Abstract
AbstractHigher-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 centralizer and normalizer of the shift dynamical system, which are natural topological invariants. Here, our focus is on shift spaces with trivial centralizers, but large normalizers. In particular, we discuss several systems where the normalizer is an infinite extension of the centralizer, including the visible lattice points and the k-free integers in some real quadratic number fields.
Highlights
Shift spaces under the action of Zd form a much-studied class of dynamical systems, both for d = 1 and for d 2
We adopt the point of view of [2, 9] to analyse both the centralizer and the normalizer of the shift space, the latter denoted by R, as this pair can be quite revealing as soon as d 2
The characterization of a number-theoretic shift space via an admissibility condition was originally observed by Sarnak for the square-free integers, and later extended to Erdos B-free numbers in [20] and generalized to the lattice setting in [32]
Summary
The characterization of a number-theoretic shift space via an admissibility condition was originally observed by Sarnak for the square-free integers, and later extended to Erdos B-free numbers in [20] and generalized to the lattice setting in [32]. Any B-free lattice system defines a shift, with faithful shift action, so its symmetry group, S(XB), contains a normal subgroup that is isomorphic with Zd , namely the one generated by the shift action itself, G.
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