Abstract
Let β > 1 \beta >1 be a Pisot unit. A family of sets { X i } 1 ≤ i ≤ q \{X_i\}_{1\leq i\leq q} defined by a β \beta -numeration system has been extensively studied as an atomic surface or Rauzy fractal. For the purpose of constructing a Markov partition, a domain X ^ = ⋃ i = 1 q X ^ i \hat X=\bigcup _{i=1}^q \hat X_i constructed by an atomic surface has appeared in several papers. In this paper we show that the domain X ^ \hat X completely characterizes the set of purely periodic β \beta -expansions.
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