# On the Existence of Polynomials with Chaotic Behaviour

Authors: Nilson C. Bernardes; Alfredo Peris
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We establish a general result on the existence of hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) polynomials on locally convex spaces. As a consequence we prove that every (real or complex) infinite-dimensional separable Frèchet space admits mixing (hence hypercyclic) polynomials of arbitrary positive degree. Moreover, every complex infinite-dimensional separable Banach space with an unconditional Schauder decomposition and every complex Frèchet space with an unconditional basis support chaotic and frequently hypercyclic polynomials of arbitrary positive degree. We also study distributional chaos for polynomials and show that every infinite-dimensional separable Banach space supports polynomials of arbitrary positive degree that have a dense distributionally scrambled linear manifold.


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## Body

## 1. Introduction

LetX be a topological vector space over 𝕂=ℝ or ℂ. Given a continuous linear operator T:X→X, recall that T is said to be hypercyclic (resp., supercyclic) if there exists a vector x∈X whose orbit {Tnx;n∈ℕ} (resp., whose projective orbit {αTnx;n∈ℕ,α∈𝕂}) is dense inX. In this case, such a vectorxis said to be hypercyclic (resp., supercyclic) for T. If, for every nonempty open set U⊂X, we have that
(1)liminfN→∞|{n≤N;Tnx∈U}|N>0
(where |A| denotes the cardinality of A⊂ℕ), thenxis said to be a frequently hypercyclic vector for T, and the operator T is called frequently hypercyclic. The operator T is chaotic if it is hypercyclic and has a dense set of periodic points. The study of hypercyclic operators and, more generally, the area of linear dynamics has experienced a great development during the last 30 years. We refer the reader to the recent books [1, 2] for an up to date account of the subject.A fundamental problem in this area was the following question on the existence of hypercyclic operators formulated by Rolewicz [3]: Does every infinite-dimensional separable Banach space admit a hypercyclic operator? Herzog [4] proved that the answer is “yes” if we replace the word “hypercyclic” by “supercyclic.” Later, the question was finally answered affirmatively (and independently) by Ansari [5] and Bernal-González [6]. Bonet and Peris [7] extended their result to Fréchet spaces, so that we have the following existence theorem: Every infinite-dimensional separable Fréchet space admits a hypercyclic operator.The situation for chaos and frequent hypercyclicity is different: hereditarily indecomposable complex Banach spaces support neither chaotic operators [8] nor frequently hypercyclic operators [9]. The recent works [10, 11] show that, under very general conditions, we have a positive result: Every complex infinite-dimensional separable Banach space with an unconditional Schauder decomposition and every complex Fréchet space with an unconditional basis support a chaotic and frequently hypercyclic operator.The study of hypercyclicity and supercyclicity for polynomials was initiated by Bernardes [12] who established the following results: (i)
no Banach spaceX≠{0} admits a hypercyclic homogeneous polynomial of degree ≥2;(ii)
every infinite-dimensional separable Banach space admits supercyclic homogeneous polynomials of arbitrary positive degree.Other results on the dynamics of homogeneous polynomials were obtained by Bernardes [13]. We refer the reader to the monograph [14] for polynomials on locally convex spaces. A map Q:X→X defined on a locally convex space X is a homogeneous polynomial of degreem (in short, m-homogeneous) if there is an m-linear continuous map A:X×⋯×X→X such that Qx=A(x,…,x) for any x∈X. A general polynomial of degreen is a map P:X→X that can be written as P=∑m=0n‍Qm, where each Qm is an m-homogeneous polynomial on X (by convenience, a 0-homogeneous polynomial is a constant map). In contrast to the case of Banach spaces, Peris [15] exhibited for eachm≥2 an example of a chaotic homogeneous polynomial of degree m on the product Fréchet space ℂℕ. More examples of hypercyclic and chaotic homogeneous polynomials were obtained by Aron and Miralles [16] on certain spaces of continuous and differentiable functions. Also, Peris [17] obtained examples of chaotic nonhomogeneous polynomials on the Banach spaces ℓp (1≤p<∞) and c0. Moreover, Martínez-Giménez and Peris [18] found different types of chaotic homogeneous and nonhomogeneous polynomials on Köthe echelon spaces. A general result on the problem of existence of hypercyclic polynomials was obtained by Martínez-Giménez and Peris [19]: Every complex infinite-dimensional separable Fréchet space admits hypercyclic polynomials of arbitrary positive degree.However, the problem remained open for real scalars even in the case of Banach spaces.In this paper, by using a different approach, we prove that every (real or complex) infinite-dimensional separable Fréchet space admits hypercyclic polynomials of arbitrary positive degree (Theorem2). In fact, we derive this theorem from a more general result (Theorem 1), which can also be applied in situations where the underlying space is not a Fréchet space (Example 5). We also prove that every complex infinite-dimensional separable Banach space with an unconditional Schauder decomposition and every complex Fréchet space with an unconditional basis support chaotic and frequently hypercyclic polynomials of arbitrary positive degree (Theorem 3). Let us mention that there exists a real Banach space with an unconditional basis that does not admit a chaotic operator [10, Theorem 4.1]. Moreover, we also consider, for the first time, the existence of (weakly) mixing polynomials and the existence of polynomials that exhibit distributional chaos. In particular, we prove that every infinite-dimensional separable Banach space supports polynomials of arbitrary positive degree that are distributionally chaotic (Theorem 4), with a dense linear manifold as a scrambled set.We should point out that in the linear case, there are many sufficient criteria for hypercyclic and chaotic properties (see, e.g., [1, 2, 20, 21]), but no criterion is available for the hypercyclic or chaotic behaviour of polynomials. This fact shows the difficulties to obtain hypercyclic or chaotic polynomials.


