  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.
