Periodicity of the univoque β-expansions

Let m ≥ 1 be an integer, 1 < β ≤ m + 1. A sequence ɛ1 ɛ2 ɛ3 with ɛi ∈ {0,1, … m} is called a β-expansion of a real number x if x=∑i∈iβi. It is known that when the base β is smaller than the generalized golden ration, any number has uncountably many expansions, while when β is larger, there are numbers which has unique expansion. In this paper, we consider the bases such that there is some number whose unique expansion is purely periodic with the given smallest period. We prove that such bases form an open interval, moreover, any two such open intervals have inclusion relationship according to the Sharkovskii ordering between the given minimal periods. We remark that our result answers an open question posed by Baker, and the proof for the case m = 1 is due to Allouche, Clarke and Sidorov.