An algebraic approach to entropy plateaus in non-integer base expansions

For a positive integer $ M $ and a real base $ q\in(1, M+1] $, let $ {\mathcal{U}}_q $ denote the set of numbers having a unique expansion in base $ q $ over the alphabet $ \{0, 1, \dots, M\} $, and let $ \mathbf{U}_q $ denote the corresponding set of sequences in $ \{0, 1, \dots, M\}^ {\mathbb{N}} $. Komornik et al. [ Adv. Math. 305 (2017), 165–196] showed recently that the Hausdorff dimension of $ {\mathcal{U}}_q $ is given by $ h(\mathbf{U}_q)/\log q $, where $ h(\mathbf{U}_q) $ denotes the topological entropy of $ \mathbf{U}_q $. They furthermore showed that the function $ H: q\mapsto h(\mathbf{U}_q) $ is continuous, nondecreasing and locally constant almost everywhere. The plateaus of $ H $ were characterized by Alcaraz Barrera et al. [ Trans. Amer. Math. Soc., 371 (2019), 3209–3258]. In this article we reinterpret the results of Alcaraz Barrera et al. by introducing a notion of composition of fundamental words, and use this to obtain new information about the structure of the function $ H $. This method furthermore leads to a more streamlined proof of their main theorem.