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Non-recurrence sets for weakly mixing linear dynamical systems

Published online by Cambridge University Press:  24 August 2012

SOPHIE GRIVAUX
SOPHIE GRIVAUX*
Affiliation:
CNRS, Laboratoire Paul Painlevé, UMR 8524, Université Lille 1, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France (email: grivaux@math.univ-lille1.fr)
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Abstract

We study non-recurrence sets for weakly mixing dynamical systems by using linear dynamical systems. These are systems consisting of a bounded linear operator acting on a separable complex Banach space $X$, which becomes a probability space when endowed with a non-degenerate Gaussian measure. We generalize some recent results of Bergelson, del Junco, Lemańczyk and Rosenblatt, and show in particular that sets $\{n_{k}\}$ such that $n_{k+1}/n_{k}\to +\infty $, or such that $n_{k}$ divides $n_{k+1}$ for each $k\ge 0$, are non-recurrence sets for weakly mixing linear dynamical systems. We also give examples, for each $r\ge 1$, of $r$-Bohr sets which are non-recurrence sets for some weakly mixing systems.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 1 , February 2014 , pp. 132 - 152
Copyright
Copyright © 2012 Cambridge University Press 

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