Unique SRB measures and transitivity for Anosov diffeomorphisms

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.