University of Mazandaran, Iran
http://cjms.journals.umz.ac.ir
ISSN: 1735-0611 CJMS.4(2)(2015), 197-204
Pointwise almost periodicity in in a generalized shift dynamical system
Fatemah Ayatollah Zadeh Shirazi1 and Meysam Miralaei 2 1 Faculty of Math., Stat. and Computer Science, College of Science,
University of Tehran, Tehran, Iran ([email protected]) 2 Department of Mathematical Sciences, Isfahan University of
Technology, Isfahan, Iran ([email protected])
Abstract. In the following text we prove that in a generalized
shift dynamical system (XΓ, σϕ) for discrete X with at least two elements, arbitrary nonempty Γ and bijectionϕ: Γ→Γ, the fol-lowing statements are equivalent:
• (XΓ, σ
ϕ) is pointwise recurrent;
• (XΓ, σϕ) is pointwise almost periodic;
• (XΓ, σϕ) is pointwise regularly almost periodic;
• (XΓ, σ
ϕ) is compactly almost periodic;
• Per(ϕ) = Γ (ϕ: Γ→Γ is pointwise periodic).
Keywords: Almost periodic, Compactly almost periodic, Com-pactly recurrent, Dynamical system, Generalized shift, Periodic, Pointwise almost periodic, Pointwise periodic, Pointwise regularly almost periodic, Pointwise recurrent, Recurrent.
2000 Mathematics subject classification: 37B05, 54H20.
1. Preliminaries
LetY be an arbitrary set. We call the collection F of subsets ofY ×Y a uniformity on Y if:
• for all α∈ F we have ∆Y ⊆α;
• for all α, β ∈ F we have α∩β ∈ F;
1Corresponding author: [email protected] Received: 11 July 2014
Revised: 25 April 2015 Accepted: 24 May 2015
• for all α∈ F and β ⊆Y ×Y withα⊆β we have β ∈ F;
• for all α∈ F there existsβ ∈ F with β◦β−1 ⊆α;
where ∆Y = {(y, y) : y ∈ Y}, α◦ β = {(x, y) : there exists z such that (x, z) ∈ α and (z, y) ∈ β}, and α−1 = {(y, x) : (x, y) ∈ α} for α, β⊆Y ×Y. We call an element of uniformity F an index. Moreover, forα∈ F and x∈Y let α[x] ={y: (x, y)∈α}.
IfF is a uniformity onY, we call (Y,F) a uniform space and equip it with topologyT ={U ⊆Y : for allx∈U there existsα∈ F withα[x]⊆U}
(topology generated byF). For nonempty set Λ if{(Yθ,Fθ) :θ∈Λ} is a collection of uniform spaces, then Y
θ∈Λ
Yθ under product topology is a
uniform space too, and we may consider the following uniformity over it:
{γ ⊆Y
θ∈Λ Yθ×
Y
θ∈Λ
Yθ: there exist θ1, . . . , θn∈Λ and
α1 ∈ Fθ1, . . . , αn∈ Fθn withγ ⊆κ(α1,···,αn)},
whereκ(α1,···,αn) is the following set:
{((xθ)θ∈Λ,(yθ)θ∈Λ)∈
Y
θ∈Λ Yθ×
Y
θ∈Λ
Yθ:∀i∈ {1, . . . , n}((xθi, yθi)∈αi)}.
The topological space W is uniformazable if there exists a uniformity
G on W such that the topology generated by G coincides with original topology ofW and in this case we callG an admissible uniformity onW. IfY is compact Hausdorff, then it admits a unique admissible uniformity
{α⊆Y×Y : ∆Y is a subset of the interior ofα}. See [7] for more details.
