Dynamical pairs with an absolutely continuous bifurcation measure

Thomas Gauthier

doi:10.5802/afst.1735

Thomas Gauthier ¹

¹ CMLS, École Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France

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

Abstract
Abstract

In this article, we study algebraic dynamical pairs $(f, a)$ parametrized by an irreducible quasi-projective curve $Λ$ 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 $λ_{0}$ , between the measure $μ_{f, a}$ and the maximal entropy measure of $f_{λ_{0}}$ . We also establish an equivalent result for dynamical pairs of $ℙ^{k}$ , 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 $λ_{0}$ , entre la mesure de bifurcation $μ_{f, a}$ et la mesure d’entropie maximale de $f_{λ_{0}}$ . Sous une hypothèse relativement générale, nous établissons également un résultat similaire pour les paires dynamiques de $ℙ^{k}$ .

Received: 2020-09-28
Accepted: 2021-03-12
Published online: 2023-08-22

DOI: 10.5802/afst.1735

Author's affiliations:

Thomas Gauthier ¹

¹ CMLS, École Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Dynamical pairs with an absolutely continuous bifurcation measure},
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     pages = {203--230},
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Thomas Gauthier. Dynamical pairs with an absolutely continuous bifurcation measure. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 2, pp. 203-230. doi : 10.5802/afst.1735. https://afst.centre-mersenne.org/articles/10.5802/afst.1735/

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