  We prove that every $C^2$ Anosov diffeomorphism in a compact and connected
Riemannian manifold has a unique SRB and physical probability measure, whose
basin of attraction covers Lebesgue almost every point in the manifold. Then,
we use structural stability of Anosov diffeomorphisms to deduce that all $C^1$
Anosov diffeomorphisms on compact and connected Riemannian manifolds are
transitive.
