Ramification current, post-critical normality and stability of holomorphic endomorphisms of $\mathbb P^k$

In the context of holomorphic families of endomorphisms of $\mathbb P^k$, we prove that stability in the sense of [BBD18] is equivalent to a summability condition for the post-critical mass and to the convergence of a suitably defined ramification current. This allows us to both simplify the approach of [BBD18] and better relate stability to post-critical normality.