Frequent Hypercyclicity Research Articles

Growth of hypercyclic functions: a continuous path between -frequent hypercyclicity and hypercyclicity

Abstract We are interested in the optimal growth in terms of Lp-averages of hypercyclic and $\mathcal{U}$ -frequently hypercyclic functions for some weighted Taylor shift operators acting on the space of analytic functions on the unit disc. We unify the results obtained by considering intermediate notions of upper frequent hypercyclicity between $\mathcal{U}$ -frequent hypercyclicity and hypercyclicity.

Frequent hypercyclicity of the tensor products of composition operators on weighted Dirichlet spaces

In this paper, we mainly study the frequent hypercyclicity of a scalar multiple of the tensor product of two linear fractional composition operators on , where is the weighted Dirichlet space. We will restrict the discussion to the situations that both φ and ψ are hyperbolic non automorphisms, hyperbolic automorphisms and parabolic automorphisms, respectively. Meanwhile, we give the equivalent conditions of to be frequently hypercyclic, hypercyclic, mixing and chaotic.

Disjoint frequent hypercyclicity of composition operators

We give a sufficient condition for two operators to be disjointly frequently hypercyclic. We apply this criterion to composition operators acting on H(D) or on the Hardy space H2(D). We simplify a result on disjoint frequent hypercyclicity of pseudo shifts of a recent paper of Martin et al. and we exhibit two disjointly frequently hypercyclic weighted shifts.

Frequently hypercyclic random vectors

We show that, under suitable conditions, an operator acting like a shift on some sequence space has a frequently hypercyclic random vector whose distribution is strongly mixing for the operator. This result will be applied to chaotic weighted shifts. We also apply it to every operator satisfying the Frequent Hypercyclicity Criterion, recovering a result of Murillo and Peris.

Shift-like operators on Lp(X)

In this article we develop a general technique which takes a known characterization of a property for weighted backward shifts and lifts it up to a characterization of that property for a large class of operators on Lp(X). We call these operators “shift-like”. The properties of interest include chaotic properties such as Li-Yorke chaos, hypercyclicity, frequent hypercyclicity as well as properties related to hyperbolic dynamics such as shadowing, expansivity and generalized hyperbolicity. Shift-like operators appear naturally as composition operators on Lp(X) when the underlying space is a dissipative measure system. In the process of proving the main theorem, we provide some results concerning when a property is shared by a linear dynamical system and its factors.

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.

Disjoint frequently hypercyclic pseudo-shifts

We obtain a Disjoint Frequent Hypercyclicity Criterion and show that it characterizes disjoint frequent hypercyclicity for a family of unilateral pseudo-shifts on c0(N) and ℓp(N), 1≤p<∞. As an application, we characterize disjoint frequently hypercyclic weighted shifts. We give analogous results for the weaker notions of disjoint upper frequent and reiterative hypercyclicity. Finally, we provide counterexamples showing that, although the frequent hypercyclicity, upper frequent hypercyclicity, and reiterative hypercyclicity coincide for weighted shifts on ℓp(N), this equivalence fails for disjoint versions of these notions.

Frequent hypercyclicity and chaoticity of Toeplitz operators and their tensor products

Frequent hypercyclicity and chaoticity of Toeplitz operators and their tensor products

Chaos and frequent hypercyclicity for composition operators

AbstractThe notions of chaos and frequent hypercyclicity enjoy an intimate relationship in linear dynamics. Indeed, after a series of partial results, it was shown by Bayart and Ruzsa in 2015 that for backward weighted shifts on $\ell _p(\mathbb {Z})$, the notions of chaos and frequent hypercyclicity coincide. It is with some effort that one shows that these two notions are distinct. Bayart and Grivaux in 2007 constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$. It was only in 2017 that Menet settled negatively whether every chaotic operator is frequently hypercylic. In this article, we show that for a large class of composition operators on $L^{p}$-spaces, the notions of chaos and frequent hypercyclicity coincide. Moreover, in this particular class, an invertible operator is frequently hypercyclic if and only if its inverse is frequently hypercyclic. This is in contrast to a very recent result of Menet where an invertible operator frequently hypercyclic on $\ell _1$ whose inverse is not frequently hypercyclic is constructed.

