  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 $q\in(1,M+1]$.
