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

For two real bases $q_0,q_1>1$, a binary sequence $i_1{i}_2\cdots \in \{0,1{\}}^{\mathrm{\infty }}$ is the $(q_0,q_1)$-expansion of the number $${\pi }_{q_0,q_1}(i_1{i}_2\cdots )=\sum _{k=1}^{\mathrm{\infty }}\frac{i_k}{q_{i_1}\cdots q_{i_k}}\mathrm{.}$$ 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.