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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 \({\mathbb Q}.\)

11J71 Distribution modulo one
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
Zbl 0494.10040