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Let X0, X be mixing one-sided subshifts of finite type such that X0 ⊆ X. We show a necessary and sufficient condition for the existence of mixing Markov shifts Y0, Y , Y0 ⊆ Y , and a conjugacy π : Y → X with π(Y0) = X0, such that the sets of letters appearing in both systems are the same, more precisely, L1(Y0) = L1(Y ).

Coding nested mixing one-sided subshifts of finite type as Markov shifts having exactly the same alphabet