Work

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

Year: 2026

Type: article

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.

Source: Bulletin of the London Mathematical Society

DOI: 10.1112/blms.70344

Authors: Jian Lu, Wolfgang Steiner, Yuru Zou

Institutions: Shenzhen University, Centre National de la Recherche Scientifique, Université Paris Cité, Institut de Recherche en Informatique Fondamentale

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,930