By a (topological) dynamical system ((Z, µ), h) or briefly (Z, h) we mean a Hausdorff uniform topological space Z (phase space) equipped with uniformity µ and a homeomorphism h : Z → Z. In dynamical system (Z, h), we call nonempty subset W invariant if h(W) =W. We call the dynamical system (Z, h) [8], [9]:
•periodic, if there existsn≥1 withhn= idZ, where idZ:Z →Z is the identity map, idZ(x) =x,x∈Z;
•pointwise periodic, if for all x∈Z there existsn≥1 with hn(x) =x;
•pointwise recurrent, if for all z∈Z and all open neighborhood U of z there existsn≥1 such thathn(z)∈U;
• pointwise almost periodic, if for all z ∈ Z and all open neighbor-hood U of z, there exists N ≥ 1 such that for all p ∈ Z there exists
n∈ {p, p+ 1, . . . , p+N −1} withhn(z)∈U;
•pointwise regularly almost periodic, if for allz∈Z and all open neigh-borhood U of z, there exists n∈ Z\ {0} such thathnm(z) ∈ U for all
• recurrent (or uniformly recurrent) if for all α ∈ µ, there existsn ≥1 with{(hn(z), z) :z∈X} ⊆α;
•almost periodic(oruniformly almost periodic), if for allα∈µ, there ex-istsN ≥1 such that for allp∈Zthere existsn∈ {p, p+1, . . . , p+N−1}
with{(hn(z), z) :z∈Z} ⊆α;
•regularly almost periodic(oruniformly regularly almost periodic), if for all α ∈ µ, there exists n∈Z\ {0} such that {(hnm(z), z) : z ∈Z, m ∈ N} ⊆α;
•compactly almost periodic(oruniformly compactly almost periodic), if for all compact subsetB ofZ,S
{hn(B) :n∈Z} is compact and for all compact invariant subsetW of Z, (W, hW) is almost periodic;
•compactly recurrent (oruniformly compactly recurrent), if for all com-pact subset B of Z, S
{hn(B) :n∈Z} is compact and for all compact invariant subset W of Z, (W, h W) is recurrent (the concept of com-pactly recurrence is introduced here, imitating the concept of comcom-pactly almost periodicity in [9]). For nonempty arbitrary sets Γ, X and map ϕ : Γ → Γ, we call σϕ : XΓ → XΓ with σϕ((xα)α∈Γ) = (xϕ(α))α∈Γ (for (xα)α∈Γ ∈ XΓ), a generalized shift [3]. Whenever Γ = N and
ϕ(n) = n+ 1 (n ∈ N), σϕ :XN → XN is the familiar one sided shift, also whenever Γ =Z and ϕ(n) =n+ 1 (n∈Z), σϕ :XZ → XZ is the
well-known two sided shift. On the other hand, if X is a topological space andXΓ is equipped with product topology, then σ
ϕ :XΓ → XΓ is continuous.
Remark 1.1. For nonempty arbitrary sets Γ,Xand mapϕ: Γ→Γ, with
|X| ≥ 2, the map σϕ : XΓ → XΓ is bijective if and only if ϕ : Γ → Γ is bijective. Hence ifX is a topological space andXΓ is equipped with product topology, thenσϕ :XΓ→XΓ is a homeomorphism if and only ifϕ: Γ→Γ is bijective.
For mapping f :A→A, we calla∈Aa periodic point of f :A→ Aif there existsn≥1 withfn(a) =a. Let Per(f) ={a∈A:ais a periodic point off :A→A}.
In the following text suppose X is a discrete topological space with at least two elements,Γis an infinite set,ϕ: Γ→Γis bijec-tive, and consider XΓ under product (pointwise convergence) topology.
2. Pointwise periodicity in generalized shift dynamical systems
Remark 2.1. For maps λ, θ : Γ → Γ, we have σλ = σθ if and only if λ=θ.
Proof. If θ 6=λ, there exists β ∈ Γ with θ(β) 6= λ(β). Choose distinct p, q∈X, and letxα=pforα 6=θ(β) andxθ(β)=q. Then for (yα)α∈Γ:= σθ((xα)α∈Γ) and (zα)α∈Γ := σλ((xα)α∈Γ) we have zβ = xθ(β) = q and yβ = xλ(β) = p (since λ(β) 6= θ(β)). So zβ =6 yβ and σθ((xα)α∈Γ) 6= σλ((xα)α∈Γ), which leads toσθ 6=σλ and completes the proof.
Remark 2.2. For mapsλ, θ: Γ→Γ, we haveσλ◦σθ =σθ◦λ.
Theorem 2.3. In the generalized shift dynamical system (XΓ, σϕ), the
following statements are equivalent:
1. (XΓ, σϕ) is periodic (i.e., there exists n ≥ 1 such that σnϕ = idXΓ);
2. (XΓ, σ
ϕ) is pointwise periodic (i.e., Per(σϕ) =XΓ);
3. ϕ: Γ→Γ is periodic (i.e., there exists m≥1 with ϕm = idΓ).
Proof. It is clear that if (XΓ, σ
ϕ) is periodic, then it is pointwise periodic. Now suppose (XΓ, σϕ) is pointwise periodic and choose distinctp, q∈X. Suppose β ∈ Γ. Let xα = p for α 6= β and xβ = q. Since σϕ is pointwise periodic, there existsn≥1 with σϕn((xα)α∈Γ) = (xα)α∈Γ. So
(xα)α∈Γ = (xϕn(α))α∈Γ and q =xβ =xϕn(β) which leads to ϕn(β) =β. Hence ϕ: Γ→Γ is pointwise periodic. For α ∈Γ let nα= min{n≥1 : ϕn(α) =α}, it’s evident that for allα∈Γ we have nα =nϕ(α) (note to the fact that ϕ : Γ → Γ is bijective). In the following claim we prove that sup{nα:α ∈Γ} is finite.
