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

Vigny, Gabriel

doi:10.5802/aif.2861

Hyperbolic measure of maximal entropy for generic rational maps of

ℙ^{k}

[Mesure hyperbolique d’entropie maximale pour les applications rationnelles génériques de

ℙ^{k}

]

Vigny, Gabriel ¹

¹ U. P. J. V. LAMFA - UMR 7352 33, rue Saint-Leu 80039 Amiens (France)

Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 645-680.

Résumé
Abstract

Soit $f$ une application rationnelle dominante de $ℙ^{k}$ telle qu’il existe $s < k$ avec $λ_{s} (f) > λ_{l} (f)$ pour tout $l$ . Sous des hypothèses raisonnables, nous montrons que, pour $A$ hors d’un ensemble pluripolaire de $Aut (ℙ^{k})$ , l’application $f \circ A$ admet une mesure hyperbolique d’entropie maximale $log λ_{s} (f)$ 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 $ℙ^{k + 1}$ . 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.

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

EuDML MR Zbl

DOI : 10.5802/aif.2861

Classification : 37Fxx, 32H04, 32Uxx
Keywords: Complex dynamics, meromorphic maps, Super-potentials, entropy, hyperbolic measure
Mot clés : dynamique complexe, applications méromorphes, super-potentiels, entropie, mesures hyperbolique

Affiliations des auteurs :

Vigny, Gabriel ¹

¹ U. P. J. V. LAMFA - UMR 7352 33, rue Saint-Leu 80039 Amiens (France)

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     title = {Hyperbolic measure of maximal entropy for generic rational maps of $\mathbb{P}^k$},
     journal = {Annales de l'Institut Fourier},
     pages = {645--680},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
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Vigny, Gabriel. Hyperbolic measure of maximal entropy for generic rational maps of $\mathbb{P}^k$. Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 645-680. doi : 10.5802/aif.2861. https://aif.centre-mersenne.org/articles/10.5802/aif.2861/

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