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 : 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
Vigny, Gabriel 1

1 U. P. J. V. LAMFA - UMR 7352 33, rue Saint-Leu 80039 Amiens (France)
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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. http://www.numdam.org/articles/10.5802/aif.2861/

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

[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 | DOI | EuDML | MR | Zbl

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

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

[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

[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 | Numdam | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | Numdam | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

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

[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 | Zbl

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

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

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

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

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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