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Let $M$ be a positive integer and $q \in(1,M+1].$ We consider expansions ofreal numbers in base $q$ over the alphabet $\{0,\ldots, M\}$. In particular, westudy 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 determinethe plateaus of $H$, and characterize the bifurcation set $\mathcal E$ of $q$'swhere the function $H$ is not locally constant. Moreover, we show that$\mathcal E$ is a Cantor set of full Hausdorff dimension. We also investigatethe topological transitivity of a naturally occurring subshift $(\mathbf{V}_q,\sigma),$ which has a close connection with open dynamical systems. Finally, weprove that the Hausdorff dimension and box dimension of $\mathcal{U}_q$coincide for all $q\in(1,M+1]$.

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