Entropy, topological transitivity, and dimensional properties of unique $q$-expansions

Abstract

Let $M$ be a positive integer and $q\in (1,M+1].$ We consider expansions of real numbers in base $q$ over the alphabet $\{0,\dots ,M\}$. In particular, we study the set ${\mathcal{U}}_q$ of real numbers with a unique $q$-expansion, and the set ${\mathbf{U}}_q$ of corresponding sequences. It was shown in (Komornik et al, 2017 Adv. Math.) that the function $H$, which associates to each $q\in (1,M+1]$ the topological entropy of ${\mathcal{U}}_q$, is a Devil's staircase. In this paper we explicitly determine the plateaus of $H$, and characterize the bifurcation set $\mathcal{E}$ of $q$'s where the function $H$ is not locally constant. Moreover, we show that $\mathcal{E}$ is a Cantor set of full Hausdorff dimension. We also investigate the topological transitivity of a naturally occurring subshift $({\mathbf{V}}_q,\ddot{I}ƒ),$ which has a close connection with open dynamical systems. Finally, we prove that the Hausdorff dimension and box dimension of ${\mathcal{U}}_q$ coincide for all $q\in (1,M+1]$.

56 pages, 7 figures. To appear in Trans. Amer. Math. Soc