Propagation of equilibrium states in stable families of endomorphisms of ${\mathbb{P}}^k(\mathbb{C})$

We prove that, within any holomorphic family of endomorphisms of ${\mathbb{P}}^k(\mathbb{C})$ in any dimension $k\ge 1$ and algebraic degree $d\ge 2$, the measurable holomorphic motion associated to dynamical stability in the sense of Berteloot-Bianchi-Dupont preserves the class of equilibrium states associated with weight functions $\psi$ satisfying $\sup \psi -\inf \psi <\log d$.