On the bifurcation set of unique expansions

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 $\{0,1,\dots ,M\}$, 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 $q\in \mathcal{B}$, \[ \lim_{δ\rightarrow 0} \dim_H(\mathcal{B}\cap(q-δ, q+δ))=\dim_H\mathcal{U}_q, \] 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.

36 pages and 1 figure. To appear in Acta Arithmetica