Equidistribution of saddle periodic points for automorphisms on compact Kähler manifolds

Let $f$ be a holomorphic automorphism of a compact Kähler manifold $X$ with simple cohomological action. We prove that the saddle periodic points of $f$ equidistribute with respect to the equilibrium measure. In particular, our result applies to every holomorphic automorphism of a compact Kähler surface with positive topological entropy.