arXiv subjects

David Walmsley

Publications and source records attributed to David Walmsley.

5 recordsLinked to original sources

arXiv2026

Chain recurrent shifts on trees

We characterize when a weighted backward shift is chain recurrent on the $\ell^p$ ($1\leq p<\infty$) and $c_0$ spaces of a directed tree. The characterization is given in terms of two divergence conditions on the weights: a forward condition on the descendants of each vertex and, in the unrooted case, a backward condition on the descendants of each ancestor. The conditions reduce, in the case of symmetric weighted shifts on symmetric trees, to the classical characterizations of chain recurrence on the sequence spaces $\ell^p(\mathbb{N})$, $\ell^p(\mathbb{Z})$, $c_0(\mathbb{N})$, and $c_0(\mathbb{Z})$.

math.FA↗

arXiv2023

Ultra hypercyclicity and its connection to mixing properties

Recently, two new topological properties for operators acting on a topological vector space were introduced: strong hypercyclicity and hypermixing. We introduce a new property called ultra hypercyclicity and compare it to strong hypercyclicity and hypermixing, as well as the classical notions of mixing, weak mixing, and hypercyclicity. We show that every ultra hypercyclic operator on Fr\'echet space must be weakly mixing, and that there exists a strongly hypercyclic operator which is not ultra hypercyclic. We also characterize, in terms of the weight sequence, the ultra hypercyclic weighted backward shifts on $c_0$ and $\ell^p$, $1\leq p<\infty$. Finally, we improve upon a necessary condition for strongly hypercyclic weighted backward shifts.

math.FA↗

arXiv2023

Strong topological transitivity, hypermixing, and their relationships with other dynamical properties

Recently, two stronger versions of dynamical properties have been introduced and investigated: strong topological transitivity, which is a stronger version of the topological transitivity property, and hypermixing, which is a stronger version of the mixing property. We continue the investigation of these notions with two main results. First, we show there are dynamical systems which are strongly topologically transitive but not weakly mixing. We then show that on $\ell^p$ or $c_0$, there is a weighted backward shift which is strongly topologically transitive but not mixing.

math.FA↗

arXiv2020

Two Families of Hypercyclic Non-Convolution Operators

Let $H(\mathbb{C})$ be the set of all entire functions endowed with the topology of uniform convergence on compact sets. Let $\lambda,b\in\mathbb{C}$, let $C_{\lambda,b}:H(\mathbb{C})\to H(\mathbb{C})$ be the composition operator $C_{\lambda,b} f(z)=f(\lambda z+b)$, and let $D$ be the derivative operator. We extend results on the hypercyclicity of the non-convolution operators $T_{\lambda,b}=C_{\lambda,b} \circ D$ by showing that whenever $|\lambda|\geq 1$, the collection of operators \begin{align*} \{\psi(T_{\lambda,b}): \psi(z)\in H(\mathbb{C}), \psi(0)=0 \text{ and } \psi(T_{\lambda,b}) \text{ is continuous}\} \end{align*} forms an algebra under the usual addition and multiplication of operators which consists entirely of hypercyclic operators (i.e., each operator has a dense orbit). We also show that the collection of operators \begin{align*} \{C_{\lambda,b}\circ\varphi(D): \varphi(z) \text{ is an entire function of exponential type with } \varphi(0)=0\} \end{align*} consists entirely of hypercyclic operators.

math.FA↗

arXiv2019

Linear Factorization of Hypercyclic Functions for Differential Operators

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 infinite product of linear polynomials: \[ f(z) = \prod_{j=1}^\infty \, \left( 1-\frac{z}{a_j}\right), \] where each $a_j$ is a nonzero complex number.

math.FA↗