# Devaney Chaos and Distributional Chaos in the Solution of Certain Partial Differential Equations

Authors: Xavier Barrachina; J. Alberto Conejero
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

The notion of distributional chaos has been recently added to the study of the linear
dynamics of operators andC0-semigroups of operators. We will study this notion of chaos for some
examples of C0-semigroups that are already known to be Devaney chaotic.


---

## Body

## 1. Introduction

During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others. These notions have been extended, as far as possible, to the setting ofC0-semigroups of linear and continuous operators. The recent monograph by Grosse-Erdmann and Peris Manguillot [1] is a good reference for researchers interested in the study of linear dynamics. In particular, it contains a chapter dedicated to analyze the dynamics of C0-semigroups. See also [2], which contains additional information on further topics in the area.In the sequel, letX be an infinite-dimensional separable Banach space. A C0-semigroup 𝒯 is a family of linear and continuous operators {Tt}t≥0⊂L(X) such that T0=Id, TtTs=Tt+s for all t,s≥0, and for all s≥0, we have limt→sTt=Ts pointwise on X.We say that aC0-semigroup 𝒯 is Devaney chaotic if it is transitive and it has a dense set of periodic points. On the one hand, a C0-semigroup 𝒯 is transitive if for any pair of nonempty open sets U,V⊂X there is some t>0 such that Tt(U)∩V≠∅. In this setting, transitivity is equivalent to the existence of some x∈X with dense orbit in X, that is, {Ttx,t≥0}¯=X, see for instance [1, Th. 1.57]. This phenomenon is usually known in operator theory as hypercyclicity, and such a vector x is said to be a hypercyclic vector for 𝒯. On the other hand, a vector x∈X is said to be a periodic point for 𝒯 if there is some t>0 such that Ttx=x.Other definitions of chaos, such as the one introduced by Li-Yorke and the one of distributional chaos introduced by Schweizer and Smítal, have been also considered. The relationships between these two notions in the Banach space setting have been recently studied in [3]. We recall that a C0-semigroup 𝒯 is said to be Li-Yorke chaotic if there exists an uncountable subset Γ⊂X, called the scrambled set, such that for every pair x,y∈Γ of distinct points, we have that
(1.1)liminft→∞||Ttx-Tty||=0,limsupt→∞||Ttx-Tty||>0.
Clearly, every hypercyclic C0-semigroup is Li-Yorke chaotic: we just have to fix a hypercyclic vector x∈X and consider Γ:={λx;|λ|≤1} as a scrambled set, as it is indicated in [4, page 84].Distributional chaos is inspired by the notion of Li-Yorke chaos. In order to define it, given a subset B⊂ℝ0+, we define its upper density as Dens¯(B):=limsupt→∞(1/t)μ(B∩[0,t]), where μ stands for the Lebesgue measure on ℝ0+.Definition 1.1.
AC0-semigroup 𝒯 in L(X) is distributionally chaotic if there are an uncountable set S⊂X and δ>0, so that for each ε>0 and each pair x,y∈S of distinct points, we have
(1.2)Dens¯({s≥0:||Tsx-Tsy||≥δ})=1,Dens¯({s≥0:||Tsx-Tsy||<ε})=1.The setS is called the scrambled set. If S is dense in X, then 𝒯 is said to be densely distributionally chaotic.A vectorx∈X is said to be distributionally irregular for 𝒯 if for every δ>0(1.3)Dens¯({s≥0:||Tsx||>δ})=1,Dens¯({s≥0:||Tsx||<δ})=1.Such vectors were considered in [5] so as to get a further insight into the phenomenon of distributional chaos, showing the equivalence between a distributionally chaotic operator and an operator having a distributionally irregular vector.The first systematic approach to distributional chaos for linear operators was taken in [6], where this phenomenon was studied in detail for backward shift operators. Later, Peris and Barrachina proved that for translation C0-semigroups on weighted Lp spaces, 1≤p<∞, Devaney chaos implies distributional chaos. However, the converse does not hold. They also provide an example of a translation C0-semigroup that is distributionally chaotic but it is neither Devaney chaotic nor hypercyclic [7].Hypercyclicity and Devaney chaos are hard to observe directly from the definition. The Hypercyclicity Criterion, in any of its forms, and the Desch-Schappacher-Webb Criterion have turn out to be powerful tools in order to verify these properties. Very recently, Albanese et al. have stated a criterion in order to show that aC0-semigroup is distributionally chaotic (and has a dense distributionally irregular manifold) [8]. Our goal is to study distributional chaos for some C0-semigroups that are already known to be Devaney chaotic. The dynamics exhibited by these C0-semigroups will motivate us to pose some open questions.


