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Rigidity times for a weakly mixing dynamical system which are not rigidity times for any irrational rotation

  • BASSAM FAYAD (a1) and ADAM KANIGOWSKI (a2)

Abstract

We construct an increasing sequence of natural numbers $(m_{n})_{n=1}^{+\infty }$ with the property that $(m_{n}{\it\theta}[1])_{n\geq 1}$ is dense in $\mathbb{T}$ for any ${\it\theta}\in \mathbb{R}\setminus \mathbb{Q}$ , and a continuous measure on the circle ${\it\mu}$ such that $\lim _{n\rightarrow +\infty }\int _{\mathbb{T}}\Vert m_{n}{\it\theta}\Vert \,d{\it\mu}({\it\theta})=0$ . Moreover, for every fixed $k\in \mathbb{N}$ , the set $\{n\in \mathbb{N}:k\nmid m_{n}\}$ is infinite. This is a sufficient condition for the existence of a rigid, weakly mixing dynamical system whose rigidity time is not a rigidity time for any system with a discrete part in its spectrum.

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[1]Adams, T.. Tower multiplexing and slow weak mixing. Preprint, 2013, arXiv:1301.0791.
[2]Bergelson, V., Del Junco, A., Lemanczyk, M. and Rosenblatt, J.. Rigidity and non-recurrence along sequences. Ergod. Th. & Dynam. Sys. (2013), doi:10.1017/etds.2013.5.
[3]Cornfield, I. P., Fomin, S. V. and Sinai, Ya.G. Ergodic Theory. Springer, New York, 1982.
[4]Fayad, B. and Thouvenot, J.-P.. On the convergence to 0 of m n𝜁[1]. Acta Arith. to appear, arXiv:1312.2510.

Rigidity times for a weakly mixing dynamical system which are not rigidity times for any irrational rotation

  • BASSAM FAYAD (a1) and ADAM KANIGOWSKI (a2)

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