Hypercyclic Vectors Research Articles

Trichotomy for the orbits of a hypercyclic operator on a Banach space

We obtain a trichotomy for the orbits of a hypercyclic operator T T on a separable Banach space X X : (1) every vector is mean asymptotic to zero; (2) generic vectors are absolutely mean irregular; (3) every hypercyclic vector is mean divergent to infinity. Examples of weighted backward shifts on ℓ p \ell ^p show that all three cases can happen.

Extending families of disjoint hypercyclic operators

In the present note, we solve two open questions posed by Salas in [33] about disjoint hypercyclic operators. First, we show that given any family T1,…,TN of disjoint hypercyclic operators, one can always select an operator T such that the extended family T1,…,TN,T of operators remains disjoint hypercyclic. In fact, we prove that the set of operators T which can extend the family of disjoint hypercyclic operators is dense in the strong operator topology in the algebra of bounded operators. Second, we show the existence of two disjoint weakly mixing operators that fail to possess a dense d-hypercyclic manifold. Thus, these operators satisfy the Disjoint Blow-up/Collapse property but fail to satisfy the Strong Disjoint Blow-up/Collapse property, a notion which was introduced by Salas as a sufficient condition for having a dense linear manifold of disjoint hypercyclic vectors.

Self-similar fractals and common hypercyclicity

We obtain a multi-dimensional generalization of the Costakis-Sambarino criterion for common hypercyclic vectors with optimal consequences on a large class of fractals. Applications include families of products of backward shifts parameterized by any Hölder continuous curve in Rd, for all d≥1.

Growth of hypercyclic entire functions for some non-convolution operators

Abstract A continuous linear operator T T defined on a Fréchet space X X is said to be hypercyclic if there exists f ∈ X f\in X such that, the orbit { T n f } \left\{{T}^{n}f\right\} is dense in X X . In this article, we consider the operators introduced by Aron and Markose, defined on the space of entire functions by T λ , b f ( z ) = f ′ ( λ z + b ) {T}_{\lambda ,b}f\left(z)=f^{\prime} \left(\lambda z+b) , λ ∈ C \ { 1 } \lambda \in {\mathbb{C}}\setminus \left\{1\right\} , b ∈ C b\in {\mathbb{C}} , and we aimed to explore the rate of growth of hypercyclic vectors for T λ , b {T}_{\lambda ,b} . We discover that T λ , b {T}_{\lambda ,b} is a weighted backward shift with respect to some basis and this fact allows us to find sharp estimates of the growth of T λ , b {T}_{\lambda ,b} -hypercyclic vectors. When ∣ λ ∣ = 1 | \lambda | =1 , the T λ , b {T}_{\lambda ,b} -hypercyclic function growth is similar to the D D -hypercyclic functions ( D D is the differentiation operator), and when ∣ λ ∣ > 1 | \lambda | \gt 1 , the T λ , b {T}_{\lambda ,b} -hypercyclic functions can grow very slowly but not arbitrarily slowly. A lower bound of this growth is found in terms of the W W -Lambert function. Finally, partial results are obtained for T λ , b {T}_{\lambda ,b} -frequently hypercyclic functions.

On the Existence of Infinite-Dimensional Closed Subspaces of Frequently Hypercyclic Vectors for $T_f$

On the Existence of Infinite-Dimensional Closed Subspaces of Frequently Hypercyclic Vectors for $T_f$

Reprint of: On spaceability of shift-like operators on Lp

We prove the spaceability of the set of hypercyclic vectors for shift-like operators. Shift-like operators appear naturally as composition operators on Lp(X), when the underlying space X is dissipative. In the process of proving the main theorem, we provide, among other results of independent interest, a characterization of weakly mixing dissipative composition operators of bounded distortion.

On spaceability of shift-like operators on Lp

We prove the spaceability of the set of hypercyclic vectors for shift-like operators. Shift-like operators appear naturally as composition operators on Lp(X), when the underlying space X is dissipative. In the process of proving the main theorem, we provide, among other results of independent interest, a characterization of weakly mixing dissipative composition operators of bounded distortion.

Transreal Explicit Construction of Universal Possible Worlds

In an earlier paper we supplied an indirect proof of the existence of universal possible worlds that have the topological property of being hypercyclic, which means they can access every world in sequences of worlds that approach arbitrarily closely to every possible world. That proof states that there are universal worlds but it does not exhibit such a world explicitly. We now explicitly construct two such universal worlds. A continuous universal world constructs possible worlds with transreal co-ordinates directly. A discrete world provides a binary hypercyclic vector which can be used to create transfloating-point co-ordinates that approximate transreal co-ordinates. We also discuss the philosophical implications of universal worlds for an omniscient observer and human science.

Commutant hypercyclicity of Hilbert space operators

An operator T on a Hilbert space H is commutant hypercyclic if there is a vector x in H such that the set {Sx : TS = ST} is dense in H. We prove that operators on finite dimensional Hilbert space, a rich class of weighted shift operators, isometries, exponentially isometries and idempotents are all commutant hypercyclic. Then we discuss on commutant hypercyclicity of 2 ? 2 operator matrices. Moreover, for each integer number n ? 2, we give a commutant hypercyclic nilpotent operator of order n on an infinite dimensional Hilbert space. Finally, we study commutant transitivity of operators and give necessary and sufficient conditions for a vector to be a commutant hypercyclic vector.

