On the Existence of Polynomials with Chaotic Behaviour

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 infinitedimensional 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.


Introduction
Let  be a topological vector space over K = R or C. Given a continuous linear operator  :  → , recall that  is said to be hypercyclic (resp., supercyclic) if there exists a vector  ∈  whose orbit {  ;  ∈ N} (resp., whose projective orbit {  ;  ∈ N,  ∈ K}) is dense in .In this case, such a vector  is said to be hypercyclic (resp., supercyclic) for .If, for every nonempty open set  ⊂ , we have that lim inf (where || denotes the cardinality of  ⊂ N), then  is 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  ̸ = {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  :  →  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  = ∑  =0   , where each   is an -homogeneous polynomial on  (by convenience, a 0homogeneous polynomial is a constant map).In contrast to the case of Banach spaces, Peris [15] exhibited for each  ≥ 2 an example of a chaotic homogeneous polynomial of degree  on the product Fréchet space C N .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 ℓ  (1 ≤  < ∞) and  0 .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.

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  ≥ 0 (resp.,  0 ≥ 0) such that It follows from the Hahn-Banach Theorem that there exists a continuous linear projection  from  onto .Define Then,  :  →  is a nonzero continuous -homogeneous polynomial such that  ( + ) =  whenever  ∈ ,  ∈ . Moreover, Indeed, if  ∈  and  =  + V, then where we used (3) and the fact that  = .
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 K 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  = 1 in Theorem 1, and for every integer  ≥ 2, we know that there exists a frequently hypercyclic polynomial  =  +  ∘  −  of degree  on  such that   = ( + ) ∘   −  for every  ∈ N. Since every periodic point of  is also periodic for , we immediately get that  is also chaotic.

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][28][29]).
We recall that, given  ⊂ N, its upper and lower densities are defined by dens () := lim sup respectively.A continuous map  :  →  on a metric space  is distributionally chaotic if there exist an uncountable set Γ ⊂  and  > 0 such that for every  > 0 and each pair of distinct points ,  ∈ Γ, we have that dens { ∈ N :  (  () ,   ()) < } = 0, In this case, the set Γ is a distributionally -scrambled set and the pair (, ) a distributionally chaotic pair.We say that  is densely distributionally chaotic if the set Γ 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 ,  ⊂ N with dens() = dens() = 1 such that lim In this case we have, by linearity, that the set Γ := span{} is a distributionally -scrambled set, for every  > 0. 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 Γ :=  is a distributionally -scrambled set, for every  > 0, if  is a distributionally irregular manifold.
Theorem 4. If  is an infinite-dimensional separable Banach space and  ∈ N,  > 1, then there is a distributionally chaotic polynomial of degree  defined on  with a dense distributionally scrambled linear manifold.
Proof.Let  ∈  be a nonzero vector and let  ⊂  be a closed subspace complement of span{}.By Theorems 24 and 35 in [27], there exist an operator T :  → , a dense subset {  ;  ∈ N} ⊂ , a bounded sequence of linearly independent distributionally irregular vectors (  )  in , and a set of integers  ⊂ N with dens() = 1 such that lim  T (  ) = 0 and lim ∈ T (  ) = 0 for each  ∈ N. Let {  ;  ∈ N} ⊂ K be a dense sequence in the scalar field and define the following countable set of vectors: where  is a bijection from N × N onto N. Extend T to an operator  :  →  by setting  = 0.By following the proof of [27, Theorem 24], we have that each nonzero  ∈  := span{ , ; ,  ∈ N} is a distributionally irregular vector for .Moreover, lim ∈    = 0 for every  ∈ .
We construct the polynomial  of degree  as in Theorem 1, and we have that for all  ∈ N. The polynomial  := +∘ has degree  if we require V = T for some nonzero  ∈ .
and we conclude that  is a distributionally scrambled dense linear manifold for the polynomial .

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.
(1) Every Hausdorff countable inductive limit of separable Banach spaces  = ind    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  = ⊕  ℓ  , 1 ≤  < ∞, and  = ⊕   = D(Ω), where  is the space of rapidly decreasing functions and D(Ω) is the space of test functions on an open set Ω ⊂ R  , 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.