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 { O}K of the biquadratic number field K = { Q}(√2,i). The group of shift maps S , which acts on the Minkowski embedding ΓK{ 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}K, we prove that Sym( X k , Γ K ) is isomorphic to the semi-direct product { Z}⁴×\!\! height 4.6pt depth -0.1pt\ { Stab}(Vₖ), where the stabilizer is explicitly determined by the unit group { O}K× and the Galois group { Gal}(K/{ Q}).