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

Abstract For two real bases , a binary sequence is the ‐expansion of the number Let be the set of all real numbers having a unique ‐expansion. When the bases are equal, that is, , Allaart and Kong (2019) established the continuity in of the Hausdorff dimension of the univoque set , building on the work of Komornik, Kong, and Li. We derive explicit formulas for the Hausdorff dimension of and the entropy of the underlying subshift for arbitrary , and prove the continuity of these quantities as functions of . 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.