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 $\{0,\dots ,M\}$. 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.

46 pages. Accepted to its publication in Ergodic Theory and Dynamical Systems