Entropy of meromorphic maps and dynamics of birational maps
Mémoires de la Société Mathématique de France, no. 122 (2010), 103 p.
The full text of recent articles is available to journal subscribers only. See the article on the journal's website

We study the dynamics of meromorphic maps for a compact Kähler manifold X. More precisely, we give a simple criterion that allows us to produce a measure of maximal entropy. We can apply this result to bound the Lyapunov exponents. Then, we study the particular case of a family of generic birational maps of k for which we construct the Green currents and the equilibrium measure. We use for that the theory of super-potentials. We show that the measure is mixing and gives no mass to pluripolar sets. Using the criterion we get that the measure is of maximal entropy. It implies finally that the measure is hyperbolic.

On étudie la dynamique des applications méromorphes sur les variétés kählériennes compactes. Plus précisément, on donne un critère simple qui permet de produire des mesures d’entropie maximale. On peut appliquer ce résultat pour borner les exposants de Lyapounov. Ensuite, on étudie le cas particulier d’une famille générique d’applications birationnelles de k pour laquelle on construit les courants de Green et la mesure d’équilibre. On utilise pour cela la théorie des super-potentiels. On montre que la mesure est mélangeante et qu’elle n’a pas de masse sur les ensembles pluripolaires. En utilisant le critère on obtient que la mesure est d’entropie maximale. Cela implique finalement que la mesure est hyperbolique.

DOI : https://doi.org/10.24033/msmf.434
Classification:  37Fxx,  32H04,  32Uxx,  37A35,  37Dxx
Keywords: Complex dynamics, meromorphic maps, super-potentials, currents, entropy, hyperbolic measure
@book{MSMF_2010_2_122__1_0,
     author = {De Th\'elin, Henry and Vigny, Gabriel},
     title = {Entropy of meromorphic maps and dynamics of birational~maps},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {122},
     year = {2010},
     doi = {10.24033/msmf.434},
     zbl = {1214.37004},
     mrnumber = {2752759},
     language = {en},
     url = {http://www.numdam.org/item/MSMF_2010_2_122__1_0}
}
De Thélin, Henry; Vigny, Gabriel. Entropy of meromorphic maps and dynamics of birational maps. Mémoires de la Société Mathématique de France, Serie 2, , no. 122 (2010), 103 p. doi : 10.24033/msmf.434. http://www.numdam.org/item/MSMF_2010_2_122__1_0/

[1] E. Bedford & J. Diller« Energy and invariant measures for birational surface maps », Duke Math. J. 128 (2005), p. 331–368. | MR 2140266 | Zbl 1076.37031

[2] E. Bedford, M. Lyubich & J. Smillie« Polynomial diffeomorphisms of 𝐂 2 . IV. The measure of maximal entropy and laminar currents », Invent. Math. 112 (1993), p. 77–125. | MR 1207478 | Zbl 0792.58034

[3] E. Bedford & J. Smillie« Polynomial diffeomorphisms of 𝐂 2 : currents, equilibrium measure and hyperbolicity », Invent. Math. 103 (1991), p. 69–99. | MR 1079840 | Zbl 0721.58037

[4] —, « Polynomial diffeomorphisms of 𝐂 2 . III. Ergodicity, exponents and entropy of the equilibrium measure », Math. Ann. 294 (1992), p. 395–420. | MR 1188127 | Zbl 0765.58013

[5] J.-Y. Briend & J. Duval« Deux caractérisations de la mesure d’équilibre d’un endomorphisme de P k (𝐂) », Publ. Math. Inst. Hautes Études Sci. 93 (2001), p. 145–159. | MR 1863737

[6] D. Burguet« A proof of Yomdin-Gromov’s algebraic lemma », Israel J. Math. 168 (2008), p. 291–316. | MR 2448063 | Zbl 1169.14038

[7] S. Cantat« Dynamique des automorphismes des surfaces K3 », Acta Math. 187 (2001), p. 1–57. | MR 1864630

[8] E. M. ChirkaComplex analytic sets, Mathematics and its Applications (Soviet Series), vol. 46, Kluwer Academic Publishers Group, 1989. | MR 1111477

[9] D. Coman & V. Guedj« Invariant currents and dynamical Lelong numbers », J. Geom. Anal. 14 (2004), p. 199–213. | MR 2051683 | Zbl 1080.37050

