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 ψ satisfying ψ - inf ψ < log d.