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