Equivalencies between beta-shifts and S-gap shifts

Authors: DAWOUD AHMADI DASTJERDI, SOMAYYEH JANGJOO SHALDEHI

Abstract: Let X_{\beta} be a \beta -shift for \beta \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 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).

Keywords: Shift of finite type, sofic, right-resolving, synchronized, finite equivalence, almost conjugacy, zeta function

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