Denseness of intermediate \(\beta\)-shifts of finite-type. (English) Zbl 1442.37027
Summary: We determine the structure of the set of intermediate \(\beta\)-shifts of finite-type. Specifically, we show that this set is dense in the parameter space
\[
\Delta := \{ (\beta , \alpha ) \in \mathbb{R}^{2} : \beta \in (1, 2) \text{ and } 0 \leq \alpha \leq 2 - \beta \}.
\]
This generalises the classical result of W. Parry [Acta Math. Acad. Sci. Hung. 11, 401–416 (1960; Zbl 0099.28103)] from 1960 for greedy \(\beta\)-shifts.
MSC:
| 37B10 | Symbolic dynamics |
| 37B51 | Multidimensional shifts of finite type |
| 11A67 | Other number representations |
| 11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |
Citations:
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