Claim. sup{nα:α∈Γ}<∞.
Proof of Claim. If sup{nα:α∈Γ}= +∞, then there exists a strinctly increasing sequence{nαk}k∈N. Let:
xα =
q α∈ {αk:k∈N},
p otherwise.
Since (XΓ, σϕ) is pointwise periodic, there exists m ≥ 1 with σϕm((xα)α∈Γ) = (xα)α∈Γ. So for all k≥ 1 we have q =xαk = xϕm(αk), which leads to ϕm(αk) ∈ {αl :l ∈ N}, using nϕm(α) =nα and the fact that{nαl}l∈Nis one to one, we concludeϕm(α) =α. Byϕm(α) =αwe have m≥nα, so sup{nα :α ∈Γ} ≤m, which is a contradiction, hence sup{nα :α∈Γ}<∞.
IfN = sup{nα :α∈Γ}, then for allα∈Γ we haveϕN!(α) =α. There-fore, ϕN!= idΓ, and ϕ: Γ→Γ is periodic.
In order to complete the proof of theorem, note to the fact that if ϕn= id
Γ, using Remarks 2.1 and 2.2 we haveσnϕ =σϕn =σid
Γ = idXΓ,
3. Pointwise almost periodicity in generalized shift dynamical systems
In this section we prove that (XΓ, σϕ) is any of pointwise recurrent, pointwise almost periodic, pointwise regularly almost periodic, com-pactly recurrent, comcom-pactly almost periodic if and only if Per(ϕ) = Γ.
Lemma 3.1. If (XΓ, σϕ) is pointwise recurrent, then Per(ϕ) = Γ.
Proof. Suppose θ∈Γ and (XΓ, σϕ) is pointwise recurrent. Choose dis-tinctp, q∈X and let:
Uα =
{p} α=θ ,
X α6=θ , and xα =
p α=θ ,
q α6=θ . (*)
Since (XΓ, σϕ) is pointwise recurrent and Y α∈Γ
Uα is an open
neighbor-hood of (xα)α∈Γ there exists n ≥ 1 with (xϕn(α))α∈Γ = σϕn(xα)α∈Γ ∈
Y
α∈Γ
Uα. In particular xϕn(θ) ∈ Uθ = {p} and xϕn(θ) =p which leads to
ϕn(θ) =θby (*), and θis periodic under ϕ.
Lemma 3.2. IfPer(ϕ) = Γ, then(XΓ, σϕ)is pointwise regularly almost
periodic.
Proof. Suppose Per(ϕ) = Γ. For w = (wα)α∈Γ ∈ XΓ if U is an open neighborhood ofw, then there existθ1, . . . , θk∈Γ such that
Y
α∈Γ
Uα⊆U,
where:
Uα =
{wα} α=θ1, . . . , θk, X otherwise.
For all i ∈ {1, . . . , k} there exists ri ≥ 1 such that ϕri(θi) = θi. For all i ∈ {1, . . . , k} and t ∈ Z we have ϕr1···rkt(θ
i) = θi, moreover if (yα)α∈Γ=σϕr1···rkt((wα)α∈Γ) = (wϕr1···rkt(α))α∈Γ, then fori= 1, . . . , kwe
have yθi = wϕr1···rkt(θi) = wθi, which leads to (yα)α∈Γ ∈
Y
α∈Γ
Uα(⊆ U)
and completes the proof.
Lemma 3.3. If Per(ϕ) = Γ, and B is a compact subset of XΓ, then for eachα∈Γthere exists finite subsetDαofX such thatB ⊆ Y
α∈Γ
Dα=:D
and Dβ =Dϕ(β) for all β ∈Γ, hence σϕ(D) =D.
Proof. Forα∈Γ let orb(ϕ, α) :={ϕn(α) :n∈Z}supposeπα:XΓ→X
and hence finite subset ofX. Thus Dα := S
{πβ(B) :β ∈orb(ϕ, α)} is finite and Dα = Dβ for all β ∈ orb(ϕ, α) in particular, Dα = Dϕ(α) =
Dϕ−1(α). Moreover, σϕ(D) = Y
α∈Γ
Dϕ(α)= Y α∈Γ
Dα=D.
Lemma 3.4. If Per(ϕ) = Γ, and B is a compact subset of XΓ, then
{σn
ϕ(B) :n∈Z} is compact.
