# Almost specification ‎and ‎renewality‎ in spacing shifts

Document Type: Research Paper

Authors

1 Department of Pure Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎University of Guilan‎, ‎Iran.

2 Department of Pure Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎University of Guilan‎, ‎Iran.

Abstract

‎Let $(\Sigma_P,\sigma_P)$ be the space of a spacing shifts where $P\subset \mathbb{N}_0=\mathbb{N}\cup\{0\}$ and $\Sigma_P=\{s\in\{0,1\}^{\mathbb{N}_0}: ‎s_i=s_j=1 \mbox{ if } |i-j|\in P \cup\{0\}\}$ and $\sigma_P$ the shift map‎.
‎We will show that $\Sigma_P$ is mixing if and only if it has almost specification property with at least two periodic points‎.
‎Moreover‎, ‎we show that if $h(\sigma_P)=0$‎, ‎then $\Sigma_P$ is almost specified and if $h(\sigma_P)>0$ and $\Sigma_P$ is almost specified‎, ‎then it is weak mixing‎.
‎Also‎, ‎some sufficient conditions for a coded $\Sigma_P$ being renewal or uniquely decipherable is given‎. ‎At last we will show that here are only two conjugacies from a transitive $\Sigma_P$ to a subshift of $\{0,1\}^{\mathbb{N}_0}$‎.

Keywords

Main Subjects

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### History

• Receive Date: 29 April 2015
• Revise Date: 30 January 2016
• Accept Date: 07 April 2016