Rigidity and non-recurrence along sequences

We study two properties of a finite measure-preserving dynamical system and a given sequence $({n}_{m} )$ of positive integers, namely rigidity and non-recurrence. Our goal is to find conditions on the sequence which ensure that it is, or is not, a rigid sequence or a non-recurrent sequence for some...

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Bibliographic Details
Published in:Ergodic theory and dynamical systems 2014-10, Vol.34 (5), p.1464-1502
Main Authors: BERGELSON, V., DEL JUNCO, A., LEMAŃCZYK, M., ROSENBLATT, J.
Format: Article
Language:English
Subjects:
Dynamical systems
Dynamics
Ergodic processes
Integers
Mathematical analysis
Mathematics
Rigidity
Sequences
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Summary:We study two properties of a finite measure-preserving dynamical system and a given sequence $({n}_{m} )$ of positive integers, namely rigidity and non-recurrence. Our goal is to find conditions on the sequence which ensure that it is, or is not, a rigid sequence or a non-recurrent sequence for some weakly mixing system or more generally for some ergodic system. The main focus is on weakly mixing systems. For example, we show that for any integer $a\geq 2$ the sequence ${n}_{m} = {a}^{m} $ is a sequence of rigidity for some weakly mixing system. We show the same for the sequence of denominators of the convergents in the continued fraction expansion of any irrational $\alpha $. We also consider the stronger property of IP-rigidity. We show that if $({n}_{m} )$ grows fast enough then there is a weakly mixing system which is IP-rigid along $({n}_{m} )$ and non-recurrent along $({n}_{m} + 1)$.
ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2013.5