Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 320961, 6 pages

http://dx.doi.org/10.1155/2013/320961

Research Article

On the Existence of Polynomials with Chaotic Behaviour

Nilson C. Bernardes Jr.¹ and Alfredo Peris²

¹Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal do Rio de Janeiro, CP 68530, 21945-970 Rio de Janeiro, RJ, Brazil

²IUMPA, Universitat Politècnica de València, Departament de Matemática Aplicada, Edifici 7A, 46022 València, Spain

Received 29 May 2013; Accepted 7 August 2013

Academic Editor: Ajda Fošner

Copyright © 2013 Nilson C. Bernardes Jr. and Alfredo Peris. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

Let be a topological vector space over or . Given a continuous linear operator , recall that is said to be hypercyclic (resp., supercyclic) if there exists a vector whose orbit (resp., whose projective orbit ) is dense in. In this case, such a vectoris said to be hypercyclic (resp., supercyclic) for . If, for every nonempty open set , we have that

(where denotes the cardinality of ), thenis said to be a frequently hypercyclic vector for , and the operator is called frequently hypercyclic. The operator 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 space 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 defined on a locally convex space is a homogeneous polynomial of degree (in short, -homogeneous) if there is an -linear continuous map such that for any . A general polynomial of degree is a map that can be written as , where each is an -homogeneous polynomial on (by convenience, a -homogeneous polynomial is a constant map). In contrast to the case of Banach spaces, Peris [15] exhibited for each an example of a chaotic homogeneous polynomial of degree on the product Fréchet space . 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 () and . 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 (Theorem 2). 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

If is a continuous map from a topological space into itself, recall that is said to be transitive (resp., mixing) if, for any pair of nonempty open subsets of , there exists (resp., ) such that

Moreover, is said to be weakly mixing if is topologically transitive on . Birkhoff transitivity theorem [2, Theorem 1.16] asserts that if is separable, completely metrizable, and has no isolated point, then is transitive if and only if has a dense orbit.

Let us now establish our main result.

Theorem 1. Let be a locally convex space over . If admits a hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) operator that has an eigenvalue of modulus less than or equal to, then admits hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) polynomials of arbitrary positive degree.

Proof. Suppose that is a hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) operator on that has an eigenvalue with . Let be an eigenvector of associated with the eigenvalue . Let , choose a vector , and put . Fix an integer and consider the-homogeneous polynomial given by

It follows from the Hahn-Banach Theorem that there exists a continuous linear projectionfromonto. Define

Then, is a nonzero continuous-homogeneous polynomial such that

Moreover,

Indeed, if and , then

where we used (3) and the fact that.

We claim that

which shows that the mapis a homeomorphism fromonto. In fact, for every,

because of (5). The second equality in (8) is analogous.

Now, we will prove by induction that

Since forwe obviously have, let us assume that (10) holds for a certain, and let us consider the case. Fix. By using (5), (6), and the induction hypothesis, we obtain

as was to be shown.

Suppose that . Then for all . Let and be disjoint open neighborhoods of 0 and , respectively, with balanced. Since is transitive, there are and such that . Hence,

a contradiction. Thus, , which proves that

By (10),

If then the operator 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]). Sinceis a fixed point of , by replacing the operator by the operator , we may assume that . In the case is hypercyclic, it is clear from (14) thatis also hypercyclic. Assume is a frequently hypercyclic vector for and fix a nonempty open set . Let be a nonempty open set such that . We have that

so that

and is a frequently hypercyclic vector for . Now, consider mixing. Let be nonempty open sets. Fix points and , and put . Since , there are open neighborhoods of and ofsuch that

Let be such that . Fix such that . Then,

proving that . This shows that is mixing. A similar reasoning shows that is weakly mixing (transitive) whenever is weakly mixing (transitive).

If then pointwise on . Hence, (14) implies that is hypercyclic if is hypercyclic, and is frequently hypercyclic if is frequently hypercyclic. Consider mixing and let be nonempty open sets. There are a nonempty open set and an open neighborhood of such that

Moreover, there are a constant and a nonempty open set such that

and there existssuch that

Let be such that and . Fix with . Then,

This shows that is mixing. Similarly, is weakly mixing (transitive) if 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 over 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. Let satisfy the hypothesis. By [10, 11] there exists a chaotic and frequently hypercyclic operator with a nonzero fixed point. Thus, with in Theorem 1, and for every integer , we know that there exists a frequently hypercyclic polynomial of degree on such that for every . Since every periodic point of is also periodic for , we immediately get that 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, given , its upper and lower densities are defined by

respectively. A continuous map on a metric space is distributionally chaotic if there exist an uncountable set and such that for every and each pair of distinct points , we have that

In this case, the set is a distributionally -scrambled set and the pair a distributionally chaotic pair. We say that is densely distributionally chaotic if the set may be chosen to be dense in .

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 on a Banach space and , is a distributionally irregular vector for if there are with such that

In this case we have, by linearity, that the set is a distributionally -scrambled set, for every . If is a linear manifold such that every nonzero is a distributionally irregular vector for , then is said to be a distributionally irregular manifold for. By linearity, we also have that is a distributionally -scrambled set, for every , if is a distributionally irregular manifold.

Theorem 4. If is an infinite-dimensional separable Banach space and , then there is a distributionally chaotic polynomial of degree defined on with a dense distributionally scrambled linear manifold.

Proof. Letbe a nonzero vector and letbe a closed subspace complement of span. By Theorems 24 and  35 in [27], there exist an operator , a dense subset , a bounded sequence of linearly independent distributionally irregular vectors in , and a set of integers with such that and for each . Let be a dense sequence in the scalar field and define the following countable set of vectors:

where is a bijection from onto . Extend to an operator by setting . By following the proof of [27, Theorem 24], we have that each nonzero is a distributionally irregular vector for . Moreover, for every .

We construct the polynomial of degreeas in Theorem 1, and we have that

for all . The polynomial has degree if we require for some nonzero . Let now with . Since , we have

Let be the projection onto with kernel . Let be such that for every . By the definition of , is a distributionally irregular vector for . Then there is with such that . As a consequence,

and we conclude that is a distributionally scrambled dense linear manifold for the polynomial .

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 Theorem 1.

Example 5.

(1)Every Hausdorff countable inductive limit of separable Banach spaces such that one of the steps is dense in admits hypercyclic polynomials of arbitrary positive degree. Actually, in Proposition 4 of [7] the existence of a hypercyclic operator such that 1 is an eigenvalue of was shown, and so the result follows from Theorem 1.(2)Every metrizable countably dimensional locally convex space has hypercyclic polynomials of any positive degree. Again, this is a consequence of the fact that there is a hypercyclic operator on with fixed nonzero vectors, as was recently observed in [31].(3)Any direct sum of separable Fréchet spaces such that is infinite dimensional for infinitely many necessarily supports hypercyclic polynomials of arbitrary positive degree. In [32] it was shown that , , and , whereis the space of rapidly decreasing functions and is the space of test functions on an open set , support a noninjective hypercyclic operator. Shkarin [33] generalized this result to the present context by showing the existence of hypercyclic operators of the form for noninjective. Therefore the hypotheses of Theorem 1 are satisfied, and we conclude the result.

Acknowledgments

The present work was done while the first author was visiting the Departament de Matemàtica Aplicada at Universitat Politècnica de València (Spain). The first author is very grateful for the hospitality. The first author was supported in part by CAPES: Bolsista, Project no. BEX 4012/11-9. The second author was supported in part by MEC and FEDER, Project MTM2010-14909, and by GVA, Projects PROMETEO/2008/101 and PROMETEOII/2013/013.

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