Misiurewicz parameters and dynamical stability of polynomial-like maps of large topological degree

Given a family of polynomial-like maps of large topological degree, we relate the presence of Misiurewicz parameters to a growth condition for the volume of the iterates of the critical set. This generalizes to higher dimensions the well-known equivalence between stability and normality of the critical orbits in dimension one. We also introduce a notion of holomorphic motion of asymptotically all repelling cycles and prove its equivalence with other notions of stability. Our results allow us to generalize the theory of stability and bifurcation developed by Berteloot, Dupont and the author for the family of all endomorphisms of $$\mathbb P^k$$ of a given degree to any arbitrary family of endomorphisms of $$\mathbb P^k$$ or polynomial-like maps of large topological degree.

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