Let be a dominant rational map of such that there exists with for all . Under mild hypotheses, we show that, for outside a pluripolar set of , the map admits a hyperbolic measure of maximal entropy with explicit bounds on the Lyapunov exponents. In particular, the result is true for polynomial maps hence for the homogeneous extension of to . 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.
Soit une application rationnelle dominante de telle qu’il existe avec pour tout . Sous des hypothèses raisonnables, nous montrons que, pour hors d’un ensemble pluripolaire de , l’application admet une mesure hyperbolique d’entropie maximale 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 à . 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.
Keywords: Complex dynamics, meromorphic maps, Super-potentials, entropy, hyperbolic measure
Mot clés : dynamique complexe, applications méromorphes, super-potentiels, entropie, mesures hyperbolique
Vigny, Gabriel 1
@article{AIF_2014__64_2_645_0, author = {Vigny, Gabriel}, 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}, number = {2}, year = {2014}, doi = {10.5802/aif.2861}, zbl = {1328.37046}, mrnumber = {3330918}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2861/} }
TY - JOUR AU - Vigny, Gabriel TI - Hyperbolic measure of maximal entropy for generic rational maps of $\mathbb{P}^k$ JO - Annales de l'Institut Fourier PY - 2014 SP - 645 EP - 680 VL - 64 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2861/ DO - 10.5802/aif.2861 LA - en ID - AIF_2014__64_2_645_0 ER -
%0 Journal Article %A Vigny, Gabriel %T Hyperbolic measure of maximal entropy for generic rational maps of $\mathbb{P}^k$ %J Annales de l'Institut Fourier %D 2014 %P 645-680 %V 64 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2861/ %R 10.5802/aif.2861 %G en %F AIF_2014__64_2_645_0
Vigny, Gabriel. Hyperbolic measure of maximal entropy for generic rational maps of $\mathbb{P}^k$. Annales de l'Institut Fourier, Volume 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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