## 2. Criteria to Determine Devaney Chaos and Distributional Chaos

The following statement of the Hypercyclicity Criterion forC0-semigroups is inspired by its version for operators in [9].Theorem 2.1 (Hypercyclicity Criterion forC0-semigroups; see [10, Th. 2.1], [11, Crit. 3.1], and [1, Th. 7.26]).
Let𝒯 be a C0-semigroup in L(X). If there are a sequence {tn}n⊂ℝ+ with limn→∞tn=∞, dense subsets Y,Z⊂X and maps Stn:Z→X,n∈ℕ such that(i)
limn→∞Ttny=0 for all y∈Y,(ii)
limn→∞Stnz=0 for all z∈Z, and(iii)
limn→∞TtnStnz=z for all z∈Z, then 𝒯 is hypercyclic.Sometimes the Hypercyclicity Criterion is hard to be applied, and in fact, it only provides one of the ingredients of Devaney chaos. Moreover, in many situations, we can have the infinitesimal generator of aC0-semigroup but we do not have the explicit representation of its operators. This is quite common when we deal with the solution C0-semigroups associated to certain partial differential equations. Desch et al. gave a criterion which permits us to state the Devaney chaos of a C0-semigroup in terms of the abundance of eigenvectors of the infinitesimal generator.Theorem 2.2 (Desch-Schappacher-Webb Criterion; see [12, 13]).
LetX be a complex separable Banach space, and let 𝒯 be a C0-semigroup in L(X) with infinitesimal generator (A,D(A)). Assume that there exist an open connected subset U⊂ℂ and a weak holomorphic function f:U→X, such that(i)
U∩iℝ≠∅,(ii)
f(λ)∈ker(λI-A) for every λ∈U, and(iii)
for anyx*∈X*, if 〈f(λ),x*〉=0 for all λ∈U, then x*=0.Then the semigroup 𝒯 is chaotic.For the case of distributional chaos, Albanese et al. obtained the following sufficient condition, inspired by the result for the discrete case given by Bermúdez et al. in [4].Theorem 2.3 (Dense Distributionally Irregular Manifold Criterion; see [8, Cor. 2]).
Let𝒯 be a C0-semigroup in L(X). Assume that there exist (i)
a dense subsetX0⊂X such that limt→∞Ttx=0 for each x∈X0, and(ii)
a Lebesgue measurable subsetB⊂ℝ0+ with Dens¯(B)=1 satisfying either ∫B∥Tt∥-1dt<∞, or X is a complex Hilbert space and ∫B‍∥Tt∥-2dt<∞.Then 𝒯  has a dense manifold whose nonzero vectors are distributionally irregular vectors. (When this happens, one says that 𝒯 has a dense distributionally irregular manifold).Furthermore, they also proved that aC0-semigroup 𝒯 is distributionally chaotic if, and only if, 𝒯 has a distributionally irregular vector [8, Th. 3.4]. Therefore, Theorem 2.3 can be also understood as a criterion for distributional chaos. In the sequel, we will apply this criterion several times in order to determine that certain C0-semigroups are distributionally chaotic.


