Exponential equidistribution of periodic points for endomorphisms of ${\mathbb{P}}^k$

Let $f$ be a holomorphic endomorphism of ${\mathbb{P}}^k$ of algebraic degree $d\ge 2$. We show that the periodic points of $f$ of period $n$ equidistribute towards the equilibrium measure of $f$ exponentially fast as $n$ tends to infinity. This quantifies a theorem of Lyubich for $k=1$ and of Briend-Duval for $k\ge 2$. A byproduct of our proof is the existence of a large number of periodic cycles in the small Julia set with large multipliers.