Recurrence properties of hypercyclic operators

By J. Bes, Quentin Menet, Alfredo Peris, Y. D. Dios. Mathematische Annalen, 2014

90 citations recorded by the source corpus.

We generalize the notions of hypercyclic operators, U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {U}$$\end{document}-frequently hypercyclic operators and frequently hypercyclic operators by introducing a new concept in linear dynamics, namely A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document}-hypercyclicity. We then state an A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document}-hypercyclicity criterion, inspired by the hypercyclicity criterion and the frequent hypercyclicity criterion, and we show that this criterion characterizes the A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {A}$$\end{document}-hypercyclicity for weighted shifts. We also investigate which density properties can the sets N(x,U)={n∈N;Tnx∈U}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N(x, U)=\{n\in \mathbb {N}\ ; \ T^nx\in U\}}$$\end{document} have for a given hypercyclic operator, and we study the new notion of reiteratively hypercyclic operators.

Subfields: Algebra and Number Theory, Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/42e8132264fd61e9ccb8d16c74a896c92deff172

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Upper frequent hypercyclicity and related notions (2016, background): “ε. Then T is upper frequently hypercyclic. In the same way we have a Reiterative Hypercyclicity Criterion when we replace upper density by upper Banach density. Remark 25. B`es, Menet, Peris and Puig [7] also state an A-Hypercyclicity Criterion, and it is important to understand the diﬀerence betwee”
Expansivity and Shadowing in Linear Dynamics (2016, background): “We refer the reader to the books [2, 19] and to the more recent papers [3, 6, 7, 8, 17, 18], where many additional references can be found.”
Linear dynamical systems on Hilbert spaces: typical properties and explicit examples (2017, background): “One can go one step further and study F-hypercyclic operators, for a given family F of subsets of N (see in particular [13] and [15]); but we will not follow this quite interesting route here.”
Frequently recurrent operators (2020, methodology): “By [9, Theorem 14], if T is reiteratively hypercyclic then every hyper-cyclic vector is reiteratively hypercyclic; and the set of hypercyclic vectors is residual.”