Equivalencies between beta-shifts and $S$-gap shifts

Let Xb be a b -shift for b \in (1, 2] and X(S) a S-gap shift for S\subseteq N \cup {0}. We show that if X\beta is SFT (resp. sofic), then there is a unique S-gap shift conjugate (resp. right-resolving almost conjugate) to this X\beta, and if X\beta is not SFT, then no S-gap shift is conjugate to X\beta. For any synchronized Xb , an X(S) exists such that Xb and X(S) have a common synchronized 1-1 a.e. extension. For a nonsynchronized X\beta, this common extension is just an almost Markov synchronized system with entropy preserving maps. We then compute the zeta function of Xb from the zeta function of that X(S).