Frequent Hypercyclicity Criterion Research Articles

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.

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 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.

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.

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.

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.

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.

Strong mixing measures for $$C_0$$ C 0 -semigroups

Our purpose is to obtain a very effective and general method to prove that certain $$C_0$$ -semigroups admit invariant strongly mixing measures. More precisely, we show that the frequent hypercyclicity criterion for $$C_0$$ -semigroups ensures the existence of invariant strongly mixing measures with full support. We will provide several examples, that range from birth-and-death models to the Black–Scholes equation, which illustrate these results.

Strong mixing measures for linear operators and frequent hypercyclicity

We construct strongly mixing invariant measures with full support for operators on F-spaces which satisfy the Frequent Hypercyclicity Criterion. For unilateral backward shifts on sequence spaces, a slight modification shows that one can even obtain exact invariant measures.

Frequently hypercyclic operators and vectors

We study frequently hypercyclic operators, a natural new concept in hypercyclicity that was recently introduced by F. Bayart and S. Grivaux. We derive a strengthened version of their Frequent Hypercyclicity Criterion, which allows us to obtain examples of frequently hypercyclic operators in a straightforward way. Moreover, Bayart and Grivaux have noted that the frequent hypercyclicity setting differs from general hypercyclicity in that the set of frequently hypercyclic vectors need not be residual. We show here that, under weak assumptions, this set is only of first category. Motivated by this we study the question of whether one may write every vector in the underlying space as the sum of two frequently hypercyclic vectors. This investigation leads us to the introduction of a new notion, that of Runge transitivity.

A probabilistic version of the Frequent Hypercyclicity Criterion

For a bounded operator $T$ on a separable infinite-dimensional Banach space $X$, we give a &#8220;random&#8221; criterion not involving ergodic theory which implies that $T$ is frequently hypercyclic: there exists a vector $x$ such that for every non-empt