Academic Report of School of Mathematical Sciences [2026] No. 032

(Series Report for High-Level University Construction No. 1291)

Title：Ordered binary shifts with a hole

Speaker：Wolfgang Steiner, Professor ( University of Cité de Paris, France)

Time：11:00-12:00, Apr. 19, 2026

Location：Huiwen Building 2433

Abstract：Let $X(a,b)$ be the set of binary sequences such that no shifted sequence lies in the interval (of sequences) [a,b]. When the interval is taken with respect to the lexicographic order, this can also be seen as the survivor set of the doubling map with a hole or of a beta-transformation with a hole at 0, or as the set of trajectories of a Lorenz map, and it is now well known for which pairs (a,b) the shift space $X(a,b)$ is non-trivial or has positive topological entropy. We consider two other orders on sequences: the alternating lexicographic order and the unimodal order, which correspond to the negative doubling map and the tent map. Glendinning (1993, 2014) has studied maps of this type, and recently Glendinning and Hege characterised positive topological entropy via renormalizations. We revisit their results on the symbolic level, describe precisely the occurring renormalizations, and give formulae for the entropy of $X(a,b)$ and for the Hausdorff dimension of the set of double base expansions given by $X(a,b)$.

Speaker Profile：Wolfgang Steiner is Austrian and has obtained his PhD with Michael Drmota in Vienna. Since 2005, he has a CNRS research position at the Institut de Recherche en Informatique Fondamentale (IRIF) in Paris. He is mainly interested in various aspects of numeration systems (dynamical, combinatorial, number theoretical, geometrical, ...), in particular of beta-expansions and continued fractions.

Faculty and students are welcome to attend!

Invited by: Yuru Zou

School of Mathematical Sciences

April 18, 2026