Periodic unique beta-expansions: the Sharkovskii ordering

Abstract

Let $\beta \in (1,2)$. Each $x\in [0,\frac{1}{\beta -1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty ε_kβ^{-k}, \] where ${\epsilon }_k\in \{0,1\}$ 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 $\be$-expansion. In the present paper we study (purely) periodic unique $\beta$-expansions and show that for each $n\ge 2$ 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 $\beta \le {\beta }_n$ 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\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.

21 pages, 3 figures