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

Rafael Alcaraz Barrera

Published 2019-03-20Version 1

We consider unique expansions of a real numbers in base $q \in (1,M+1]$. We study some dynamical properties of the naturally ocurring subshift $(\mathbf{V}_q, \sigma)$. In particular, we characterise the set of bases $q \subset (1,M+1]$ such that $(\mathbf{V}_q, \sigma)$ has the specification property and the set of bases $q \in (1, M+1]$ 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.