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On several dynamical properties of shifts acting on directed trees

Part of: General theory of linear operators Normed linear spaces and Banach spaces; Banach lattices Special classes of linear operators Graph theory

Published online by Cambridge University Press:  09 January 2025

Evgeny Abakumov  and
Arafat Abbar Open the ORCID record for Arafat Abbar [Opens in a new window]
Evgeny Abakumov
Affiliation:
Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, LAMA UMR8050 F-77447 Marne-la-Vallée, France e-mail: evgueni.abakoumov@univ-eiffel.fr
Arafat Abbar*
Affiliation:
Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, LAMA UMR8050 F-77447 Marne-la-Vallée, France e-mail: evgueni.abakoumov@univ-eiffel.fr
*
e-mail: abbar.arafat@gmail.com
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Abstract

This article explores the notions of $\mathcal {F}$-transitivity and topological $\mathcal {F}$-recurrence for backward shift operators on weighted $\ell ^p$-spaces and $c_0$-spaces on directed trees, where $\mathcal {F}$ represents a Furstenberg family of subsets of $\mathbb {N}_0$. In particular, we establish the equivalence between recurrence and hypercyclicity of these operators on unrooted directed trees. For rooted directed trees, a backward shift operator is hypercyclic if and only if it possesses an orbit of a bounded subset that is weakly dense.

Keywords

Backward shiftdirected treehypercyclicityorbital limit pointsrecurrence

MSC classification

Primary: 47A16: Cyclic vectors, hypercyclic and chaotic operators
Secondary: 05C05: Trees 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 46B45: Banach sequence spaces

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Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 30
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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