Hausdorff dimension of double base expansions and binary shifts with a hole

Abstract: For two real bases $q_0, q_1 > 1$, a binary sequence $i_1 i_2 \cdots \in \{0,1\}^\infty$ is called the $(q_0,q_1)$-expansion of the number

\[ \pi_{q_0,q_1}(i_1 i_2 \cdots) = \sum_{k=1}^\infty \frac{i_k}{q_{i_1} \cdots q_{i_k}}. \] Let $\mathcal{U}_{q_0,q_1}$ denote the set of all real numbers having a unique $(q_0,q_1)$-expansion. When the two bases coincide, i.e., $q_0 = q_1 = q$, it was shown by Allaart and Kong (2019) that the Hausdorff dimension of the univoque set $\mathcal{U}_{q,q}$ varies continuously in $q$, building on earlier work of Komornik, Kong, and Li (2017). In this talk, we will derive explicit formulas for the Hausdorff dimension of $\mathcal{U}_{q_0,q_1}$ and for the topological entropy of the associated subshift for arbitrary $q_0, q_1 > 1$. We will also establish the continuity of these quantities as functions of the pair $(q_0,q_1)$.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper )

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