On the full centraliser of Erd\H{o}s $\cB$-free shifts

TL;DR

This paper investigates the symmetries of $ ext{B}$-free shift systems, proving that all self-homeomorphisms commuting with some power of the shift are essentially shifts, thus extending known symmetry results.

Contribution

It extends the understanding of symmetries of $ ext{B}$-free shifts by characterizing the full centraliser and normaliser, showing they are as restricted as the automorphism group.

Findings

Full centraliser consists only of shifts for the class considered.

Full normaliser equals the semi-direct product of the centraliser and reflection.

Symmetry group is essentially generated by shifts and reflection.

Abstract

The sets of $\cB$-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 $\cB$-free shift (otherwise known as its automorphism group) is trivial. We extend this result to the full centraliser, which effectively means to show that all self-homeomorphisms of the $\cB$-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 semi-direct product of the centraliser with the cyclic group of order two generated by reflection.