Hyperbolic measure of maximal entropy for generic rational maps of k  [ Mesure hyperbolique d’entropie maximale pour les applications rationnelles génériques de k  ]
Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 645-680.

Soit f une application rationnelle dominante de 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( k ), l’application fA admet une mesure hyperbolique d’entropie maximale logλ 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 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 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( k ), the map fA admits a hyperbolic measure of maximal entropy logλ 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 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.

DOI : https://doi.org/10.5802/aif.2861
Classification : 37Fxx,  32H04,  32Uxx
Mots clés : dynamique complexe, applications méromorphes, super-potentiels, entropie, mesures hyperbolique
@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'institut Fourier},
     volume = {64},
     number = {2},
     year = {2014},
     doi = {10.5802/aif.2861},
     zbl = {1328.37046},
     mrnumber = {3330918},
     language = {en},
     url = {archive.numdam.org/item/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, Tome 64 (2014) no. 2, pp. 645-680. doi : 10.5802/aif.2861. http://archive.numdam.org/item/AIF_2014__64_2_645_0/

[1] Bedford, Eric; Diller, Jeffrey Energy and invariant measures for birational surface maps, Duke Math. J., Volume 128 (2005) no. 2, pp. 331-368 | Article | MR 2140266 | Zbl 1076.37031

[2] Bedford, Eric; Smillie, John Polynomial diffeomorphisms of C 2 . III. Ergodicity, exponents and entropy of the equilibrium measure, Math. Ann., Volume 294 (1992) no. 3, pp. 395-420 | Article | EuDML 165007 | MR 1188127 | Zbl 0765.58013

[3] Buff, Xavier Courants dynamiques pluripolaires, Ann. Fac. Sci. Toulouse Math. (6), Volume 20 (2011) no. 1, pp. 203-214 | Article | EuDML 219725 | Numdam | MR 2830397 | Zbl 1234.37037

[4] De Thélin, Henry Sur les exposants de Lyapounov des applications méromorphes, Invent. Math., Volume 172 (2008) no. 1, pp. 89-116 | Article | MR 2385668 | Zbl 1139.37037

[5] De Thélin, Henry; Vigny, Gabriel Entropy of meromorphic maps and dynamics of birational maps, Mém. Soc. Math. Fr. (N.S.) (2010) no. 122, pp. vi+98 | Numdam | Zbl 1214.37004

[6] Demailly, J.-P. Complex analytic and differential geometry (1997) (http://www-fourier.ujf-grenoble.fr/~demailly/books.html)

[7] Diller, Jeffrey; Dujardin, Romain; Guedj, Vincent Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory, Ann. Sci. Éc. Norm. Supér. (4), Volume 43 (2010) no. 2, pp. 235-278 | EuDML 272177 | Numdam | MR 2662665 | Zbl 1197.37059

[8] Diller, Jeffrey; Dujardin, Romain; Guedj, Vincent Dynamics of meromorphic mappings with small topological degree II: Energy and invariant measure, Comment. Math. Helv., Volume 86 (2011) no. 2, pp. 277-316 | Article | MR 2775130 | Zbl 1297.37022

[9] Diller, Jeffrey; Guedj, Vincent Regularity of dynamical Green’s functions, Trans. Amer. Math. Soc., Volume 361 (2009) no. 9, pp. 4783-4805 | Article | MR 2506427 | Zbl 1172.32004

[10] Dinh, Tien-Cuong; Nguyên, Viêt-Anh; Sibony, Nessim Dynamics of horizontal-like maps in higher dimension, Adv. Math., Volume 219 (2008) no. 5, pp. 1689-1721 | Article | MR 2458151 | Zbl 1149.37025

[11] Dinh, Tien-Cuong; Sibony, Nessim Dynamique des applications d’allure polynomiale, J. Math. Pures Appl. (9), Volume 82 (2003) no. 4, pp. 367-423 | Article | MR 1992375 | Zbl 1033.37023

[12] Dinh, Tien-Cuong; Sibony, Nessim Dynamics of regular birational maps in k , J. Funct. Anal., Volume 222 (2005) no. 1, pp. 202-216 | Article | MR 2129771 | Zbl 1067.37055