Proof. Consider D = Y α∈Γ
Dα ⊇B as in Lemma 3.3. By the Tykhonoff
theoremDis a compact and hence closed subset ofXΓ. Usingσϕ(D) = Dand B⊆Dwe have{σn
ϕ(B) :n∈Z} ⊆D=D, thus{σϕn(B) :n∈Z}
is compact.
Lemma 3.5. If Per(ϕ) = Γ, and B is a compact invariant subset of
(XΓ, σϕ), then (B, σϕ B) is regularly almost periodic. In particular (B, σϕ B) is almost periodic and recurrent.
Proof. For H⊆Γ let
βH :={((xα)α∈Γ,(yα)α∈Γ) : (xα, yα)α∈Γ∈
Y
λ∈Γ
αλ} (*)
whereαλ ={(w, w) :w∈X}forλ∈H andαλ =X×Xforλ∈Γ\H. Consider D= Y
α∈Γ
Dα ⊇ B as in Lemma 3.3. We recall that since Dλs
and Y λ∈Λ
Dλ are compact Hausdorff, they admit a unique uniformity.
If α is an index of Y λ∈Λ
Dλ, then there exist λ1, . . . , λm ∈ Γ such that
β{λ1,...,λm}∩(D×D) ⊆α. Since Per(ϕ) = Γ, there exists n ∈N such thatϕn(λi) =λi for all i∈ {1, . . . , m}. For all x= (xλ)λ∈Γ∈D,k∈Z,
and for y= (yλ)λ∈Γ=σϕkn(x) = (xϕkn(λ))λ∈Γ we have:
∀i∈ {1, . . . , m} yλi =xϕkn(λi) =xλi ∈Dλi,
which leads to (x, σknϕ (x)) = (x, y) ∈ β{λ1,...,λm} ∩(D ×D). Hence (x, σϕkn(x))∈α for all x ∈D, k∈ Zand (D, σϕ D) is regularly almost periodic, therefore (B, σϕ B) is regularly almost periodic.
Corollary 3.6. By Lemmas 3.4 and 3.5, if Per(ϕ) = Γ, then (XΓ, σϕ)
is compactly almost periodic and compactly recurrent.
Theorem 3.7 (Main Theorem). The following statements are equiva-lent:
• (XΓ, σϕ) is pointwise regularly almost periodic; • (XΓ, σϕ) is compactly almost periodic;
• (XΓ, σϕ) is compactly recurrent;
• Per(ϕ) = Γ (ϕ: Γ→Γ is pointwise periodic).
Proof. Use Lemmas 3.1, 3.2, Corollary 3.6 and the fact that if (XΓ, σϕ) is compactly almost periodic, then it is pointwise almost periodic and if (XΓ, σ
ϕ) is pointwise almost periodic or pointwise regularly almost periodic, then it is pointwise recurrent.
Uniformly almost periodicity in generalized shift dynamical systems. LetX is finite, thenXΓis compact, hence by Lemma 3.5 and Theorem 3.7 the following statements are equivalent:
• (XΓ, σϕ) is pointwise recurrent;
• (XΓ, σϕ) is pointwise almost periodic;
• (XΓ, σϕ) is pointwise regularly almost periodic;
• (XΓ, σϕ) is recurrent;
• (XΓ, σ
ϕ) is almost periodic;
• (XΓ, σϕ) is regularly almost periodic;
• Per(ϕ) = Γ.
However even for infinite X if we equip XΓ with uniformity generated by basis {βH : H is a finite subset of Γ} (βHs are defined in (*) in Lemma 3.5), then the above statements are equivalent, using a similar method described in this section and Lemma 3.5.
Example 3.8. Define ϕ :N → N with ϕ(1) = 1, ϕ(k) = k+ 1
when-ever 2n ≤ k ≤ 2n+1−1 and ϕ(2n+1 −1) = 2n for n ∈ N. Then ϕ is pointwise periodic and it is not periodic, hence (XN, σ
ϕ) is pointwise al-most periodic and satisfies all equivalent conditions of Theorem 3.7, but (XN, σϕ) is not pointwise periodic and does not satisfy any of equivalent conditions of Theorem 2.3.
For more details on generalized shifts’ properties see [5] (dynamical prop-erties), [1], [6] (algebraic entropy of generalized shifts) and [2] (topolog-ical entropy).
4. Acknowledgement
The authors are grateful to the research division of the University of Tehran, for the grant which supported this research under the ref. no. 6103027/1/07.
and Dynamic Systems 6-7 November 2013, University of Mazandaran, Babolsar, Iran” [4].
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