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 O₊ of the biquadratic number field K = Q (2, i). The group of shift maps S, which acts on the Minkowski embedding ₊ 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 O₊, we prove that Sym (X k, Γ K) is isomorphic to the semi-direct product Z^4\!\! height 4. 6pt depth -0. 1pt\ Stab (V₊), where the stabilizer is explicitly determined by the unit group O₊^ and the Galois group Gal (K/ Q).
Santos et al. (Fri,) studied this question.