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Linear dynamics and recurrence properties defined via essential idempotents of $\unicode[STIX]{x1D6FD}\mathbb{N}$

Published online by Cambridge University Press:  13 July 2016

YUNIED PUIG DE DIOS
YUNIED PUIG DE DIOS*
Affiliation:
Università degli Studi di Milano, Dipartimento di Matematica “Federigo Enriques”, Via Saldini 50, 20133 Milano, Italy email puigdedios@gmail.com
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Abstract

Consider $\mathscr{F}$, a non-empty set of subsets of $\mathbb{N}$. An operator $T$ on $X$ satisfies property ${\mathcal{P}}_{\mathscr{F}}$ if, for any non-empty open set $U$ in $X$, there exists $x\in X$ such that $\{n\geq 0:T^{n}x\in U\}\in \mathscr{F}$. Let $\overline{{\mathcal{B}}{\mathcal{D}}}$ be the collection of sets in $\mathbb{N}$ with positive upper Banach density. Our main result is a characterization of a sequence of operators satisfying property ${\mathcal{P}}_{\overline{{\mathcal{B}}{\mathcal{D}}}}$, for which we have used a deep result of Bergelson and McCutcheon in the vein of Szemerédi’s theorem. It turns out that operators having property ${\mathcal{P}}_{\overline{{\mathcal{B}}{\mathcal{D}}}}$ satisfy a kind of recurrence described in terms of essential idempotents of $\unicode[STIX]{x1D6FD}\mathbb{N}$. We will also discuss the case of weighted backward shifts. Finally, we obtain a characterization of reiteratively hypercyclic operators.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 1 , February 2018 , pp. 285 - 300
Copyright
© Cambridge University Press, 2016 

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