Propagation of equilibrium states in stable families of endomorphisms of Pᵏ (C)

We prove that, within any holomorphic family of endomorphisms of Pᵏ (C) in any dimension k 1 and algebraic degree d 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 satisfying - < d.