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The class of normal subshifts includes irreducible infinite topologicalMarkov shifts, irreducible infinite sofic shifts, synchronized systems, Dyckshifts, $\beta$-shifts, substitution minimal shifts, and so on. We willcharacterize one-sided topological conjugacy classes of normal subshifts interms of the associated $C^*$-algebras and its gauge actions with potentials.

One-sided topological conjugacy of normal subshifts and gauge actions on the associated $C^*$-algebras