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.
Categories: math.NT math.DS