Symmetries of power-free integers in number fields and their shift spaces

TL;DR

This paper characterizes the automorphisms of the ring of integers in number fields that preserve $k$th power-free integers, explores their symmetries in associated shift spaces, and shows these systems are topologically distinct.

Contribution

It provides a complete description of the automorphisms preserving power-free integers and analyzes the symmetry groups and dynamical properties of their associated shift spaces.

Findings

Automorphisms are compositions of field automorphisms and units.

Symmetry groups generate the extended symmetry of the shift spaces.

Different number fields yield non-conjugate shift systems.

Abstract

We describe the group of $\mathbb Z$-linear automorphisms of the ring of integers of a number field $K$ that preserve the set $V_{K,k}$ of $k$th power-free integers: every such map is the composition of a field automorphism and the multiplication by a unit. We show that those maps together with translations generate the extended symmetry group of the shift space $\mathbb D_{K,k}$ associated to $V_{K,k}$. Moreover, we show that no two such dynamical systems $\mathbb D_{K,k}$ and $\mathbb D_{L,l}$ are topologically conjugate and no one is a factor system of another. We generalize the concept of $k$th power-free integers to sieves and study the resulting admissible shift spaces.