Conservative Anosov diffeomorphisms of the two torus without an absolutely continuous invariant measure

We construct examples of $C^1$ Anosov diffeomorphisms on ${\mathbb{T}}^2$ which are of Krieger type ${\mathrm{I}\mathrm{I}\mathrm{I}}_1$ with respect to Lebesgue measure. This shows that the Gurevic Oseledec phenomena that conservative $C^{1+\alpha }$ Anosov diffeomorphisms have a smooth invariant measure does not hold true in the $C^1$ setting.