Growth of frequently or log-frequently hypercyclic functions
type: Item
instance of: scholarly article
title: Growth of frequently or log-frequently hypercyclic functions
zbMATH Open document ID: 1494.47011
author: Augustin Mouze
author: Vincent Munnier
published in: Journal of Functional Analysis
publication date: 2021-09-14T00:00:00Z
review text: Let \(H(\mathbb{D})\) denote the space analytic functions on the open unit disc equipped with the standard topology of uniform convergence on compact sets. The Taylor shift operator \(T\) on \(H(\mathbb{D})\) is defined as follows for \(z\in\mathbb{D}\) and \(f(z)=\sum_{k\geq0}a_kz^k\): \[ T(f)(z)=T\left(\sum_{k\geq0}a_kz^k\right)=\sum_{k\geq0}a_{k+1}z^k. \] The dynamical properties of this operator (and many others) have been intensively studied in recent years. One of the most studied dynamical notion is that of frequent hypercyclicity introduced in [\textit{F. Bayart} and \textit{S. Grivaux}, Trans. Am. Math. Soc. 358, No. 11, 5083--5117 (2006; Zbl 1115.47005)]. On an infinite-dimensional Fréchet space \(F\), an operator \(T\) is called frequently hypercyclic if there exists a vector \(x\in F\) (also called frequently hypercyclic) such that for any non-empty open set \(U\subseteq F\), the classical lower density of the return sets \(N(x,U):=\{n\in\mathbb{N}:T^nx\in U\}\) is positive where the lower density of a subset \(A\subseteq\mathbb{N}\) is defined as \[ \underline{d}(A):=\liminf_{N\to\infty}\frac{\#\left(A\cap[0;N]\right)}{N+1}. \] One may also define a weaker notion of log-frequent hypercyclicity by replacing the lower density in the previous definition by the so-called log-density \(\underline{d}_{log}\) given by \[ \underline{d}_{log}(A):=\liminf_{N\to\infty}\frac{1}{\log(N)}\sum_{n\in A\cap[0;N]}\frac{1}{n}. \] In particular, when dealing with frequently hypercyclic operators on spaces of analytic functions, a classical subject of interest is the rate of growth of frequently hypercyclic functions. Recall that for \(f\in H(\mathbb{D})\) and \(1\leq p<\infty\), \[ M_p(r,f):=\left(\frac{1}{2\pi}\int_{0}^{2\pi}\left\vert f(re^{i\theta})\right\vert^p d\theta\right)^{\frac{1}{p}}\text{ and }M_{\infty}(r,f):=\sup_{\theta\in[0,2\pi]}{\left\vert f(re^{i\theta})\right\vert}. \] The rate of growth of frequently hypercyclic functions for the Taylor shift on \(H(\ma…
reviewed by: Romuald Ernst
Mathematics Subject Classification ID: 47A16
Mathematics Subject Classification ID: 30E25
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 7395024
zbMATH Keywords: frequently hypercyclic operator
zbMATH Keywords: rate of growth
zbMATH Keywords: boundary behavior
MaRDI profile type: Publication
full work available at URL: j.jfa.2021.109171
OpenAlex ID: W3175030610
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DOI: 10.1016/J.JFA.2021.109171
description: scientific article; zbMATH DE number 7395024