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

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  • 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.

    Mathematics Subject Classification: Primary: 11A63; Secondary: 37B10, 37B40, 68R15.


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  • Figure 1.  The labeled graph $ \mathcal G_ \mathbf{a} = (G, \mathcal L_ \mathbf{a}) $ with labeling $ \mathcal L_ \mathbf{a}: E\to L_ \mathbf{a}: = \big\{ \mathbf{a}, \mathbf{a}^+, \overline{ \mathbf{a}}, \overline{ \mathbf{a}^+}\big\} $, and the labeled graph $ \mathcal G^* = (G, \mathcal L^*) $ with labeling $ \mathcal L^*: E\to\left\{{0, 1}\right\} $

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