Equivalencies between beta-shifts and S-gap shifts

Let X_{beta} be a beta -shift for beta in (1, 2] and X(S) a S-gap shift for Ssubseteq 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 X_{beta} , an X(S) exists such that X_{beta} 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 X_{beta} from the zeta function of that X(S).