[10] H. De Thélin« Sur la construction de mesures selles », Ann. Inst. Fourier (Grenoble) 56 (2006), p. 337–372. | Numdam | MR 2226019

[11] —, « Sur les exposants de Lyapounov des applications méromorphes », Invent. Math. 172 (2008), p. 89–116. | MR 2385668

[12] —, « Ahlfors’ currents in higher dimension », in appear in Ann. Fac. Sci. Toulouse. | Numdam | Zbl 1195.32004

[13] J.-P. Demailly« Monge-Ampère operators, Lelong numbers and intersection theory », in Complex analysis and geometry, Univ. Ser. Math., Plenum, 1993, p. 115–193. | MR 1211880 | Zbl 0792.32006

[14] —, « Complex analytic and algebraic geometry », http://www-fourier.ujf-grenoble.fr/~demailly/books.html, 1997.

[15] J. Diller« Dynamics of birational maps of 𝐏 2 », Indiana Univ. Math. J. 45 (1996), p. 721–772. | MR 1422105 | Zbl 0874.58022

[16] J. Diller, R. Dujardin & V. Guedj« Dynamics of meromorphic maps with small topological degree I: from cohomology to currents », preprint arXiv:0803.0955, to appear in Indiana Univ. Math. J. | Zbl 1234.37039

[17] —, « Dynamics of meromorphic maps with small topological degree II: Energy and invariant measure », preprint arXiv:0805.3842, to appear in Comment. Math. Helvet.

[18] —, « Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory », preprint arXiv:0806.0146, to appear in Ann. Ecole Norm. Sup. | Numdam | Zbl 1197.37059

[19] J. Diller & V. Guedj« Regularity of dynamical Green’s functions », Trans. Amer. Math. Soc. 361 (2009), p. 4783–4805. | MR 2506427 | Zbl 1172.32004

[20] T.-C. Dinh & C. Dupont« Dimension de la mesure d’équilibre d’applications méromorphes », J. Geom. Anal. 14 (2004), p. 613–627. | MR 2111420

[21] T.-C. Dinh & N. Sibony« Dynamique des applications d’allure polynomiale », J. Math. Pures Appl. 82 (2003), p. 367–423. | MR 1992375

[22] —, « Regularization of currents and entropy », Ann. Sci. École Norm. Sup. 37 (2004), p. 959–971. | Numdam | MR 2119243 | Zbl 1074.53058

[23] —, « Dynamics of regular birational maps in k », J. Funct. Anal. 222 (2005), p. 202–216. | MR 2129771 | Zbl 1067.37055

[24] —, « Green currents for holomorphic automorphisms of compact Kähler manifolds », J. Amer. Math. Soc. 18 (2005), p. 291–312. | MR 2137979 | Zbl 1066.32024

[25] —, « Une borne supérieure pour l’entropie topologique d’une application rationnelle », Ann. of Math. 161 (2005), p. 1637–1644.

[26] —, « Decay of correlations and the central limit theorem for meromorphic maps », Comm. Pure Appl. Math. 59 (2006), p. 754–768. | MR 2172806 | Zbl 1137.37023

[27] —, « Distribution des valeurs de transformations méromorphes et applications », Comment. Math. Helv. 81 (2006), p. 221–258. | MR 2208805

[28] —, « Geometry of currents, intersection theory and dynamics of horizontal-like maps », Ann. Inst. Fourier (Grenoble) 56 (2006), p. 423–457. | Numdam | MR 2226022 | Zbl 1089.37036

[29] —, « Pull-back currents by holomorphic maps », Manuscripta Math. 123 (2007), p. 357–371. | MR 2314090 | Zbl 1128.32020

[30] —, « Equidistribution towards the Green current for holomorphic maps », Ann. Sci. Éc. Norm. Supér. 41 (2008), p. 307–336. | Numdam | Zbl 1160.32029

[31] —, « Super-potentials of positive closed currents, intersection theory and dynamics », Acta Math. 203 (2009), p. 1–82. | MR 2545825 | Zbl 1227.32024

[32] —, « Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms », preprint arXiv:0804.0860. | Zbl 1202.32033

[33] R. Dujardin« Laminar currents and birational dynamics », Duke Math. J. 131 (2006), p. 219–247. | MR 2219241 | Zbl 1099.37037