## 2. Existence of Hypercyclic, Chaotic, and Frequently Hypercyclic Polynomials

IfT is a continuous map from a topological space X into itself, recall that T is said to be transitive (resp., mixing) if, for any pair U,V of nonempty open subsets of X, there exists n≥0 (resp., n0≥0) such that
(2)Tn(U)∩V≠∅(resp.,∀n≥n0).
Moreover, T is said to be weakly mixing if T×T is topologically transitive on X×X. Birkhoff transitivity theorem [2, Theorem 1.16] asserts that if X is separable, completely metrizable, and has no isolated point, then T is transitive if and only if T has a dense orbit.Let us now establish our main result.Theorem 1.
LetX be a locally convex space over 𝕂. If X admits a hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) operator that has an eigenvalue of modulus less than or equal to1, then X admits hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) polynomials of arbitrary positive degree.Proof.
Suppose thatT is a hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) operator on X that has an eigenvalue λ with |λ|≤1. Let u∈X∖{0} be an eigenvector of T associated with the eigenvalue λ. Let Y≔span{u}, choose a vector v∈X∖Y, and put Z≔span{u,v}. Fix an integer m≥2 and consider them-homogeneous polynomial Q:Z→X given by
(3)Q(αu+βv)≔βmu.
It follows from the Hahn-Banach Theorem that there exists a continuous linear projectionFfromXontoZ. Define
(4)P≔Q∘F.
Then, P:X→X is a nonzero continuousm-homogeneous polynomial such that
(5)P(x+y)=Pxwheneverx∈X,y∈Y.
Moreover,
(6)T∘P=λP.
Indeed, if x∈X and Fx=αu+βv, then
(7)T(Px)=T(Q(αu+βv))=T(βmu)=βmTu=λβmu=λPx,
where we used (3) and the fact thatTu=λu.
We claim that(8)(I-P)∘(I+P)=I=(I+P)∘(I-P),
which shows that the mapI+Pis a homeomorphism fromXontoX. In fact, for everyx∈X,
(9)(I-P)((I+P)(x))=(I-P)(x+Px)=x+Px-P(x+Px)=x,
because of (5). The second equality in (8) is analogous.
Now, we will prove by induction that(10)(T+P∘T-λP)n=(I+P)∘Tn-λnP∀n∈ℕ.
Since forn=1we obviously haveT+P∘T-λP=(I+P)∘T-λP, let us assume that (10) holds for a certainn∈ℕ, and let us consider the casen+1. Fixx∈X. By using (5), (6), and the induction hypothesis, we obtain
(11)(T+P∘T-λP)n+1(x)=((I+P)∘Tn-λnP)((T+P∘T-λP)(x))=(I+P)(Tn+1x+Tn(P(Tx))-λTn(Px))-λnP(Tx+P(Tx)-λPx)=(I+P)(Tn+1x+λnP(Tx)-λn+1Px)-λnP(Tx)=Tn+1x+λnP(Tx)-λn+1Px+P(Tn+1x)-λnP(Tx)=(I+P)(Tn+1x)-λn+1Px,
as was to be shown.
Suppose thatP∘T=λP. Then P∘Tn=λnP for all n≥0. Let W and U be disjoint open neighborhoods of 0 and u, respectively, with W balanced. Since T is transitive, there are n≥0 and y∈P-1(W) such that Tny∈P-1(U). Hence,
(12)P(Tny)∈U,P(Tny)=λnPy∈W,
a contradiction. Thus, P∘T≠λP, which proves that
(13)R≔T+P∘T-λPis a polynomial of degreem.
By (10),
(14)Rn=(I+P)∘Tn-λnP∀n∈ℕ.
