Dynamical pairs with an absolutely continuous bifurcation measure

Gauthier, Thomas

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 32 (2023) no. 2, pp. 203-230

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 bifurcation 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 mild assumption.

Dans cet article, nous étudions les paires dynamiques $(f,a)$ algébriques paramétrées par une courbe quasi-projective irréductible possédant une mesure de bifurcation absolument continue. Nous prouvons que, si la famille $f$ n’est pas isotriviale et si la paire $(f,a)$ est instable, c’est équivalent au fait que la famille $f$ soit une famille d’exemples de Lattès flexibles. A cette fin, nous montrons la densité des paramètres transversalement prérépulsifs dans le lieu de bifurcation de la paire $(f,a)$, ainsi qu’une propriété de similarité, en un paramètre transversalement prérépulsif ${\lambda }_0$, entre la mesure de bifurcation ${\mu }_{f,a}$ et la mesure d’entropie maximale de $f_{{\lambda }_0}$. Sous une hypothèse relativement générale, nous établissons également un résultat similaire pour les paires dynamiques de ${\mathbb{P}}^k$.

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