  Given a positive integer $M$, for $q\in(1, M+1]$ let ${\mathcal{U}}_q$ be the
set of $x\in[0, M/(q-1)]$ having a unique $q$-expansion with the digit set
$\{0, 1,\ldots, M\}$, and let $\mathbf{U}_q$ be the set of corresponding
$q$-expansions. Recently, Komornik et al.~(Adv. Math., 2017) showed that the
topological entropy function $H: q \mapsto h_{top}(\mathbf{U}_q)$ is a Devil's
staircase in $(1, M+1]$. Let $\mathcal{B}$ be the bifurcation set of $H$
defined by
  \[
  \mathcal{B}=\{q\in(1, M+1]: H(p)\ne H(q)\quad\textrm{for any}\quad p\ne q\}.
  \]
  In this paper we analyze the fractal properties of $\mathcal{B}$, and show
that for any $q\in \mathcal{B}$,
  \[
  \lim_{\delta\rightarrow 0} \dim_H(\mathcal{B}\cap(q-\delta,
q+\delta))=\dim_H\mathcal{U}_q,
  \] where $\dim_H$ denotes the Hausdorff dimension. Moreover, when
$q\in\mathcal{B}$ the univoque set $\mathcal{U}_q$ is dimensionally
homogeneous, i.e., $
  \dim_H(\mathcal{U}_q\cap V)=\dim_H\mathcal{U}_q $ for any open set $V$ that
intersect $\mathcal{U}_q$.
  As an application we obtain a dimensional spectrum result for the set
$\mathcal{U}$ containing all bases $q\in(1, M+1]$ such that $1$ admits a unique
$q$-expansion. In particular, we prove that for any $t>1$ we have \[
  \dim_H(\mathcal{U}\cap(1, t])=\max_{ q\le t}\dim_H\mathcal{U}_q. \] We also
consider the variations of the sets $\mathcal{U}=\mathcal{U}(M)$ when $M$
changes.