## 3. Distributionally ChaoticC0-Semigroups

In this section, we consider several examples ofC0-semigroups that are already known to be Devaney chaotic, and we will study when they exhibit distributional chaos. These examples will be considered on the following spaces:
(3.1)Lρp(I,ℂ)={f∈ℳ(I,ℂ):||f||p,ρ=(∫I|f(s)|pρ(s)ds)1/p<∞},with1≤p<∞,
where I is an interval on ℝ and ρ a weight function. If ρ(x)=1, then we will simply denote it as Lp(I,ℂ), 1≤p<∞. The hypothesis on ρ may be different on each example.In [14], Takeo considered the following first order abstract Cauchy problem on Lp(I,ℂ), 1≤p<∞:
(3.2)∂u∂t=ζ(x)∂u∂x+h(x)u,u(0,x)=f(x),x∈I,
where ζ and h are bounded continuous functions defined on I. This ordinary differential equation has been used to model the dynamics of a population of cells under simultaneous proliferation and maturation [15]. When ζ(x) is constant and equal to 1 and I=ℝ0+, the solution C0-semigroup {Tt}t≥0 of (3.2) is defined as
(3.3)Ttf(x)=exp(∫xx+th(s)ds)f(x+t),∀x,t≥0,f∈Lp(ℝ0+,ℂ).Theorem 3.1.
Ifh(x) is a real function and there is a measurable set B⊂ℝ0+ such that Dens¯(B)=1 and ∫B‍exp(-p∫0xh(s)ds)dx<∞, then the C0-semigroup {Tt}t≥0 defined in (3.3) is distributionally chaotic on Lp(ℝ0+,ℂ), 1≤p<∞.Proof.
If we defineρ(x)=exp(-p∫0x‍h(s)ds), then the operators of {Tt}t≥0 can be rewritten as
(3.4)Ttf(x)=(ρ(x)ρ(x+t))1/pf(x+t).
This function ρ(x) is an admissible weight function in the sense of [12, Def. 4.1], which ensures that the left translation semigroup {τt}t≥0 defined as
(3.5)τtf(x)=f(x+t),forx,t≥0,f∈Lρp(ℝ0+,ℂ)
is a C0-semigroup on Lρp(ℝ0+,ℂ).
Let us defineϕ(f)(x)=(ρ(x))1/pf(x) and consider the following commutative diagram:
(3.6)
The hypothesis onB let us conclude that {τt}t≥0  is distributionally chaotic on Lρp(ℝ0+,ℂ), see [7, Th. 2.3]. Therefore, the conclusion is obtained since distributional chaos is preserved under conjugacy [6, Th. 2].Remark 3.2.
The previous result can be compared with the characterizations of hypercyclicity and Devaney chaos for the translationC0-semigroup on the spaces Lρp(ℝ0+,ℂ), 1≤p<∞: The translation C0-semigroup {τt}t≥0 is hypercyclic on Lρp(ℝ0+,ℂ) if and only if, liminfx→∞ρ(x)=0 [12], and {τt}t≥0 is Devaney chaotic on it if, and only if, ∫0∞‍ρ(x)dx<∞ [16, 17]. Using conjugacy, these results can be transferred to the C0-semigroup {Tt}t≥0 [1, Ex. 7.5.2].
On the one hand, ifh(x) is constant and equal to 1, then we have that {Tt}t≥0 is Devaney chaotic and distributionally chaotic on Lp(ℝ0+,ℂ). On the other hand, taking B=[0,2]∪⋃n∈ℕ‍[n2+1,(n+1)2[, h(x)=1 if x∈B and h(x)=-1 elsewhere, we have Dens¯(B)=1 and ∫B‍ρ(x)dx<∞. Therefore {Tt}t≥0 is distributionally chaotic on Lp(ℝ0+,ℂ). It is also hypercyclic since ρ(n2)=e-(n2-2n+4)p for n≥2, which yields that liminfx→∞ρ(x)=0 [14, Th. 2.2]. However, it cannot be Devaney chaotic since ∫ℝ0+‍ρ(x)dx=∞.
To sum up, we have an example of aC0-semigroup that is hypercyclic, distributionally chaotic, but it is not Devaney chaotic. This example can be compared with the example provided in [7, Ex. 2.] of a distributionally chaotic translation C0-semigroup that is neither hypercyclic nor chaotic.Now, let us consider another example of aC0-semigroup whose dynamical behavior was already discussed in [14]: Let ρ:[0,1]→ℝ+ be a continuous function such that there exist constants M≥1,  ω∈ℝ, and γ<0 such that
