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