By a classical result by Mané-Sad-Sullivan and Lyubich , the stability of a family of rational maps is determined by the stability of the repelling cycles. More precisely, the so-called \lambda-lemma ensures that, once one can follow holomorphically with the parameter a dense subset of the Julia set J_{\lam_0} at a given parameter \lambda_0, then every point of J_{\lam_0} can be followed holomorphically.
In this talk we show that in stable families (in the sense of Berteloot-Bianchi-Dupont) of endomorphisms of P^k all measures of sufficiently large entropy move holomorphically with the parameter. As a consequence,
almost all points (with respect to any measure with large entropy at any parameter) in the Julia set can be followed holomorphically without intersections.