[13] Dinh, Tien-Cuong; Sibony, Nessim Une borne supérieure pour l’entropie topologique d’une application rationnelle, Ann. of Math. (2), Volume 161 (2005) no. 3, pp. 1637-1644 | Article | MR 2180409 | Zbl 1084.54013

[14] Dinh, Tien-Cuong; Sibony, Nessim Distribution des valeurs de transformations méromorphes et applications, Comment. Math. Helv., Volume 81 (2006) no. 1, pp. 221-258 | Article | MR 2208805 | Zbl 1094.32005

[15] Dinh, Tien-Cuong; Sibony, Nessim Geometry of currents, intersection theory and dynamics of horizontal-like maps, Ann. Inst. Fourier (Grenoble), Volume 56 (2006) no. 2, pp. 423-457 | Article | Numdam | MR 2226022 | Zbl 1089.37036

[16] Dinh, Tien-Cuong; Sibony, Nessim Pull-back currents by holomorphic maps, Manuscripta Math., Volume 123 (2007) no. 3, pp. 357-371 | Article | MR 2314090 | Zbl 1128.32020

[17] Dinh, Tien-Cuong; Sibony, Nessim Super-potentials of positive closed currents, intersection theory and dynamics, Acta Math., Volume 203 (2009) no. 1, pp. 1-82 | Article | MR 2545825 | Zbl 1227.32024

[18] Dinh, Tien-Cuong; Sibony, Nessim Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms, J. Algebraic Geom., Volume 19 (2010) no. 3, pp. 473-529 | Article | MR 2629598 | Zbl 1202.32033

[19] Dujardin, Romain Hénon-like mappings in 2 , Amer. J. Math., Volume 126 (2004) no. 2, pp. 439-472 | Article | MR 2045508 | Zbl 1064.37035

[20] Dujardin, Romain Laminar currents and birational dynamics, Duke Math. J., Volume 131 (2006) no. 2, pp. 219-247 | Article | MR 2219241 | Zbl 1099.37037

[21] Dupont, Christophe Large entropy measures for endomorphisms of ℂℙ k , Israel J. Math., Volume 192 (2012) no. 2, pp. 505-533 | Article | MR 3009733

[22] Federer, Herbert Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969, pp. xiv+676 | MR 257325 | Zbl 0176.00801

[23] Gromov, M. Convex sets and Kähler manifolds, Advances in differential geometry and topology, World Sci. Publ., Teaneck, NJ, 1990, pp. 1-38 | MR 1095529 | Zbl 0770.53042

[24] Gromov, Mikhaïl On the entropy of holomorphic maps, Enseign. Math. (2), Volume 49 (2003) no. 3-4, pp. 217-235 | MR 2026895 | Zbl 1080.37051

[25] Guedj, Vincent Entropie topologique des applications méromorphes, Ergodic Theory Dynam. Systems, Volume 25 (2005) no. 6, pp. 1847-1855 | Article | MR 2183297 | Zbl 1087.37015

[26] Guedj, Vincent Ergodic properties of rational mappings with large topological degree, Ann. of Math. (2), Volume 161 (2005) no. 3, pp. 1589-1607 | Article | MR 2179389 | Zbl 1088.37020

[27] Kifer, Yuri; Liu, Pei-Dong Random dynamics, Handbook of dynamical systems. Vol. 1B, Elsevier B. V., Amsterdam, 2006, pp. 379-499 | MR 2186245 | Zbl 1130.37301

[28] Ledrappier, François; Walters, Peter A relativised variational principle for continuous transformations, J. London Math. Soc. (2), Volume 16 (1977) no. 3, pp. 568-576 | Article | MR 476995 | Zbl 0388.28020

[29] Russakovskii, Alexander; Shiffman, Bernard Value distribution for sequences of rational mappings and complex dynamics, Indiana Univ. Math. J., Volume 46 (1997) no. 3, pp. 897-932 | Article | MR 1488341 | Zbl 0901.58023