Some sufficient conditions for transitivity of Anosov diffeomorphisms

Given a C2− Anosov diffeomorphism f:M→M, we prove that the Jacobian condition Jfn(p)=1, for every point p such fn(p)=p, implies transitivity. As application in celebrated theory of Sinai-Ruelle-Bowen, this result allows us to state classical theorem Livsic-Sinai without directly assuming transitivity as general hypothesis. A special consequence our is C2-Anosov diffeomorphism, which regular, indeed transitive.