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

Alcaraz Barrera, Rafael; Baker, Simon; Kong, Derong

doi:10.1090/tran/7370

Entropy, topological transitivity, and dimensional properties of unique $q$-expansions
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by Rafael Alcaraz Barrera, Simon Baker and Derong Kong PDF

Trans. Amer. Math. Soc. 371 (2019), 3209-3258 Request permission

Abstract:

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 by Komornik, Kong, and Li 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 $\mathscr {E}$ of $q$’s where the function $H$ is not locally constant. Moreover, we show that $\mathscr {E}$ 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 $q\in (1,M+1]$.

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