If|λ|=1 then the operator λ-1T is also hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) by [22] (resp., [23] and [2, Proposition 12.15], [24, Remark 5], and [1, Theorems 6.28 and 6.30]). Sinceuis a fixed point of λ-1T, by replacing the operator T by the operator λ-1T, we may assume that λ=1. In the case T is hypercyclic, it is clear from (14) thatRis also hypercyclic. Assume a∈X is a frequently hypercyclic vector for T and fix a nonempty open set U⊂X. Let V⊂X be a nonempty open set such that (I+P)(V)⊂U+Pa. We have that
(15){n∈ℕ;Tna∈V}⊂{n∈ℕ;Rna∈U},
so that
(16)liminfN→∞|{n≤N;Rna∈U}|N>0,
and a is a frequently hypercyclic vector for R. Now, consider T mixing. Let U,V⊂X be nonempty open sets. Fix points y∈U and z∈V, and put w≔z+Py. Since w-Py=z∈V, there are open neighborhoods U′ of y and V′ ofwsuch that
(17)U′⊂U,V′-P(U′)⊂V.
Let n∈ℕ be such that Tn(U′)∩(I+P)-1(V′)≠∅. Fix x∈U′ such that Tnx∈(I+P)-1(V′). Then,
(18)x∈U,Rnx=(I+P)(Tnx)-Px∈V′-P(U′)⊂V,
proving that Rn(U)∩V≠∅. This shows that R is mixing. A similar reasoning shows that R is weakly mixing (transitive) whenever T is weakly mixing (transitive).
If|λ|<1 then λnP→0 pointwise on X. Hence, (14) implies that R is hypercyclic if T is hypercyclic, and R is frequently hypercyclic if T is frequently hypercyclic. Consider T mixing and let U,V⊂X be nonempty open sets. There are a nonempty open set V′ and an open neighborhood W of 0 such that
(19)V′+W⊂V.
Moreover, there are a constant C>0 and a nonempty open set U′ such that
(20)U′⊂U,P(U′)⊂{αu;|α|<C},
and there existsε>0such that
(21){αu;|α|<ε}⊂W.
Let n∈ℕ be such that |λ|nC<ε and Tn(U′)∩(I+P)-1(V′)≠∅. Fix x∈U′ with Tnx∈(I+P)-1(V′). Then,
(22)x∈U,Rnx=(I+P)(Tnx)-λnPx∈V′+W⊂V.
This shows that R is mixing. Similarly, R is weakly mixing (transitive) if T is weakly mixing (transitive).Since every infinite-dimensional separable Fréchet space admits a mixing operator with a nonzero fixed point [2, Theorem 8.13], the previous theorem immediately implies the following result.Theorem 2.
Every infinite-dimensional separable Fréchet space over𝕂 admits mixing (hence hypercyclic) polynomials of arbitrary positive degree.Also, the existence of chaotic and frequently hypercyclic polynomials of arbitrary degree can be ensured under very general assumptions by the results in [10, 11].Theorem 3.
Every complex infinite-dimensional separable Banach space with an unconditional Schauder decompositionand every complex Fréchet space with an unconditional basis support chaotic and frequently hypercyclic polynomials of arbitrary positive degree.Proof.
LetX satisfy the hypothesis. By [10, 11] there exists a chaotic and frequently hypercyclic operator T:X→X with a nonzero fixed point. Thus, with λ=1 in Theorem 1, and for every integer m≥2, we know that there exists a frequently hypercyclic polynomial R=T+P∘T-P of degree m on Xsuch that Rn=(I+P)∘Tn-P for every n∈ℕ. Since every periodic point of T is also periodic for R, we immediately get that R is also chaotic.


