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Rigidity and non-recurrence along sequences

Published online by Cambridge University Press:  16 April 2013

V. BERGELSON ,
A. DEL JUNCO ,
M. LEMAŃCZYK  and
J. ROSENBLATT
V. BERGELSON
Affiliation:
Department of Mathematics, Ohio State University, Columbus, OH 43210, USA email [email protected]
A. DEL JUNCO
Affiliation:
Department of Mathematics, University of Toronto, Toronto, M5S 3G3, Canada email [email protected]
M. LEMAŃCZYK
Affiliation:
Faculty of Math and Computer Science, Nicolaus Copernicus University, Toruń, Poland email [email protected]
J. ROSENBLATT
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA email [email protected]
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Abstract

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 $({n}_{m} + 1)$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 5 , October 2014 , pp. 1464 - 1502
Copyright
Copyright ©2013 Cambridge University Press 

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