Rigidity and non-recurrence along sequences
By Vitaly Bergelson, A. DEL JUNCO, Mariusz Lemanczyk, Joseph M. Rosenblatt. Ergodic Theory and Dynamical Systems, 2011
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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 $({n}_{m} + 1)$.
Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/4e73d4055f4d55ef965e0f8b503cc960ec026afe
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Citations in the corpus
- Non-ergodic Z-periodic billiards and infinite translation surfaces (2011, background): “ew product Tψ: X×G→X×G is ergodic if and only if EG(ψ) = G. The cocycle is a coboundary if and only if EG(ψ) = {0}. We also recall the following characterization of coboundaries. Proposition 3.3 (see [5]). If T : (X,µ) →(X,µ) is an ergodic automorphism then the cocycle ψ: X→Gfor T is a coboundary if”