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

Vigny, Gabriel

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

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

Let $f$ be a dominant rational map of ${\mathbb{P}}^k$ such that there exists $s<k$ with ${\lambda }_s\left(f\right)>{\lambda }_l\left(f\right)$ for all $l$. Under mild hypotheses, we show that, for $A$ outside a pluripolar set of $\mathrm{Aut}\left({\mathbb{P}}^k\right)$, the map $f\circ A$ admits a hyperbolic measure of maximal entropy $log{\lambda }_s\left(f\right)$ 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.

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

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