One-sided topological conjugacy of normal subshifts and gauge actions on the associated $C^{\ast }$-algebras

The class of normal subshifts includes irreducible infinite topological Markov shifts, irreducible infinite sofic shifts, synchronized systems, Dyck shifts, $\beta$-shifts, substitution minimal shifts, and so on. We will characterize one-sided topological conjugacy classes of normal subshifts in terms of the associated $C^{\ast }$-algebras and its gauge actions with potentials.