On the specification property and synchronization 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 [0, {M}{q 1} over the alphabet {0, \ldots M\ In particular, we study some dynamical properties of the natural occurring subshift \boldsymbol{{V}}_q, \sigma related to unique expansions in such base q. We characterize the set of $q \in \mathcal {V} \subset (1,M+1] such that \boldsymbol{{V}}_q, \sigma has the specification property and the set of $q \in \mathcal {V} such that \boldsymbol{{V}}_q, \sigma is a synchronized 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 as subsets of \mathcal {V} giving similar results to those shown by Blanchard 10 and Schmeling in 36 in the context of \beta transformations.
Ergodic Theory and Dynamical Systems