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Rigidity and non-recurrence along sequences
Published online by Cambridge University Press: 16 April 2013
Abstract
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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 
$({n}_{m} + 1)$.
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