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 Latt\`es 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.