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

We describe the group of Z-linear automorphisms of the ring of integers of a number field K that preserve the set V₊, ₊ of kth 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 D₊, ₊ associated to V₊, ₊. Moreover, we show that no two such dynamical systems D₊, ₊ and D₋, ₋ are topologically conjugate and no one is a factor system of another. We generalize the concept of kth power-free integers to sieves and study the resulting admissible shift spaces.