AN ALGEBRAIC APPROACH TO ENTROPY PLATEAUS IN NON-INTEGER BASE EXPANSIONS

For a positive integer M and a real base q is an element of (1, M + 1], let U-q denote the set of numbers having a unique expansion in base q over the alphabet {0, 1, ... , M}, and let U-q denote the corresponding set of sequences in {0, 1, ... M}(N). Komornik et al. [Adv. Math. 305 (2017), 165-196] showed recently that the Hausdorff dimension of Uq is given by h(U-q)/log q, where h(U-q) denotes the topological entropy of U-q. They furthermore showed that the function H : q -> h(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.