Related papers: Frequent hypercyclicity of random entire functions…

We study the rate of growth of entire functions that are frequently hypercyclic with respect to some upper weighted densities for the differentiation operator. The statements obtained show the link between the minimal growth of frequently…

On the Fr\'{e}chet space of entire functions $H(\mathbb{C})$, we show that every nonscalar continuous linear operator $L:H(\mathbb{C})\to H(\mathbb{C})$ which commutes with differentiation has a hypercyclic vector $f(z)$ in the form of the…

We find some sufficient conditions for a system of partial derivatives of an entire function to be complete in the space $H(\mathbb{C}^d)$ of all entire functions of $d$ variables. As an appliation of this result we describe new classes of…

We study the rate of growth of entire functions that are distributionally irregular for the differentiation operator D. More specifically, given $p \in [1,\infty ]$ and $b \in (0,a)$, where $a = \frac{1}{2 max\{2,p\}}$, we prove that there…

We investigate the conjugate indicator diagram or, equivalently, the indicator function of (frequently) hypercyclic functions of exponential type for differential operators. This gives insights into growth conditions of these functions on…

A criterion to obtain frequent hypercyclicity for a sequence of convolution operators on the space of entire functions on the complex plane is provided. The criterion involves that the generating functions of the operators do not vanish on…

We introduce and study the notion of hereditary frequent hypercyclicity, which is a reinforcement of the well known concept of frequent hypercyclicity. This notion is useful for the study of the dynamical properties of direct sums of…

We solve a problem posed by A. Bonilla and K.-G. Grosse-Erdmann by constructing an entire function $f$ that is frequently hypercyclic with respect to the differentiation operator, and satisfies $M_f(r)\leq\displaystyle ce^r r^{-1/4}$, where…

A theorem of Godefroy and Shapiro states that non-trivial convolution operators on the space of entire functions on $\mathbb{C}^n$ are hypercyclic. Moreover, it was shown by Bonilla and Grosse-Erdmann that they have frequently hypercyclic…

For the Dunkl operator $\Lambda_\alpha$ $(\alpha > -1/2)$ on the space of entire functions on the complex space C, the critical rate of growth for the integral means $M_p(f,r)$ of their hypercyclic functions $f$ is obtained. The rate of…

We solve several problems on frequently hypercyclic operators. Firstly, we characterize frequently hypercyclic weighted shifts on $\ell^p(\mathbb Z)$, $p\geq 1$. Our method uses properties of the difference set of a set with positive upper…

It is proved that, if $(P_n)$ is a sequence of polynomials with complex coefficients having unbounded valences and tending to infinity at sufficiently many points, then there is an infinite dimensional closed subspace of entire functions,…

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…

Let (k(n)) n=1,2,... be a strictly increasing sequence of positive integers . We consider a specific sequence of differential operators Tk(n),{\lambda} , n=1,2,... on the space of entire functions , that depend on the sequence (k(n))…

In this article, we study exponents which preserve complete monotonicity of functions on lattices. We prove that for any completely monotone function $f$ on a finite lattice, $f^\alpha$ is completely monotone for all $\alpha\geq c$, where…

We study multiply recurrent and hypercyclic operators as a special case of $\mathcal F$-hypercyclicity, where $\mathcal F$ is the family of subsets of the natural numbers containing arbitrarily long arithmetic progressions. We prove several…

For an analytic function $f(z)$ on the unit disk $|z|<1$ with $f(0)=f'(0)-1=0$ and $f(z)\ne0, 0<|z|<1,$ we consider the power deformation $f_c(z)=z(f(z)/z)^c$ for a complex number $c.$ We determine those values $c$ for which the operator…

Value distribution and uniqueness problems of difference operator of an entire function have been investigated in this article. This research shows that a finite ordered entire function $ f $ when sharing a set $ \mathcal{S}=\{\alpha(z),…

Let $D$ be the differentiation operator $Df=f'$ acting on the Fr\'echet space $\H$ of all entire functions in one variable with the standard (compact-open) topology. It is known since 1950's that the set $H(D)$ of hypercyclic vectors for…

In this paper, we introduce the notions of $f$-frequent hypercyclicity and ${\mathcal F}$-hypercyclicity for $C$-distribution semigroups in separable Fr\'echet spaces. We particularly analyze the classes of $q$-frequently hypercyclic…