[34] C. Favre« Points périodiques d’applications birationnelles de 𝐏 2 », Ann. Inst. Fourier (Grenoble) 48 (1998), p. 999–1023. | Numdam | MR 1656005 | Zbl 0924.58083

[35] C. Favre & M. Jonsson« Brolin’s theorem for curves in two complex dimensions », Ann. Inst. Fourier (Grenoble) 53 (2003), p. 1461–1501. | Numdam | MR 2032940 | Zbl 1113.32005

[36] J. E. Fornæss & N. Sibony« Complex dynamics in higher dimension. I », Astérisque 222 (1994), p. 201–231. | MR 1285389 | Zbl 0813.58030

[37] —, « Complex dynamics in higher dimensions », in Complex potential theory (Montreal, PQ, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 439, Kluwer Acad. Publ., 1994, p. 131–186. | MR 1332961 | Zbl 0811.32019

[38] J. E. Fornaess & N. Sibony« Complex dynamics in higher dimension. II », in Modern methods in complex analysis (Princeton, NJ, 1992), Ann. of Math. Stud., vol. 137, Princeton Univ. Press, 1995, p. 135–182. | MR 1369137 | Zbl 0847.58059

[39] M. Gromov« Entropy, homology and semialgebraic geometry », Astérisque 145-146 (1987), p. 225–240, Séminaire Bourbaki, vol. 1985/86, exposé no 663. | Numdam | MR 880035

[40] —, « Convex sets and Kähler manifolds », in Advances in differential geometry and topology, World Sci. Publ., Teaneck, NJ, 1990, p. 1–38. | Zbl 0770.53042

[41] —, « On the entropy of holomorphic maps », Enseign. Math. 49 (2003), p. 217–235. | Zbl 1080.37051

[42] V. Guedj« Courants extrémaux et dynamique complexe », Ann. Sci. École Norm. Sup. 38 (2005), p. 407–426. | Numdam | MR 2166340

[43] —, « Entropie topologique des applications méromorphes », Ergodic Theory Dynam. Systems 25 (2005), p. 1847–1855. | MR 2183297 | Zbl 1087.37015

[44] V. Guedj & N. Sibony« Dynamics of polynomial automorphisms of 𝐂 k », Ark. Mat. 40 (2002), p. 207–243. | MR 1948064 | Zbl 1034.37025

[45] A. Katok & B. HasselblattIntroduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge Univ. Press, 1995. | MR 1326374 | Zbl 0878.58020

[46] S. ŁojasiewiczIntroduction to complex analytic geometry, Birkhäuser, 1991. | MR 1131081

[47] R. Mañé« A proof of Pesin’s formula », Ergodic Theory Dynam. Systems 1 (1981), p. 95–102. | MR 627789 | Zbl 0489.58018

[48] A. Russakovskii & B. Shiffman« Value distribution for sequences of rational mappings and complex dynamics », Indiana Univ. Math. J. 46 (1997), p. 897–932. | MR 1488341 | Zbl 0901.58023

[49] N. Sibony« Quelques problèmes de prolongement de courants en analyse complexe », Duke Math. J. 52 (1985), p. 157–197. | MR 791297 | Zbl 0578.32023

[50] —, « Dynamique des applications rationnelles de 𝐏 k », in Dynamique et géométrie complexes (Lyon, 1997), Panoramas & Synthèses, vol. 8, Soc. Math. France, 1999.

[51] Y. T. Siu« Analyticity of sets associated to Lelong numbers and the extension of closed positive currents », Invent. Math. 27 (1974), p. 53–156. | MR 352516 | Zbl 0289.32003

[52] H. Skoda« Prolongement des courants, positifs, fermés de masse finie », Invent. Math. 66 (1982), p. 361–376. | MR 662596 | Zbl 0488.58002

[53] G. Vigny« Dirichlet-like space and capacity in complex analysis in several variables », J. Funct. Anal. 252 (2007), p. 247–277. | MR 2357357 | Zbl 1130.32017

[54] P. WaltersAn introduction to ergodic theory, Graduate Texts in Math., vol. 79, Springer, 1982. | MR 648108 | Zbl 0475.28009

[55] Y. Yomdin« Volume growth and entropy », Israel J. Math. 57 (1987), p. 285–300. | MR 889979 | Zbl 0641.54036