(3.7)ρ(x)≤Meωtρ(eγtx),∀x∈[0,1],t>0.
With such a function ρ, we can consider the spaces Lρp([0,1],ℂ), for 1≤p<∞. The family of operators {St}t≥0 with Stf(x)=f(eγtx), t≥0 defines a C0-semigroup on them [14].Theorem 3.3.
Ifγ<0, then the C0-semigroup {St}t≥0 is distributionally chaotic on Lρp([0,1],ℂ), 1≤p<∞.Proof.
Let us apply Theorem2.3. Take X0={f∈𝒞([0,1],ℂ):f(0)=0}. This set is dense in Lρp([0,1],ℂ) and, clearly, limt→∞Stf=0 for every f∈X0, which fulfills condition (i) in Theorem 2.3.
Let us prove that∫0∞‍∥St∥p,ρ-1dt is finite: Fix t>0 and a continuous function g on [0,1] with ∥g∥p,ρ=1, for instance g(x)=1/ρ(x)1/p.
There is somet0>0 such that for t>t0, we have (∫0eγt‍ρ(x)dx)1/p≤t-2/∥g∥∞. For these t>t0, define
(3.8)gt(x)={g(e-γtx),if0≤x≤eγt,0,elsewhere.
Since ∥gt∥p,ρ≤t-2 and Stgt=g, then ∥St∥p,ρ≥t2 for t≥t0. So that ∫t0∞‍∥St∥p,ρ-1dt is convergent, which yields the conclusion.Remark 3.4.
The assumptionγ<0 forces w>0: If not, take any x∈[0,1]. Taking limits when t→∞ in the inequality ρ(x)/ρ(eγtx)≤Meωt we have ρ(x)/ρ(0)≤0, which is a contradiction because ρ is a positive continuous function.Remark 3.5.
An alternative proof provided by an anonymous referee is the following: Ifρ:[0,1]→ℝ+ is a continuous weight function which is admissible in the sense of (3.7), then ψ:ℝ0+→ℝ0+ defined as ψ(x):=ρ(eγx)eγx is an admissible weight function in the sense of [12, Def. 4.1]. Therefore, taking ϕ:Lψp(ℝ0+,ℂ)→Lρp([0,1],ℂ) defined as ϕ(f)(x):=f(log(x)/γ), we have the following commutative diagram:
(3.9)
If γ<0, then ∫0∞‍ψ(x)dx<∞. So that, by conjugacy, {St}t≥0 is hypercyclic, Devaney and distributionally chaotic, see Remark 3.2.We return to the initial value problem stated in (3.2). Consider the case when I=[0,1], ζ(x):=γx, γ<0, and h∈𝒞([0,1],ℂ). Under these hypotheses, the C0-semigroup {T~t}t≥0 defined as(3.10)T~tf(x)=exp(∫0th(eγ(t-r)x)dr)f(eγtx)fort≥0,x∈[0,1],
gives the solution C0-semigroup to (3.2) on Lp([0,1],ℂ), 1≤p<∞ [14, Th. 3.4]. The particular case when γ=-1 and h(x)=-1/2 was studied using the Wiener measure in [15].Theorem 3.6.
Ifγ<0 and min{ℜ(h(x)):x∈[0,1]}>γ/p, then the C0-semigroup {T~t}t≥0 defined in (3.10) is distributionally chaotic on Lp([0,1],ℂ), 1≤p<∞.Proof.
We apply again Theorem2.3: Condition (i) holds in the same way as in the proof of Theorem 3.3 taking X0={f∈𝒞([0,1]),ℂ):f(0)=0}.
In order to verify condition (ii), letα∈ℝ be such that min{ℜ(h(x)):x∈[0,1]}>α>γ/p. For every t>0, we define ft as a function with ||ft||p=1 and supp(ft)⊂[0,eγt]. Using it, we have the following estimations for ||T~t||p:
(3.11)∥T~t∥p≥∥T~tft∥p≥eαt(∫01|ft(eγtx)|pdx)1/p=e(α-γ/p)t(∫0eγt|ft(y)|pdy)1/p=e(α-γ/p)t
so that ∫0∞‍∥T~t∥p-1dt is finite, which yields the conclusion.Under the hypothesis of the last theorem, Takeo proved that{T~t}t≥0 is Devaney chaotic by applying the Desch-Schappacher-Webb Criterion [14]. Independently, Brzeźniak and Dawidowicz also proved that {T~t}t≥0 is Devaney chaotic when γ=-1 and h(x)=λ∈ℝ with λ>-1/p, that is known as the von Foerster-Lasota equation [18, Theorems  8.3 and 8.4]. Furthermore, they also showed that for λ≤-1/p the orbits of all elements tend to 0, which makes chaos disappear. Therefore, we can affirm that Devaney chaos coincides exactly with distributional chaos for the same values of λ. As we will see later, this is due to the fact that Devaney chaos can be obtained here from the Desch-Schappacher-Webb Criterion. This can be easily seen if we reformulate Theorem 2.3 in terms of the infinitesimal generator of the C0-semigroup. The following result is a continuous version of [4, Cor. 31].Theorem 3.7.
