标题： 双基展开和带有空洞的二进制移位的Hausdorff维数 显示英文标题

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

摘要： 对于两个实数底数$q_0, q_1 > 1$，一个二进制序列$i_1 i_2 \cdots \in \{0,1\}^\infty$是数\[ \pi_{q_0,q_1}(i_1 i_2 \cdots) = \sum_{k=1}^\infty \frac{i_k}{q_{i_1} \cdots q_{i_k}}. \]的$(q_0,q_1)$-展开。设$U_{q_0,q_1}$为所有具有唯一$(q_0,q_1)$-展开的实数的集合。 当底数相等时，即$q_0 = q_1 = q$，Allaart 和 Kong（2019）建立了无歧义集$U_{q,q}$的豪斯多夫维数在$q$中的连续性，这是建立在 Komornik、Kong 和 Li（2017）工作的基础上的。 我们推导了任意$q_0, q_1 > 1$下的$U_{q_0,q_1}$的豪斯多夫维数和底层子移位的熵的显式公式，并证明了这些量作为$(q_0, q_1)$的函数的连续性。 我们的结果还涉及由二进制移位带洞描述的一般动力系统，包括但不限于带洞的倍射映射和（线性）洛伦兹映射。 显示英文摘要

摘要： 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 \[ \pi_{q_0,q_1}(i_1 i_2 \cdots) = \sum_{k=1}^\infty \frac{i_k}{q_{i_1} \cdots q_{i_k}}. \] Let $U_{q_0,q_1}$ 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.