*Accepted Paper*

**Inserted:** 9 jul 2022

**Last Updated:** 4 jul 2024

**Journal:** International Mathematics Research Notices (IMRN)

**Year:** 2022

**Abstract:**

We prove that, in stable families of endomorphisms of $\mathbb{P}^k(\mathbb{C})$, all invariant measures with strictly positive Lyapunov exponents at a given parameter can be followed holomorphically with the parameter. As a consequence, almost all points (with respect to any such measure at any parameter) in the Julia set can be followed holomorphically without intersections. An analogous result holds for families of polynomial-like maps whose topological degree dominates the other dynamical degrees. This generalizes previous results for the measure of maximal entropy. We also prove a sufficient condition for the positivity of the Lyapunov exponents of a measure in term of the metric entropy, generalizing an analogous result by de Th\'elin and Dupont for the endomorphisms of $\mathbb{P}^k(\mathbb{C})$ to the setting of polynomial-like map. Our work provides a parallel in this setting to the probabilistic stability of H\'enon maps by Berger-Dujardin-Lyubich.