Absolute continuity of hyperbolic invariant measures for endomorphisms

We prove that, for a C2 non-invertible but non-degenerate map f on a compact Riemannian manifold without boundary, a hyperbolic invariant measure μ is absolutely continuous with respect to the Lebesgue measure on the manifold if, under a condition on the Jacobian of the map, the measure satisfies two entropy formulae for positive exponents [13] and negative exponents [7], respectively. This implies that the entropy production eμ(f) = 0 if and only if μ << Leb