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 [F. Berteloot et al., Ann. Sci. École Norm. Sup. 51 (2018)] 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 the above-mentioned paper and better relate stability to post-critical normality.