Examples of type {rm III}₁ inhomogenous Markov shifts supported on topological Markov shifts

We construct inhomogenous Markov measures for which the shift is of Kreiger type ${\rm III}_{1}$. These measures are fully supported on a toplogical markov shift space of the hyperbolic toral automorphism $f(x,y)=\left(\{x+y\},x\right):\mathbb{T}^{2}\to\mathbb{T}^{2}$ and are used in a subsequent paper in the construction of $C^{1}$ Anosov diffeomorphisms of $\mathbb{T}^{2}$ without an absolutely continuous invariant measure.