Frequent Hypercyclicity Research Articles

The Specification Property for $$C_0$$-Semigroups

We study one of the strongest versions of chaos for continuous dynamical systems, namely the specification property. We extend the definition of specification property for operators on a Banach space to strongly continuous one-parameter semigroups of operators, that is, $$C_0$$ -semigroups. In addition, we study the relationships of the specification property for $$C_0$$ -semigroups (SgSP) with other dynamical properties: mixing, Devaney’s chaos, distributional chaos, and frequent hypercyclicity. Concerning the applications, we provide several examples of semigroups which exhibit the SgSP with particular interest on solution semigroups to certain linear PDEs, which range from the hyperbolic heat equation to the Black–Scholes equation.

FREQUENTLY HYPERCYCLIC BILATERAL SHIFTS

AbstractIt is not known whether the inverse of a frequently hypercyclic bilateral weighted shift on c0(ℤ) is again frequently hypercyclic. We show that the corresponding problem for upper frequent hypercyclicity has a positive answer. We characterise, more generally, when bilateral weighted shifts on Banach sequence spaces are (upper) frequently hypercyclic.

Disjoint Hypercyclicity and Topological Entropy of Composition Operators

In a previous paper, we characterized the Devaney chaos, frequent hypercyclicity and dense distributional chaos of composition operators induced by continuous self-maps on the real line. The present paper further investigates the disjoint hypercyclicity and topological entropy of these operators. It is shown that the composition operator is [Formula: see text]-transitive if and only if it is Cesàro-hypercyclic, if and only if it is supercyclic, if and only if it has the specification property on the whole space. Furthermore, sufficient and necessary conditions for a pair of composition operators to be disjoint hypercyclic (disjoint mixing, respectively) are obtained. Finally, sufficient conditions for the composition operator to admit infinite topological entropy are provided.

Upper frequent hypercyclicity and related notions

Enhancing a recent result of Bayart and Ruzsa we obtain a Birkhoff-type characterization of upper frequently hypercyclic operators and a corresponding Upper Frequent Hypercyclicity Criterion. As an application we characterize upper frequently hypercyclic weighted backward shifts on sequence spaces, which in turn allows us to come up with various counter-examples in linear dynamics that are substantially simpler than those previously obtained in the literature. More generally, we introduce the notion of upper Furstenberg families $$\mathcal {A}$$ and show that our main results hold for the corresponding $$\mathcal {A}$$ -hypercyclic operators.

Frequently hypercyclic operators with irregularly visiting orbits

We prove that a bounded operator T on a separable Banach space X satisfying a strong form of the Frequent Hypercyclicity Criterion (which implies in particular that the operator is universal in the sense of Glasner and Weiss) admits frequently hypercyclic vectors with irregularly visiting orbits, i.e. vectors x∈X such that the set NT(x,U)={n≥1;Tnx∈U} of return times of x into U under the action of T has positive lower density for every non-empty open set U⊆X, but there exists a non-empty open set U0⊆X such that NT(x,U0) has no density.

A quantitative interpretation of the frequent hypercyclicity criterion

We give a quantitative interpretation of the frequent hypercyclicity criterion. Actually we show that an operator which satisfies the frequent hypercyclicity criterion is necessarily $A$-frequently hypercyclic, where $A$ refers to some weighted densities sharper than the natural lower density. In that order, we exhibit different scales of weighted densities that are of interest to quantify the ‘frequency’ measured by the frequent hypercyclicity criterion. Moreover, we construct an example of a unilateral weighted shift which is frequently hypercyclic but not $A$-frequently hypercyclic on a particular scale.

Chaotic Dynamics of Composition Operators on the Space of Continuous Functions

In this paper, the chaotic dynamics of composition operators on the space of real-valued continuous functions is investigated. It is proved that the hypercyclicity, topologically mixing property, Devaney chaos, frequent hypercyclicity and the specification property of the composition operator are equivalent to each other and are stronger than dense distributional chaos. Moreover, the composition operator [Formula: see text] exhibits dense Li–Yorke chaos if and only if it is densely distributionally chaotic, if and only if the symbol [Formula: see text] admits no fixed points. Finally, the long-time behaviors of the composition operator with affine symbol are classified in detail.

Linear dynamics of semigroups generated by differential operators

Abstract 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 of C0-semigroups of linear and continuous operators. We will review some of these notions and we will discuss basic properties of the dynamics of C0-semigroups. We will also study in detail the dynamics of the translation C0-semigroup on weighted spaces of integrable functions and of continuous functions vanishing at infinity. Using the comparison lemma, these results can be transferred to the solution C0-semigroups of some partial differential equations. Additionally, we will also visit the chaos for infinite systems of ordinary differential equations, that can be of interest for representing birth-and-death process or car-following traffic models.

Linear chaos and frequent hypercyclicity

We answer one of the main current questions in Linear Dynamics by constructing a chaotic operator which is not U \mathcal {U} -frequently hypercyclic and thus not frequently hypercyclic. This operator also gives us an example of a chaotic operator which is not distributionally chaotic and an example of a chaotic operator with only countably many unimodular eigenvalues. We complement this result by showing that every chaotic operator is reiteratively hypercyclic.

Q-Frequent hypercyclicity in spaces of operators

We provide conditions for a linear map of the form $$C_{R,T}(S)=RST$$ to be q-frequently hypercyclic on algebras of operators on separable Banach spaces. In particular, if R is a bounded operator satisfying the q-frequent hypercyclicity criterion, then the map $$C_{R}(S)=RSR^*$$ is shown to be q-frequently hypercyclic on the space $$\mathcal {K}(H)$$ of all compact operators and the real topological vector space $$\mathcal {S}(H)$$ of all self-adjoint operators on a separable Hilbert space H. Further we provide a condition for $$C_{R,T}$$ to be q-frequently hypercyclic on the Schatten von Neumann classes $$S_p(H)$$ . We also characterize frequent hypercyclicity of $$C_{M^*_\varphi ,M_\psi }$$ on the trace-class of the Hardy space, where the symbol $$M_\varphi $$ denotes the multiplication operator associated to $$\varphi $$ .

