Periodic unique beta-expansions: the Sharkovskii ordering

Let $\beta\in(1,2)$. Each $x\in[0,\frac{1}{\beta-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty \epsilon_k\beta^{-k}, \] where $\epsilon_k\in\{0,1\}$ for all $k$ (a $\beta$-expansion of $x$). If $\beta>\frac{1+\sqrt5}{2}$, then, as is well known, there always exist $x\in(0,\frac1{\beta-1})$ which have a unique $\be$-expansion. In the present paper we study (purely) periodic unique $\beta$-expansions and show that for each $n\ge2$ there exists $\beta_n\in[\frac{1+\sqrt5}{2},2)$ such that there are no unique periodic $\beta$-expansions of smallest period $n$ for $\beta\le\beta_n$ 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 Sharkovski\u{\i} 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.