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It is shown that for a non-singular conservative shift on a topologicallymixing Markov subshift with Doeblin Condition the only possible absolutelycontinuous shift-invariant measure is a Markov measure. Moreover, if it is notequivalent to a homogeneous Markov measure then the shift is of Krieger-type$\mathrm{III}_1$. A criterion for equivalence of Markov measures is included.

On Absolutely Continuous Invariant Measures and Krieger-Type of Markov Subshifts