The aim of this paper is to establish an equivalent criterion for certain expansive diffeomorphisms of the 2-torus to admit an invariant Borel probability measure that is absolutely continuous with respect to the Riemannian volume. Our result is closely related to the well known Livšic-Sinai theorem for Anosov diffeomorphisms.
"Absolutely continuous invariant measures for expansive diffeomorphisms of the 2-torus." Hiroshima Math. J. 37 (3) 491 - 517, November 2007. https://doi.org/10.32917/hmj/1200529814