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

Let M be a positive integer and q ∈ (1,M +1]. We consider expansions of real numbers in base q over the alphabet {0,…,M}. Inparticular, we study the set Uq of real numbers with a unique q-expansion, and the set Uq of corresponding sequences. It was shown by Komornik, Kong, and Li that the function H, which associates to each q ∈ (1,M+1] the topological entropy of Uq, is a Devil’s staircase. In this paper we explicitly determine the plateaus of H, and characterize the bifurcation set E of q’s where the function H is not locally constant. Moreover, we show that E is a Cantor set of full Hausdorff dimension. We also investigate the topological transitivity of a naturally occurring subshift (Vq,σ), which has a close connection with open dynamical systems. Finally, we prove that the Hausdorff dimension and box dimension of Uq coincide for all q ∈ (1,M +1].