Let $f$ be a non-invertible holomorphic endomorphism of the projective space $\mathbb{P}^k$. We show that the periodic points equidistribute towards the equilibrium measure of $f$ exponentially fast as the period tends to infinity. A byproduct of our proof is the existence of a large number of periodic cycles in the small Julia set with large multipliers. The talk is based on a joint work with Henry de Th'elin and Lucas Kaufmann.