On the full centraliser of Erdős $\mathcal B$-free shifts
Volume 176 / 2024
Colloquium Mathematicum 176 (2024), 121-134
MSC: Primary 37B10; Secondary 11A07, 52C23
DOI: 10.4064/cm9428-9-2024
Published online: 17 October 2024
Abstract
The sets of $\mathcal{B} $-free integers are considered with respect to (reversing) symmetries. It is well known that, for a large class of them, the centraliser of the associated $\mathcal{B} $-free shift (otherwise known as its automorphism group) is trivial. We extend this result to the full centraliser, which effectively shows that all self-homeomorphisms of the $\mathcal{B} $-free shift that commute with some power of the shift are shifts themselves. This also leads to the result that the full normaliser agrees with the normaliser for this class, which is the semidirect product of the centraliser with the cyclic group of order 2 generated by reflection.