## 3. Distributional Chaos for Polynomials on Banach Spaces

The original notion of chaos in the mathematical literature comes from Li and Yorke’s paper [25] on the study of dynamics of interval maps, and it concentrates on local aspects of dynamics of pairs. Schweizer and Smítal introduced the concept of distributional chaos in [26] as a natural extension of the notion of chaos given by Li and Yorke.There are several recent studies of distributional chaos for linear operators on Banach or Fréchet spaces (see, e.g., [21, 27–29]).We recall that, givenA⊂ℕ, its upper and lower densities are defined by
(23)dens¯(A)≔limsupn→∞card(A∩[1,n])n,dens_(A)≔liminfn→∞card(A∩[1,n])n,
respectively. A continuous map f:X→X on a metric space X is distributionally chaotic if there exist an uncountable set Γ⊂X and ε>0 such that for every τ>0 and each pair of distinct points x,y∈Γ, we have that
(24)dens_{n∈ℕ:d(fn(x),fn(y))<ε}=0,dens¯{n∈ℕ:d(fn(x),fn(y))<τ}=1.
In this case, the set Γ is a distributionallyε-scrambled set and the pair (x,y) a distributionally chaotic pair. We say that f is densely distributionally chaotic if the set Γ may be chosen to be dense in X.Inspired by the notion of irregular vector introduced by Beauzamy [30] for operators on Banach spaces, in [27] the authors defined the following notion: given an operator T∈B(X) on a Banach space X and x∈X, x is a distributionally irregular vector for T if there are A,B⊂ℕ with dens¯(A)=dens¯(B)=1 such that
(25)limn∈ATnx=0,limn∈B∥Tnx∥=∞.
In this case we have, by linearity, that the set Γ≔span{x}is a distributionally ε-scrambled set, for every ε>0. If M⊂X is a linear manifold such that every nonzero x∈M is a distributionally irregular vector for T, then M is said to be a distributionally irregular manifold forT. By linearity, we also have that Γ≔M is a distributionally ε-scrambled set, for every ε>0, if M is a distributionally irregular manifold.Theorem 4.
IfX is an infinite-dimensional separable Banach space and m∈ℕ,m>1, then there is a distributionally chaotic polynomial of degree m defined on X with a dense distributionally scrambled linear manifold.Proof.
Letu∈Xbe a nonzero vector and letY⊂Xbe a closed subspace complement of span{u}. By Theorems 24 and  35 in [27], there exist an operator T~:Y→Y, a dense subset {ym;m∈ℕ}⊂Y, a bounded sequence of linearly independent distributionally irregular vectors (um)m in Y, and a set of integers A⊂ℕ with dens¯(A)=1 such that limnT~n(ym)=0 and limn∈AT~n(um)=0 for each m∈ℕ. Let {rn;n∈ℕ}⊂𝕂 be a dense sequence in the scalar field and define the following countable set of vectors:
(26)zn,m≔rnu+ym+1nmuϕ(n,m)(n,m∈ℕ),
where ϕ is a bijection from ℕ×ℕ onto ℕ. Extend T~ to an operator T:X→X by setting Tu=0. By following the proof of [27, Theorem 24], we have that each nonzero z∈M≔span{zn,m;n,m∈ℕ} is a distributionally irregular vector for T. Moreover, limn∈ATnz=0 for every z∈M.
We construct the polynomialP of degreemas in Theorem 1, and we have that
(27)(T+P∘T)n=(I+P)∘Tn,
for all n∈ℕ. The polynomial R≔T+P∘T has degree m if we require v=T~y for some nonzero y∈Y. Let now z1,z2∈M with z1≠z2. Since limn∈ATnz1=limn∈ATnz2=0, we have
(28)limn∈A∥Rnz1-Rnz2∥=limn∈A∥(I+P)(Tnz1)-(I+P)(Tnz2)∥=0.
Let G:X→Y be the projection onto Y with kernel span{u}. Let C>0 be such that ∥x∥≥C∥Gx∥ for every x∈X. By the definition of M, y≔Gz1-Gz2≠0 is a distributionally irregular vector for T~. Then there is B⊂ℕ with dens¯(B)=1 such that limn∈B∥T~ny∥=∞. As a consequence,
(29)limn∈B∥Rnz1-Rnz2∥≥Climn∈B∥G(Rnz1-Rnz2)∥=Climn∈B∥T~ny∥=∞,
and we conclude that M is a distributionally scrambled dense linear manifold for the polynomial R.


## 4. Hypercyclic Polynomials on Non-Fréchet Spaces

In this section we will present the existence of hypercyclic polynomials of arbitrary positive degree on certain general locally convex spaces which are not Fréchet as an application of Theorem1.Example 5.
(1)
Every Hausdorff countable inductive limit of separable Banach spacesE=indnEn such that one of the steps Em is dense in E admits hypercyclic polynomials of arbitrary positive degree. Actually, in Proposition 4 of [7] the existence of a hypercyclic operator T:E→E such that 1 is an eigenvalue of T was shown, and so the result follows from Theorem 1.
(2)
Every metrizable countably dimensional locally convex spaceE has hypercyclic polynomials of any positive degree. Again, this is a consequence of the fact that there is a hypercyclic operator T on E with fixed nonzero vectors, as was recently observed in [31].
(3)
Any direct sumE=⊕nEn of separable Fréchet spaces such that En is infinite dimensional for infinitely many n necessarily supports hypercyclic polynomials of arbitrary positive degree. In [32] it was shown that E=⊕nℓp, 1≤p<∞, and E=⊕ns=𝒟(Ω), wheresis the space of rapidly decreasing functions and 𝒟(Ω) is the space of test functions on an open set Ω⊂ℝm, support a noninjective hypercyclic operator. Shkarin [33] generalized this result to the present context by showing the existence of hypercyclic operators of the form T=I+S for S noninjective. Therefore the hypotheses of Theorem 1 are satisfied, and we conclude the result.



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