  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.
