Equivalencies between beta-shifts and S-gap shifts

Abstract

Let $X_{\beta }$ be a sofic $\beta$-shift for $\beta \in (1,2]$. We show that there is an $S$-gap shift $X(S)$ such that $X_{\beta }$ and $X(S)$ are right-resolving almost conjugate. Conversely, a condition on $S\subseteq \mathbb{N}\cup \{0\}$ is given such that for this $S$, there is a $\beta$ such that $X(S)$ and $X_{\beta }$ have the same equivalency. We show that if $X_{\beta }$ is SFT, then there is an $S$-gap shift conjugate to this $X_{\beta }$; however, if $X_{\beta }$ is not SFT, then no $S$-gap shift is conjugate to $X_{\beta }$. Also we will investigate the existence of these sort of equivalencies for non-sofics.