LetX be a complex Banach space and let 𝒯 be a C0-semigroup in L(X) with infinitesimal generator (A,D(A)). If the following conditions hold:(i)
there is a dense subsetX0⊂X with limt→∞Ttx=0, for each x∈X0, and(ii)
there is someλ∈σp(A) with ℜ(λ)>0,then 𝒯 has a dense distributionally irregular manifold. In particular, 𝒯 is distributionally chaotic.Proof.
Fixt>0. On the one hand, if condition (i) holds, then we have limn→∞Ttnx=0 for every x∈X0. On the other hand, by the point spectral mapping theorem for C0-semigroups, since λ∈σp(A), then eλt∈σp(Tt). Therefore r(Tt)≥|eλt|>1 and, by [4, Cor. 31], Tt admits a dense distributionally irregular manifold. By [8, Rem. 2], this is equivalent to say that 𝒯 admits a dense distributionally irregular manifold. Furthermore, 𝒯 is distributionally chaotic [8, Prop. 2].Remark 3.8.
Clearly, the conditions in Theorem3.7 hold whenever the Desch-Schappacher-Webb Criterion can be applied. Therefore, among others, the following C0-semigroups that are known to be Devaney chaotic are also distributionally chaotic (and have a dense distributionally irregular manifold): [19], [20, Th. 3.1], [12, Ex. 4.12], [21], [22, Th. 1], and [23, Th. 2.1 and Th. 2.3]. See also [1, Ch. 7] for an improved version of the proof of these last two examples.Finally, Brzeźniak and Dawidowicz also studied in [18] Devaney chaos for the case γ=-1 and h(x)=λ∈ℝ in certain subspaces of Hölder continuous functions on [0,1]. For α∈]0,1], 0<r≤1, we define the space Crα([0,1]) of functions f:[0,1]→ℝ such that
(3.12)∥f∥α,r≔supx,y∈[0,1]0<|x-y|<r|f(x)-f(y)||x-y|α<∞.
For α∈]0,1[, let us consider Vα([0,1]) the space of functions
(3.13){f∈C1α([0,1]):limr→0+∥f∥α,r=0,f(0)=0}.
In [18], it is shown that Vα([0,1]) is a separable Banach space endowed with the norm ||f||α,1. Furthermore, following a constructive approach, it is proved that if γ=-1 and h(x)=λ>α, then {T~t}t≥0 is Devaney chaotic there. We will prove that in this case {T~t}t≥0 is also distributionally chaotic.Theorem 3.9.
Ifγ=-1 and h(x)=λ>α, then the C0-semigroup {T~t}t≥0 defined in (3.10) is distributionally chaotic on Vα([0,1]),α∈]0,1[.Proof.
We will apply Theorem2.3 again. Since {T~t}t≥0 is Devaney chaotic, then there is a dense set of points with bounded orbit. Therefore {T~t}t≥0 is weakly mixing [1, Th. 7.23], and any non-trivial operator T~t is weakly mixing, too [11, Th. 2.4]. Fix t>0. By [9, Th. 2.3], T~t satisfies the Hypercyclicity Criterion. So that, there is a dense set X0⊂Vα([0,1]) such that limn→∞T~tnx=0 for all x∈X0. Using the local equicontinuity of {T~t}t≥0, we have limt→∞T~tx=0 for every x∈X0 and condition (i) holds.
In order to verify condition (ii), take0<ε<(λ-α)/2 such that α+ε<1. Let us define fε(x)=xα+ε, 0≤x≤1. Since |xα+ε-yα+ε|≤|(x-y)α+ε| for all x,y∈[0,1], then we can easily see that ||fε||α,1=1 and fε∈Vα([0,1]). We also get that ||T~tfε||α,1=e(λ-α-ε)t and hence ∫0∞‍dt/∥T~t∥α,1≤∫0∞‍dt/∥T~tfε∥α,1<∞.


