Symmetries of a dynamical system arising from a biquadratic number field

We investigate the symmetries of a symbolic dynamical system (X_k, Γ_K) of number-theoretic origin. Specifically, we analyze the shift space X_k, defined as the closure of the set V_k 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 X_k that interact with the shift action: the automorphism group Aut(X_k, Γ_K), consisting of homeomorphisms that commute with every element of S, and the extended symmetry group Sym(X_k, Γ_K), which includes homeomorphisms that map the shift action to itself via an automorphism of S. While Aut(X_k, Γ_K) 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(X_k, Γ_K) 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}).