Disjoint and common hypercyclic algebras

Over time the concept of hypercyclicity has been explored in different manners and contexts, gaining new forms and applications. In particular when the space has an adjacent structure, we can always look for sets of hypercyclic vectors compatible with that framework. In this paper we deal with hypercyclic operators acting on Fréchet sequence algebras and give criteria for the existence of common and disjoint hypercyclic algebras.

Rank One Perturbation of Unitary Operators with Full Measure of Hypercyclic Vectors

Rank One Perturbation of Unitary Operators with Full Measure of Hypercyclic Vectors

Common frequent hypercyclicity

We provide with criteria for a family of sequences of operators to share a frequently universal vector. These criteria are inspired by the classical Frequent Hypercyclicity Criterion and by a recent criterion due to Grivaux, Matheron and Menet where periodic points play the central role. As an application, we obtain for any operator T in a specific class of operators acting on a separable Banach space, a necessary and sufficient condition on a subset Λ of the complex plane for the family {λT:λ∈Λ} to have a common frequently hypercyclic vector. In passing, this allows us to exhibit frequently hypercyclic weighted shifts which do not possess common frequently hypercyclic vectors. We also provide with criteria for families of the recently introduced operators of C-type to share a common frequently hypercyclic vector. Further, we prove that the same problem of common α-frequent hypercyclicity may be vacuous, where the notion of α-frequent hypercyclicity extends that of frequent hypercyclicity replacing the natural density by more general weighted densities. Finally, it is already known that any operator satisfying the classical Frequent Universality Criterion is α-frequently universal for any sequence α satisfying a suitable condition. We complement this result by showing that for any such operator, there exists a vector x which is α-frequently universal for T, with respect to all such densities α.

Common hypercyclic vectors and dimension of the parameter set

We investigate the existence of a common hypercyclic vector for a family $(T_\lambda)_{\lambda\in \Lambda}$ of hypercyclic operators acting on the same Banach space $X$. We give positive and negative results involving the dimension of $\Lambda$ and the regularity of each map $\lambda\in \Lambda\mapsto T_\lambda^n x$, $x\in X$, $n\in\mathbb N$.

Common hypercyclic vectors for unilateral weighted shifts on ℓ2

Each w∈ℓ∞ defines a bacwards weighted shift Bw:ℓ2→ℓ2. A vector x∈ℓ2 is \textit{hypercyclic} for Bw if the set of forward iterates of x is dense in ℓ2. For each such w, the set HC(w) consisting of all vectors hypercyclic for Bw is Gδ. The set of \textit{common hypercyclic vectors} for a set W⊆ℓ∞ is the set HC∗(W)=⋂w∈WHC(w). We show that HC∗(W) can be made arbitrarily complicated by making W sufficiently complex, and that even for a Gδ set W the set HC∗(W) can be non-Borel. Finally, by assuming the continuum hypothesis or Martin's axiom, we are able to construct a set W such that HC∗(W) does not have the property of Baire.

Common hypercyclic algebras for families of products of backward shifts

In this paper, we generalize to the context of algebras some recent results on the existence of common hypercyclic vectors for families of products of backward shift operators. We also give, in a multi-dimensional setting, a positive answer to a question raised by F. Bayart, D. Papathanasiou and the author about the existence of a common hypercyclic algebra on ℓ1(N) with the convolution product for the family of backward shifts (Bw(λ))λ>0 induced by the weights wn(λ)=1+λ/n.

Baire theorem and hypercyclic algebras

The question of whether a hypercyclic operator T acting on a Fréchet algebra X admits or not an algebra of hypercyclic vectors (but 0) has been addressed in the recent literature. In this paper we give new criteria and characterisations in the context of convolution operators acting on H(C) and backward shifts acting on a general Fréchet sequence algebra. Analogous questions arise for stronger properties like frequent hypercyclicity. In this trend we give a sufficient condition for a weighted backward shift to admit an upper frequently hypercyclic algebra and we find a weighted backward shift acting on c0 admitting a frequently hypercyclic algebra for the coordinatewise product. The closed hypercyclic algebra problem is also covered.

Algebrable sets of hypercyclic vectors for convolution operators

We show that several convolution operators on the space of entire functions, such as the MacLane operator, support a dense hypercyclic algebra that is not finitely generated. Birkhoff’s operator also has this property on the space of complex-valued smooth functions on the real line.

Algebras of frequently hypercyclic vectors

AbstractWe show that the multiples of the backward shift operator on the spaces , , or c0, when endowed with coordinatewise multiplication, do not possess frequently hypercyclic algebras. More generally, we characterize the existence of algebras of ‐hypercyclic vectors for these operators. We also show that the differentiation operator on the space of entire functions, when endowed with the Hadamard product, does not possess frequently hypercyclic algebras. On the other hand, we show that for any frequently hypercyclic operator T on any Banach space, is algebrable for a suitable product, and in some cases it is even strongly algebrable.

Algebrability of the set of hypercyclic vectors for backward shift operators

We study the existence of algebras of hypercyclic vectors for weighted backward shifts on Fréchet sequence spaces that are algebras when endowed with coordinatewise multiplication or with the Cauchy product. As a particular case, we obtain that the sets of hypercyclic vectors for Rolewicz's and MacLane's operators are algebrable.

Common upper frequent hypercyclicity

Considering a family of upper frequently hypercyclic operators we care about the existence of vectors which are upper frequently hypercyclic for any operator of this family. We establish sufficient conditions for a family of operators to have these vectors called common upper frequently hypercyclic vectors. Using this result, we then give a construction of such vectors. Finally we derive some applications to families of weighted shifts.