## 4. Discussion and Conclusions

Consider the initial value problem of (3.2) on L1(ℝ0+,ℂ) with ζ(x)=1 and h(x)=kxk-1/(1+xk). Here, the solution C0-semigroup {Tt}t≥0 is defined as
(4.1)Ttf(x)=1+(x+t)k1+xkf(x+t),x,t≥0.TheC0-semigroup {Tt}t≥0 defined in (4.1) is distributionally chaotic on L1(ℝ0+,ℂ) by Theorem 3.1. The hypercyclicity of this C0-semigroup for k=2 was obtained by El Mourchid in [24] and the Devaney chaos by Grosse-Erdmann and Peris in [1, Prop. 7.34]. In this case, the point spectrum of the infinitesimal generator is the closed left half plane. This inhibits the Desch-Schappacher-Webb Criterion to be applied in the way it has been formulated. Nevertheless, El Mourchid observed that the hypercyclic behavior of this C0-semigroup is essentially due to the imaginary eigenvalues of its infinitesimal generator [24], see also [1, Ex. 7.5.1]. In fact, the Desch-Schappacher-Webb Criterion can be strengthened and reformulated as follows.Theorem 4.1 (see [24, Th. 2.1] and [1, Th. 7.31]).
LetX be a complex separable Banach space, and let 𝒯 be a C0-semigroup on X with infinitesimal generator (A,D(A)). If there are a<b and continuous functions fj:[a,b]→X,j∈J, with(1)
fj(s)∈ker(isI-A) for every s∈[a,b], j∈J, and(2)
span{fj(s);s∈[a,b],j∈J} is dense in X,then the semigroup 𝒯 is Devaney chaotic.To sum up, we have seen that even when we apply this stronger version of the Desch-Schappacher-Webb Criterion for theC0-semigroup in (4.1), asking only for an abundance of eigenvalues of real part equal to zero, then we can also prove that there is a dense distributionally irregular manifold. Therefore, we can pose the following problem.Problem 1.
Do the hypothesis in Theorem4.1 imply the existence of a dense distributionally irregular manifold for 𝒯? If not, is there at least a distributionally irregular vector for 𝒯?By the equivalence between aC0-semigroup with a distributionally irregular vector and a distributionally chaotic C0-semigroup, [8, Th. 3.4], the former problem can also be presented as follows.Problem 2.
Do the hypothesis in Theorem4.1 imply that 𝒯 is distributionally chaotic?These questions could have a positive answer, but it is still unknown whether Devaney chaos implies distributional chaos onC0-semigroups.Problem 3.
Are there examples of Devaney chaoticC0-semigroups which are not distributionally chaotic?AC0-semigroup is said to be frequently hypercyclic if there exists some x∈X such that for every nonempty open set U⊂X, the set Ux:={s≥0:Tsx∈U} has positive lower density, that is liminft→∞(1/t)μ(Ux∩[0,t]) is positive. In [25], Mangino and Peris observed that with the same arguments used in [12, 24], one can show that the Desch-Schappacher-Webb Criterion implies frequent hypercyclicity. They also provide the Frequent Hypercyclicity Criterion for C0-semigroups [25, Th. 2.2]. So that, one can raise the following question.Problem 4.
Do the hypothesis of the Frequent Hypercyclicity Criterion forC0-semigroups imply distributional chaos?The hypothesis in Theorem4.1 also yields the mixing property for the C0-semigroup 𝒯, see [1]. We recall that a C0-semigroup is topologically mixing if for any pair of nonempty open sets U,V⊂X there is some t0>0 such that Tt(U)∩V≠∅ for all t≥t0. Clearly, topological mixing implies transitivity (i.e., hypercyclicity), but it is strictly stronger than it. Topologically mixing translation C0-semigroups on the weighted Lρp-spaces considered in this paper are characterized by the condition limt→∞ρ(t)=0 [26, Th. 4.3].On the one hand, the aforementioned example of Peris and Barrachina [7, Ex. 2.7] provides an example of a distributionally chaotic C0-semigroup that it is not topologically mixing. On the other hand, in [27], there is an example of a backward shift operator on a weighted sequence space ℓp(v), 1≤p<∞, that is topologically mixing but it is not distributionally chaotic. This operator will provide us an analogous counterexample in the frame of C0-semigroups. We thank A. Peris for this counterexample.Example 4.2.
Consider the sequence(nk)k defined as nk=(k!)3, k∈ℕ, and define the function ρ:ℝ0+→ℝ0+ as ρ(t)=1 if 0≤t. This function is an admissible weight in the sense of [12, Def. 4.1] and makes the translation semigroup {τt}t≥0 to be a C0-semigroup. On the one hand, since limt→∞ρ(t)=0, then the translation C0-semigroup is topologically mixing. On the other hand, if the translation C0-semigroup was distributionally chaotic, by [7, Th. 2.10], the backward shift operator, defined as B(x1,x2,…)=(x2,x3,…), would be distributionally chaotic on the space ℓ1(v):={(xn)n:∑j∈ℕ‍|xj|vj<∞} with (vn)n=(ρ(n))n, which is a contradiction as it is indicated in [27].



---

*Source: 457019-2012-12-25.xml*