Symmetries of a dynamical system arising from a biquadratic number field.

We investigate the symmetries of a symbolic dynamical system (Xk, ΓK) of number-theoretic origin. Specifically, we analyze the shift space Xk, defined as the closure of the set Vk of k-free points within the ring of integers {\cal O}_{K} of the biquadratic number field K = {\bb Q}(\sqrt{2},i). The group of shift maps S, which acts on the Minkowski embedding \Gamma_{K}\cong{\bb Z}^{4} by translations, serves as the fundamental action of the system. Our focus is on the homeomorphisms of Xk that interact with the shift action: the automorphism group Aut(Xk, ΓK), consisting of homeomorphisms that commute with every element of S, and the extended symmetry group Sym(Xk, ΓK), which includes homeomorphisms that map the shift action to itself via an automorphism of S. While Aut(Xk, ΓK) is known to be trivial (consisting solely of the shifts themselves), we demonstrate that the extended symmetry group possesses a much richer structure. By leveraging the divisibility and growth properties of {\cal O}_{K}, we prove that Sym(Xk, ΓK) is isomorphic to the semi-direct product {\bb Z}^{4}\times\!\!\hbox{\vrule height 4.6pt depth -0.1pt}\ {\rm Stab}(V_{k}), where the stabilizer is explicitly determined by the unit group {\cal O}_{K}^{\times} and the Galois group {\rm Gal}(K/{\bb Q}).