Frequently Hypercyclic Taylor Shifts

It is known that unimodular eigenvalues of an operator can give information about its dynamical behaviour. Recently, some situations have been characterized in which the Taylor shift operator is hypercyclic. The aim of this article is to use an eigenvalue criterion to find assumptions that guarantee the frequent hypercyclicity of the Taylor shift operator. As a conclusion, we also obtain holomorphic functions with smooth boundary values.

Hypercyclicity of the tensor products of multiplication operators on Hardy spaces

In this paper, we give a necessary and sufficient condition for the tensor product of two adjoints of multiplication hypercyclic operators to be mixing and chaotic. These results partially answer a question posed by Félix Martínez-Giménez and Alfredo Peris in 2003. We also prove that the tensor product of an operator satisfying the Frequent Hypercyclicity Criterion with the identity is frequently hypercyclic, thereby strengthening a previous result of Martínez-Giménez and Peris.

Operators with the specification property

We study a version of the specification property for linear dynamics. Operators having the specification property are investigated, and relationships with other well known dynamical notions such as mixing, Devaney chaos, and frequent hypercyclicity are obtained.

Recurrence properties of hypercyclic operators

We generalize the notions of hypercyclic operators, $\mathfrak{U}$-frequently hypercyclic operators and frequently hypercyclic operators by introducing a new notion of hypercyclicity, called $\mathcal{A}$-frequent hypercyclicity. We then state an $\mathcal{A}$-Frequent Hypercyclicity Criterion, inspired from the Hypercyclicity Criterion and the Frequent Hypercyclicity Criterion, and we show that this criterion characterizes the $\mathcal{A}$-frequent hypercyclicity for weighted shifts. We finish by investigating which kind of properties of density can have the sets ${N(x, U)=\{n\in \mathbb{N}:T^nx\in U\}}$ for a given hypercyclic operator and study the new notion of reiteratively hypercyclic operators.

Dynamics of composition operators on weighted Bergman spaces

We discuss the frequent hypercyclicity and hypercyclicity of multiples of the linear fractional composition operators on Aαp and give equivalent conditions of the frequent hypercyclicity and hypercyclicity. We also characterize disjoint hypercyclicity and disjoint mixing behavior of finitely many linear fractional composition operators on the weighted Bergman spaces Aαp.

A remark on the Frequent Hypercyclicity Criterion for weighted composition semigroups and an application to the linear von Foerster-Lasota equation

We generalize a result for the translation $C_0$-semigroup on $L^p(\R_+,\mu)$ about the equivalence of being chaotic and satisfying the Frequent Hypercyclicity criterion due to Mangino and Peris to certain weighted composition $C_0$-semigroups. Such $C_0$-semigroups appear in a natural way when dealing with initial value problems for linear first order partial differential operators. We apply our result to the linear von Foerster-Lasota equation arising in mathematical biology. Weighted composition $C_0$-semigroups on Sobolev spaces are also considered.

Hereditarily hypercyclic subspaces

We say that a sequence of operators $(T_n)$ possesses hereditarily hypercyclic subspaces along a sequence $(n_k)$ if for any subsequence $(m_k)\subset(n_k)$, the sequence $(T_{m_k})$ possesses a hypercyclic subspace. While so far no characterization of the existence of hypercyclic subspaces in the case of Fr\'{e}chet spaces is known, we succeed to obtain a characterization of sequences $(T_n)$ possessing hereditarily hypercyclic subspaces along $(n_k)$, under the assumption that the sequence $(T_n)$ satisfies the Hypercyclicity Criterion along $(n_k)$. We also obtain a characterization of operators possessing a hypercyclic subspace under the assumption that $T$ satisfies the Frequent Hypercyclicity Criterion.

$q$-Frequently hypercyclic operators

We introduce $q$-frequently hypercyclic operators and derive a sufficient criterion for a continuous operator to be $q$-frequently hypercyclic on a locally convex space. Applications are given to obtain $q$-frequently hypercyclic operators with respect to the norm-, $F$-norm- and weak*- topologies. Finally, the frequent hypercyclicity of the non-convolution operator $T_\mu$ defined by $T_\mu(f)(z)=f'(\mu z)$, $|\mu|\geq 1$ on the space $H(\mathbb{C})$ of entire functions equipped with the compact-open topology is shown.

Frequently hypercyclic translation semigroups

Frequent hypercyclicity for translation $C_0$-semigroups on weighted spaces of continuous functions is studied. The results are achieved by establishing an analogy between frequent hypercyclicity for translation semigroups and for weighted pseudo-shifts a

Hypercyclic behaviour of multiples of composition operators on weighted Banach spaces of holomorphic functions

Let $S(\mathbb{D})$ be the collection of all the holomorphic self-maps of open unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$, and $C_{\varphi}$, the composition operator induced by $\varphi\in S(\mathbb{D})$. For $\alpha>0,\;\lambda\in \mathbb{C}$, we give some sufficient and necessary conditions for the hypercyclicity of multiples of composition operators $\lambda C_\varphi$ acting on the weighted Banach spaces of entire functions $H_{\alpha,0}^\infty$. Moreover, we obtain a partial characterization for the frequent hypercyclicity of $\lambda C_\varphi$ on $H_{\alpha,0}^\infty$.