A generalized Livšic--Sinai Theorem for endomorphisms

The classical theorem of Livšic and Sinai states that a transitive $C^2$ Anosov diffeomorphism whose Jacobian along every periodic orbit equals one admits an invariant volume form. Recently, it was observed that the periodic Jacobian condition alone already implies transitivity, rendering the transitivity assumption unnecessary. We extend this rigidity phenomenon to the non-invertible setting. We prove that if a $C^2$ Anosov endomorphism satisfies the natural periodic Jacobian condition $J(f^n(p))=\mathrm{°}(f{)}^n$ for every periodic point $p,$ such that $f^n(p)=p,$ then the system is automatically transitive and preserves a $C^1$ volume form. As a key ingredient, we establish a $C^1$ version of the Livšic cohomological theorem for hyperbolic endomorphisms.