Dynamical pairs with an absolutely continuous bifurcation measure

Abstract

In this article, we study algebraic dynamical pairs $(f,a)$ parametrized by an irreducible quasi-projective curve $\Lambda$ having an absolutely continuous bifurcation measure. We prove that, if $f$ is non-isotrivial and $(f,a)$ is unstable, this is equivalent to the fact that $f$ is a family of Lattès maps. To do so, we prove the density of transversely prerepelling parameters in the bifucation locus of $(f,a)$ and a similarity property, at any transversely prerepelling parameter ${\lambda }_0$, between the measure ${\mu }_{f,a}$ and the maximal entropy measure of $f_{{\lambda }_0}$. We also establish an equivalent result for dynamical pairs of ${\mathbb{P}}^k$, under an additional assumption.

v3, final version, accepted for publication in the Annales de la Faculté des sciences de Toulouse