Almost specification ‎and ‎renewality‎ in spacing shifts

Abstract

‎Let $(Sigma_P,sigma_P)$ be the space of a spacing shifts where $Psubset mathbb{N}_0=mathbb{N}cup{0}$ and $Sigma_P={sin{0,1}^{mathbb{N}_0}: ‎s_i=s_j=1 mbox{ if } |i-j|in P cup{0}}$ and $sigma_P$ the shift map‎.<br /> ‎We will show that $Sigma_P$ is mixing if and only if it has almost specification property with at least two periodic points‎.<br /> ‎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.<br /> ‎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 ${0,1}^{mathbb{N}_0}$‎.