Frequent hypercyclicity of random holomorphic functions for Taylor shifts and optimal growth

Frequent hypercyclicity of random holomorphic functions for Taylor shifts and optimal growth

$$\mathcal{U}$$-frequent hypercyclicity notions and related weighted densities

We study dynamical notions lying between $$\mathcal{U}$$ -frequent hypercyclicity and reiterative hypercyclicity by investigating weighted upper densities between the unweighted upper density and the upper Banach density. While chaos implies reiterative hypercyclicity, we show that chaos does not imply $$\mathcal{U}$$ -frequent hypercyclicity with respect to any weighted upper density. Moreover, we show that if T is $$\mathcal{U}$$ -frequently hypercyclic (resp. reiteratively hypercyclic) then the n-fold product of T is still U-frequently hypercyclic (resp. reiteratively hypercyclic) and that this implication is also satisfied for each of the considered $$\mathcal{U}$$ -frequent hypercyclicity notions.

Frequent hypercyclicity of weighted composition operators on the space of smooth functions

Frequent hypercyclicity of weighted composition operators on the space of smooth functions

Chaos and frequent hypercyclicity for weighted shifts

AbstractBayart and Ruzsa [Difference sets and frequently hypercyclic weighted shifts. Ergod. Th. &amp; Dynam. Sys.35 (2015), 691–709] have recently shown that every frequently hypercyclic weighted shift on $\ell ^p$ is chaotic. This contrasts with an earlier result of Bayart and Grivaux [Frequently hypercyclic operators. Trans. Amer. Math. Soc.358 (2006), 5083–5117], who constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$ . We first generalize the Bayart–Ruzsa theorem to all Banach sequence spaces in which the unit sequences form a boundedly complete unconditional basis. We then study the relationship between frequent hypercyclicity and chaos for weighted shifts on Fréchet sequence spaces, in particular, on Köthe sequence spaces, and then on the special class of power series spaces. We obtain, rather curiously, that every frequently hypercyclic weighted shift on $H(\mathbb {D})$ is chaotic, while $H(\mathbb {C})$ admits a non-chaotic frequently hypercyclic weighted shift.

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.

Strongly Mixing Convolution Operators on Fréchet Spaces of Entire Functions of a Given Type and Order

We show that convolution operators on certain spaces of entire functions of a given type and order on Banach spaces are strongly mixing with respect to an invariant Borel probability measure with full support (a stronger property than frequent hypercyclicity). Based on results of S. Muro, D. Pinasco and M. Savransky we also show the existence of frequently hypercyclic entire functions of exponential growth, and the existence of frequently hypercyclic subspaces for such convolution operators.

Linear Dynamical Systems

This expository survey is dedicated to recent developments in the area of linear dynamics. Topics include frequent hypercyclicity, $\mathcal{U}$-frequent hypercyclicity, reiterative hypercyclicity, operators of C-type, Li-Yorke and distributional chaos, and hypercyclic algebras.

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.

Frequent universality criterion and densities

We improve a recent result by giving the optimal conclusion both to the frequent universality criterion and the frequent hypercyclicity criterion using the notion of $A$-densities, where $A$ refers to some weighted densities sharper than the natural lower density. Moreover, we construct an operator which is logarithmically frequently hypercyclic but not frequently hypercyclic.

A bridge between [formula omitted]-frequent hypercyclicity and frequent hypercyclicity

Given A the family of weights a=(an)n decreasing to 0 such that the series ∑n=0∞an diverges, we show that the supremum on A of lower weighted densities coincides with the unweighted upper density and that the infimum on A of upper weighted densities coincides with the unweighted lower density. We then investigate the notions of U-frequent hypercyclicity and frequent hypercyclicity associated to these weighted densities. We show that there exists an operator which is U-frequently hypercyclic for each weight in A but not frequently hypercyclic, although the set of frequently hypercyclic vectors always coincides with the intersection of sets of U-frequently hypercyclic vectors for each weight in A.

Dynamical properties of random walks

In this paper, we study dynamical properties such as hypercyclicity, supercyclicity, frequent hypercyclicity and chaoticity for transition operators associated to countable irreducible Markov chains. As particular cases, we consider simple random walks on [Formula: see text] and [Formula: see text].