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29 March 2024
 
  » arxiv » 1108.3414

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Sofic S-gap Shifts, Entropy Function and Bowen-Franks Groups
D. Ahmadi Dastjerdi ; S. Jangjoo ;
Date 17 Aug 2011
AbstractLet $S=s_n$ be an increasing finite or infinite subset of $mathbb N igcup 0$ and $X(S)$ the $S$-gap shift associated to $S$. Let $f_S(x)=1-sum 1/ (x^{s_n+1})$ be the entropy function which will be vanished at $2^{h(X(S))}$ where $h(X(S))$ is the entropy of the system. Suppose $X(S)$ is sofic with adjacency matrix $A$ and the characteristic polynomial $chi_{A}$. We will show that $chi_A(x)=Q_S(x)f_S(x)$ where $Q_S(x)=x^{max S+1}$ when $|S|<infty$ and if $|S|=infty$, then $Q_S(x)=x^p-x^q$ for a positive integer $p$ and an integer $q$. The integers $p$ and $q$ will be explicitly determined. If $X(S)$ is SFT, then $zeta(t)=1/(f_S(t^{-1}))$ or $zeta(t)=1/((1-t)f_S(t^{-1}))$ when $|S|<infty$ or $|S|=infty$ respectively. Here $zeta$ is the zeta function of $X(S)$. We also will compute the Bowen-Franks groups of a sofic $S$-gap.
Source arXiv, 1108.3414
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