On the specification property and synchronization of unique q-expansions

Given a positive integer M and q is an element of (1, M + 1] we consider expansions in base q for real numbers x is an element of [0, M/q - 1] over the alphabet {0,..., M}. In particular, we study some dynamical properties of the natural occurring subshift (V-q, sigma) related to unique expansions in such base q. We characterize the set of q is an element of V subset of (1, M + 1] such that (V-q, sigma) has the specification property and the set of q is an element of V such that (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 V giving similar results to those shown by Blanchard [ 10] and Schmeling in [ 36] in the context of beta-transformations.