Summary: Let \(\beta\in (1,2)\). Each \(x\in [0,\frac 1{\beta-1}]\) can be represented in the form \(x=\sum_{k=1}^\infty \varepsilon_k\beta^{-k}\), where \(\varepsilon_k\in\{0,1\}\) for all \(k\) (a \(\beta\)-expansion of \(x\)). If \(\beta>\frac{1+\sqrt 5}{2}\), then, as is well known, there always exist \(x\in (0,\frac 1{\beta-1})\) which have a unique \(\beta\)-expansion.
In the present paper we study (purely) periodic unique \(\beta\)-expansions and show that for each \(n\geq 2\) there exists \(\beta_n\in [\frac{1+\sqrt 5}{2},2)\) such that there are no unique periodic \(\beta\)-expansions of smallest period \(n\) for \(\beta\leq \beta_n\) and at least one such expansion for \(\beta>\beta_n\).
Furthermore, we prove that \(\beta_k<\beta_m\) if and only if \(k\) is less than \(m\) in the sense of the Sharkovskiĭ ordering. We give two proofs of this result, one of which is independent, and the other one links it to the dynamics of a family of trapezoidal maps.