On the bifurcation set of unique expansions
Volume 188 / 2019
Abstract
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. (2017) showed that the topological entropy function $H: q \mapsto h_{\rm top}(\mathbf{U}_q)$ is a devil’s staircase in $(1, M+1]$.
Let $\mathscr{B}$ be the bifurcation set of $H$ defined by \[ \mathscr{B}=\{q\in(1, M+1]: H(p)\ne H(q)\ \textrm{for any} p\ne q\}. \] We analyze the fractal properties of $\mathscr{B}$ and show that for any $q\in \mathscr{B}$, \[ \lim_{\delta\rightarrow 0} \dim_H(\mathscr{B}\cap(q-\delta, q+\delta))=\dim_H\mathcal{U}_q, \] where $\dim_H$ denotes the Hausdorff dimension. Moreover, when $q\in\mathscr{B}$ 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 intersects $\mathcal{U}_q$.
As an application we obtain a dimensional spectrum result for the set $\mathscr{U}$ containing all bases $q\in(1, M+1]$ such that $1$ admits a unique $q$-expansion. In particular, we prove that for any $t \gt 1$ we have \[ \dim_H(\mathscr{U}\cap(1, t])=\max_{ q\le t}\dim_H\mathcal{U}_q. \] We also consider the variations of the sets $\mathscr{U}=\mathscr{U}(M)$ when $M$ varies.
