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 the $(q_0,q_1)$-expansion of the number \[ π_{q_0,q_1}(i_1 i_2 \cdots) = \sum_{k=1}^\infty \frac{i_k}{q_{i_1} \cdots q_{i_k}}. \] Let $U_{q_0,q_1}$ be the set of all real numbers having a unique $(q_0,q_1)$-expansion. When the bases are equal, i.e., $q_0 = q_1 = q$, Allaart and Kong (2019) established the continuity in $q$ of the Hausdorff dimension of the univoque set $U_{q,q}$, building on the work of Komornik, Kong, and Li (2017). We derive explicit formulas for the Hausdorff dimension of $U_{q_0,q_1}$ and the entropy of the underlying subshift for arbitrary $q_0, q_1 > 1$, and prove the continuity of these quantities as functions of $(q_0, q_1)$. Our results also concern general dynamical systems described by binary shifts with a hole, including, in particular, the doubling map with a hole and (linear) Lorenz maps.

Figures (2)

• Figure 1: Numerical values of $\dim_H \mathcal{U}_{q_0,q_1}$ for $1 < q_0, q_1 \le 2.5$, represented by colours. The black curve bordering the domain with $\dim_H \mathcal{U}_{q_0,q_1} = 0$ is $q_1 = \mathcal{K}(q_0)$, given by KomSteZou2022. We have $\dim_H \mathcal{U}_{q_0,q_1} = 1$ if and only if $(q_0{-}1)(q_1{-}1) = 1$ (thin white curve).
• Figure 2: The maps $T_0, T_1$ restricted to the interval $[0,\frac{1}{q_1-1}]$, with $q_0,q_1$ as in Example \ref{['ex:1']}.

Theorems & Definitions (57)

• Theorem 1.1
• Theorem 1.2
• Example 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3