Entropy, topological transitivity, and dimensional properties of unique $q$-expansions

Let $M$ be a positive integer and $q \in(1,M+1].$ We consider expansions of real numbers in base $q$ over the alphabet $\{0,\ldots, M\}$. In particular, we study the set $\mathcal{U}_{q}$ of real numbers with a unique $q$-expansion, and the set $\mathbf{U}_q$ of corresponding sequences. It was shown in (Komornik et al, 2017 Adv. Math.) that the function $H$, which associates to each $q\in(1, M+1]$ the topological entropy of $\mathcal{U}_q$, is a Devil's staircase. In this paper we explicitly determine the plateaus of $H$, and characterize the bifurcation set $\mathcal E$ of $q$'s where the function $H$ is not locally constant. Moreover, we show that $\mathcal E$ is a Cantor set of full Hausdorff dimension. We also investigate the topological transitivity of a naturally occurring subshift $(\mathbf{V}_q, \sigma),$ which has a close connection with open dynamical systems. Finally, we prove that the Hausdorff dimension and box dimension of $\mathcal{U}_q$ coincide for all $q\in(1,M+1]$.