1∏i+kMλt=1

at,Mλ

ei+kMλ)‖

≤ ∑mλi=1 ‖(xi

∏it=1 dt,Mλ

)(∑∞

k=11∏i+kMλ

t=1at,Mλ

ei+kMλ)‖

≤ ∑mλi=1 ‖(

∑∞k=1

1∏i+kMλt=1

at,Mλ

ei+kMλ)‖

≤ εas λ = 1, 2, ..., n and by this, the proof is complete.

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n-tuples and chaoticity 657

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Received: September, 2011