Bifurcation sets arising from non-integer base expansions

Given a positive integer $M$ and $q\in (1,M+1]$, let ${\mathcal{U}}_q$ be the set of $x\in [0,M\mathrm{/}(q-1)]$ having a unique $q$-expansion: there exists a unique sequence $(x_i)=x_1{x}_2\dots$ with each $x_i\in \{0,1,\dots ,M\}$ such that $$x=\frac{x_1}{q}+\frac{x_2}{q^2}+\frac{x_3}{q^3}+\cdots \mathrm{.}$$ Denote by ${\mathbf{U}}_q$ the set of corresponding sequences of all points in ${\mathcal{U}}_q$. It is well-known that the function $H:q\mapsto h({\mathbf{U}}_q)$ is a Devil’s staircase, where $h({\mathbf{U}}_q)$ denotes the topological entropy of $\mathbf U_q$. In this paper we {give several characterizations of} the bifurcation set $$\mathcal{B}:=\{q\in (1,M+1]:H(p)\mathrm{\ne }H(q)\text{ for any }p\mathrm{\ne }q\}\mathrm{.}$$ Note that $\mathcal{B}$ is contained in the set ${\mathcal{U}}^R$ of bases $q\in (1,M+1]$ such that $1\in {\mathcal{U}}_q$. By using a transversality technique we also calculate the Hausdorff dimension of the difference $\mathcal B\backslash$\mathcal{U}$^R$. Interestingly this quantity is always strictly between $0$ and $1$. When $M=1$ the Hausdorff dimension of $\mathcal B\backslash$\mathcal{U}$^R$is$\frac{\log 2}{3\log \lambda^*}\approx 0.368699$, where ${\lambda }^{\ast }$ is the unique root in $(1,2)$ of the equation $x^5-x^4-x^3-2x^2+x+1=0$.