In this paper we extend the notion of a locally hypercyclic operator to that of a locally hypercyclic tuple of operators. We then show that the class of hypercyclic tuples of operators forms a proper subclass to that of locally hypercyclic tuples of operators. What is rather remarkable is that in every finite dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$, a pair of commuting matrices exists which forms a locally hypercyclic, non-hypercyclic tuple. This comes in direct contrast to the case of hypercyclic tuples where the minimal number of matrices required for hypercyclicity is related to the dimension of the vector space. In this direction we prove that the minimal number of diagonal matrices required to form a hypercyclic tuple on $\mathbb{R}^n$ is $n+1$, thus complementing a recent result due to Feldman.
We characterize disjoint and simultaneously hypercyclic tuples of unilateral pseudo-shift operators on $\ell^p(\mathbb{N})$. As a consequence, complementing the results of Bernal and Jung, we give a characterization for simultaneously hypercyclic tuples of unilateral weighted shifts. We also give characterizations for unilateral pseudo-shifts that satisfy the Disjoint and Simultaneous Hypercyclicity Criterions. Contrary to the disjoint hypercyclicity case, tuples of weighted shifts turn out to be simultaneously hypercyclic if and only if they satisfy the Simultaneous Hypercyclicity Criterion.
We give a short proof of the existence of disjoint hypercyclic tuples of operators of any given length on any separable infinite dimensional Frechet space. A similar argument provides disjoint dual hypercyclic tuples of operators of any length on any infinite dimensional Banach space with separable dual.
A tuple (T_1,...,T_k) of (n x n) matrices over R is called hypercyclic if for some x in R^n the set {T^{m_1} T^{m_2}...T^{m_k} x : m_1,m_2,...,m_k in N} is dense in R^n. We prove that the minimum number of (n x n) matrices in Jordan form over R which form a hypercyclic tuple is n+1. This answers a question of Costakis, Hadjiloucas and Manoussos.
For any countable group, and also for any locally compact second countable, compactly generated topological group, G, we show the existence of a "universal" hypercyclic (i.e. topologically transitive) representation on a Hilbert space, in the sense that it simultaneously models every possible ergodic probability measure preserving free action of G.
We give a characterization of hypercyclic using (locally hypercyclic) of semigroup G of affine maps of C^n. We prove the existence of a G-invariant open subset of C^n in which any locally hypercyclic orbit is dense in C^n.
According to Grivaux, the group $GL(X)$ of invertible linear operators on a separable infinite dimensional Banach space $X$ acts transitively on the set $Σ(X)$ of countable dense linearly independent subsets of $X$. As a consequence, each $A\in Σ(X)$ is an orbit of a hypercyclic operator on $X$. Furthermore, every countably dimensional normed space supports a hypercyclic operator. We show that for a separable infinite dimensional Fréchet space $X$, $GL(X)$ acts transitively on $Σ(X)$ if and only if $X$ possesses a continuous norm. We also prove that every countably dimensional metrizable local…
Bonet, Frerick, Peris and Wengenroth constructed a hypercyclic operator on the locally convex direct sum of countably many copies of the Banach space $\ell_1$. We extend this result. In particular, we show that there is a hypercyclic operator on the locally convex direct sum of a sequence $\{X_n\}_{n\in\N}$ of Fréchet spaces if and only if each $X_n$ is separable and there are infinitely many $n\in\N$ for which $X_n$ is infinite dimensional. Moreover, we characterize inductive limits of sequences of separable Banach spaces which support a hypercyclic operator.
In this paper, we characterize hypercyclic sequences of weighted translation operators on an Orlicz space in the context of locally compact hypergroups.
In this article we answer a question raised by N. Feldman in \cite{Feldman} concerning the dynamics of tuples of operators on $\mathbb{R}^n$. In particular, we prove that for every positive integer $n\geq 2$ there exist $n$ tuples $(A_1, A_2, ..., A_n)$ of $n\times n$ matrices over $\mathbb{R}$ such that $(A_1, A_2, ..., A_n)$ is hypercyclic. We also establish related results for tuples of $2\times 2$ matrices over $\mathbb{R}$ or $\mathbb{C}$ being in Jordan form.
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 $T_μ$ defined by $T_μ(f)(z) = f'(μz)$, $μ\ge1$ on the space $H(\mathbb{C})$ of entire functions equipped with the compact-open topology is shown.
We give a characterization of hypercyclic finitely generated abelian semigroups of matrices on C^n using the extended limit sets (the J-sets). Moreover we construct for any n\geq 2 an abelian semigroup G of GL(n;C) generated by n + 1 diagonal matrices which is locally hypercyclic but not hypercyclic and such that JG(e_k) = C^n for every k = 1; : : : ; n, where (e_1; : : : ; e_n) is the canonical basis of C^n. This gives a negative answer to a question raised by Costakis and Manoussos.
Let $G$ be a locally compact group, and let $1\le p < \infty$. Consider the weighted $L^p$-space $L^p(G,ω)=\{f:\int|fω|^p<\infty\}$, where $ω:G\to \mathbb R$ is a positive measurable function. Under appropriate conditions on $ω$, $G$ acts on $L^p(G,ω)$ by translations. When is this action hypercyclic, that is, there is a function in this space such that the set of all its translations is dense in $L^p(G,ω)$? H. Salas (1995) gave a criterion of hypercyclicity in the case $G=\mathbb Z$ . Under mild assumptions, we present a corresponding characterization for a general locally compact group $G$. …
We say that a (countably dimensional) topological vector space $X$ is orbital if there is $T\in L(X)$ and a vector $x\in X$ such that $X$ is the linear span of the orbit ${T^nx:n=0,1,...}$. We say that $X$ is strongly orbital if, additionally, $x$ can be chosen to be a hypercyclic vector for $T$. Of course, $X$ can be orbital only if the algebraic dimension of $X$ is finite or infinite countable. We characterize orbital and strongly orbital metrizable locally convex spaces. We also show that every countably dimensional metrizable locally convex space $X$ does not have the invariant subset prop…
Let $B$, $I$ be the unweighted backward shift and the identity operator respectively on $l^{\infty}(\mathbb{N})$, the space of bounded sequences over the complex numbers endowed with the supremum norm. We prove that $I+λB$ is locally topologically transitive if and only if $|λ|>2$. This, shows that a classical result of Salas, which says that backward shift perturbations of the identity operator are always hypercyclic, or equivalently topologically transitive, on $l^p(\mathbb{N})$, $1\leq p<+\infty$, fails to hold for the notion of local topological transitivity on $l^{\infty}(\mathbb{N})$. We…
Motivated by the theory of weighted shifts on directed trees and its multivariable counterpart, we address the question of identifying commutant and reflexivity of the multiplication $d$-tuple $\mathscr M_z$ on a reproducing kernel Hilbert space $\mathscr H$ of $E$-valued holomorphic functions on $Ω$, where $E$ is a separable Hilbert space and $Ω$ is a bounded domain in $\mathbb C^d$ admitting bounded approximation by polynomials. In case $E$ is a finite dimensional cyclic subspace for $\mathscr M_z$, under some natural conditions on the $B(E)$-valued kernel associated with $\mathscr H$, the c…
We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space, and study generalized Ruelle operators and $ C^{\ast} $-algebras associated to these groupoids. We provide a new characterization of $ 1 $-cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $-tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle-Perron-Frobenius theory for dynamical systems of several commuting operators ($ k $-Ruelle triples and c…
