Eigenvalues and the stabilized automorphism group

We study the stabilized automorphism group of minimal and, more generally, certain transitive dynamical systems. Our approach involves developing new algebraic tools to extract information about the rational eigenvalues of these systems from their stabilized automorphism groups. In particular, we prove that if two minimal system have isomorphic stabilized automorphism groups and each has at least one non-trivial rational eigenvalue, then the systems have the same rational eigenvalues. Using these tools, we also extend Schmieding's result on the recovery of entropy from the stabilized automorphism group to include irreducible shifts of finite type.