Hyperbolic measure of maximal entropy for generic rational maps of $\mathbb{P}^k$

[Mesure hyperbolique d’entropie maximale pour les applications rationnelles génériques de $\mathbb{P}^k$]

Vigny, Gabriel

Annales de l'Institut Fourier, Tome 64 (2014), p. 645-680 / Harvested from Numdam

Résumé

Soit $f$ une application rationnelle dominante de $\mathbb{P}^k$ telle qu’il existe $s <k$ avec $\lambda _s(f)>\lambda _l(f)$ pour tout $l$. Sous des hypothèses raisonnables, nous montrons que, pour $A$ hors d’un ensemble pluripolaire de $\mathrm{Aut} (\mathbb{P}^k)$, l’application $f\circ A$ admet une mesure hyperbolique d’entropie maximale $\log \lambda _s(f)$ avec des bornes explicites sur les exposants de Lyapunov. En particulier, le résultat est vrai pour les applications polynomiales et donc pour l’extension homogène de $f$ à $\mathbb{P}^{k+1}$. Cela donne de nombreux exemples où la dynamique non uniformément hyperbolique est prouvée.

Un des outils principaux est l’approximation du graphe d’une application méromorphe par un courant positive fermé lisse. Cela permet de faire les calculs dans un cadre lisse et on utilise la théorie des super-potentiels pour passer à la limite.

Abstract

Let $f$ be a dominant rational map of $\mathbb{P}^k$ such that there exists $s <k$ with $\lambda _s(f)>\lambda _l(f)$ for all $l$. Under mild hypotheses, we show that, for $A$ outside a pluripolar set of $\mathrm{Aut} (\mathbb{P}^k)$, the map $f\circ A$ admits a hyperbolic measure of maximal entropy $\log \lambda _s(f)$ with explicit bounds on the Lyapunov exponents. In particular, the result is true for polynomial maps hence for the homogeneous extension of $f$ to $\mathbb{P}^{k+1}$. This provides many examples where non uniform hyperbolic dynamics is established.

One of the key tools is to approximate the graph of a meromorphic function by a smooth positive closed current. This allows us to do all the computations in a smooth setting, using super-potentials theory to pass to the limit.

Détail

Publié le : 2014-01-01

MR 3330918 | Zbl 06387288 | Zbl 1328.37046

DOI : https://doi.org/10.5802/aif.2861
Classification:  37Fxx,  32H04,  32Uxx
Mots clés: dynamique complexe, applications méromorphes, super-potentiels, entropie, mesures hyperbolique

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