Number-theoretic positive entropy shifts with small centraliser and large normaliser

Abstract

Higher-dimensional binary shifts of number-theoretic origin with positive topological entropy are considered. We are particularly interested in analysing their symmetries and extended symmetries. They form groups, known as the topological centraliser and normaliser of the shift dynamical system, which are natural topological invariants. Here, our focus is on shift spaces with trivial centralisers, but large normalisers. In particular, we discuss several systems where the normaliser is an infinite extension of the centraliser, including the visible lattice points and the $k$-free integers in some real quadratic number fields.