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

Jian Lu; Wolfgang Steiner; Yuru Zou

paper

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

arXiv:2509.04227

Abstract

For two real bases $q_{0}, q_{1} > 1$ , a binary sequence $i_{1} i_{2} \dots \in {0, 1}^{\infty}$ 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.