Entropy, Topological transitivity, and Dimensional properties of unique $q$-expansions

Rafael Alcaraz Barrera, Simon Baker, Derong Kong

Published 2016-09-07Version 1

Let $M$ be a positive integer and $q \in(1,M+1].$ We consider expansions of real numbers in base $q$ over the alphabet $\{0,\ldots, M\}$. In particular, we study the set $\mathcal{U}_{q}$ of real numbers with a unique $q$-expansion, and the set $\mathcal{U}_q'$ of corresponding sequences. It was shown by Komornik, Kong and Li (2015) 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 set $\mathcal{E}$ of $q$'s where the function $H$ is increasing. Moreover, we show that the set $\mathcal{E}$ is a Cantor set of full Hausdorff dimension. We also investigate the topological transitivity of a naturally occurring subshift $(\widetilde{\mathcal{V}}_q', \sigma),$ 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 $q\in(1,M+1]$.