Periodic unique beta-expansions: the Sharkovskii ordering

Jean-Paul Allouche, Matthew Clarke, Nikita Sidorov

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 $β>…