On the specification property and synchronisation of unique $q$-expansions

Abstract: Given a positive integer $M$ and $q \in (1, M+1]$ we consider expansions in base $q$ for real numbers $x \in \left[0, {M}/{q-1}\right]$ over the alphabet ${0, \ldots, M}$. In particular, we study some dynamical properties of the natural occurring subshift $(\mathbf{V}_q, \sigma)$ related to unique expansions in such base $q$. We characterise the set of $q \in (1,M+1]$ such that $(\mathbf{V}_q, \sigma)$ has the specification property and the set of $q \in (1,M+1]$ such that $(\mathbf{V}_q, \sigma)$ is a synchronised subshift. Such properties are studied by analysing the combinatorial and dynamical properties of the quasi-greedy expansion of $q$. We also calculate the size of such classes giving similar results to those shown by Schmeling in (Ergodic Theory and Dynamical Systems, 17:675--694, 6 1997) in the context of $\beta$-transformations.