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On periodic $$\beta$$-expansions of Pisot numbers and Rauzy fractals. (English) Zbl 0991.11040
Let $$\beta$$ be a real number greater than 1 and $$T_{\beta}x=\beta x - [\beta x],$$ where $$[\dots ]$$ stands for the integer part. Then each $$x \in [0,1)$$ has a $$\beta$$-greedy expansion by the series $$\sum _{k=1} ^{\infty} [\beta T_{\beta}^{k-1}x] \beta^{-k}.$$ Let $$\beta$$ be a Pisot number, namely, all its conjugates over $$\mathbb Q$$ except for $$\beta$$ itself are strictly inside the unit circle. It is known [see, e.g., K. Schmidt, Bull. Lond. Math. Soc. 12, 269-278 (1980; Zbl 0494.10040)] that $$x \in {\mathbb Q}(\beta) \cap [0,1)$$ iff the $$\beta$$-greedy expansion of $$x$$ is periodic. Let $$x \in {\mathbb Q}(\beta) \cap [0,1),$$ where $$\beta$$ is a cubic Pisot number. The authors find the respective ‘if and only if’ condition on $$x$$ for its $$\beta$$-greedy expansion to be purely periodic. It depends on the other two conjugates of $$x$$ and on the minimal polynomial of $$\beta$$ over $${\mathbb Q}.$$

##### MSC:
 11J71 Distribution modulo one 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
##### Keywords:
Pisot numbers; Rauzy fractals; greedy expansion