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Periodic unique beta-expansions: the Sharkovskii ordering

Let $β\in(1,2)$. Each $x\in[0,\frac{1}{β-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty ε_kβ^{-k}, \] where $ε_k\in\{0,1\}$ for all $k$ (a $β$-expansion of $x$). If $β>\frac{1+\sqrt5}{2}$, then, as is well known, there always exist $x\in(0,\frac1{β-1})$ which have a unique $\be$-expansion. In the present paper we study (purely) periodic unique $β$-expansions and show that for each $n\ge2$ there exists $β_n\in[\frac{1+\sqrt5}{2},2)$ such that there are no unique periodic $β$-expansions of smallest period $n$ for $β\leβ_n$ and at least one such expansion for $β>β_n$. Furthermore, we prove that $β_k<β_m$ if and only if $k$ is less than $m$ in the sense of the Sharkovski\uı 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.