La mesure d’équilibre d’un endomorphisme de k ()  [ The equilibrium measure of an endomorphism of k () ]
Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque no. 307  (2006), Talk no. 939, p. 33-69

Let f be a holomorphic endomorphism of k (). I will present a geometric construction, due to Briend and Duval, of a probability measure μ having the following properties: μ reflects the distribution of preimages of points outside an algebraic exceptional set, repelling periodic points of f equidistribute with respect to μ and μ is the unique measure of maximal entropy of f.

Soit f un endomorphisme holomorphe de k (). Je présenterai une construction géométrique, due à Briend et Duval, d’une mesure de probabilité μ ayant les propriétés suivantes : μ reflète la distribution des préimages des points en dehors d’un ensemble exceptionnel algébrique, les points périodiques répulsifs de f s’équidistribuent par rapport à μ et μ est l’unique mesure d’entropie maximale de f.

Classification:  32H50
Keywords: holomorphic dynamics, equilibrium measure, exceptional set, entropy
@incollection{SB_2004-2005__47__33_0,
     author = {Buff, Xavier},
     title = {La mesure d'\'equilibre d'un endomorphisme de $\mathbb {P}^k(\mathbb {C})$},
     booktitle = {S\'eminaire Bourbaki : volume 2004/2005, expos\'es 938-951},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {307},
     year = {2006},
     note = {talk:939},
     pages = {33-69},
     zbl = {1138.32009},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2004-2005__47__33_0}
}
Buff, Xavier. La mesure d’équilibre d’un endomorphisme de $\mathbb {P}^k(\mathbb {C})$, in Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Talk no. 939, pp. 33-69. http://www.numdam.org/item/SB_2004-2005__47__33_0/

[1] E. Bedford, M. Lyubich & J. Smillie - “Polynomial diffeomorphisms of 𝐂 2 (V), The measure of maximal entropy and laminar currents”, Invent. Math. 112 (1993), p. 77-125. | Article | MR 1207478 | Zbl 0792.58034

[2] E. Bedford & J. Smillie - “Polynomial diffeomorphisms of 𝐂 2 : Currents, equilibrium measure and hyperbolicity”, Invent. Math. 87 (1990), p. 69-99. | MR 1079840 | Zbl 0721.58037

[3] -, “Polynomial diffeomorphisms of 𝐂 2 (III)”, Math. Ann. 294 (1992), p. 395-420. | MR 1188127

[4] E. Bedford & B. Taylor - “A new capacity for plurisubharmonic functions”, Acta Math. 149 (1982), p. 1-39. | Article | MR 674165 | Zbl 0547.32012

[5] J.-Y. Briend, S. Cantat & M. Shishikura - “Linearity of the exceptional set for maps of k (), Math. Ann. 330 (2004), p. 39-43. | MR 2091677 | Zbl 1056.32018

[6] J.-Y. Briend & J. Duval - “Exposants de Liapounoff et distribution des points périodiques d’un endomorphisme de ℂℙ k , Acta Math. 182 (1999), p. 143-157. | Article | MR 1710180 | Zbl 1144.37436

[7] -, “Deux caractérisations de la mesure d’équilibre d’un endomorphisme de k (), Publ. Math. Inst. Hautes Études Sci. 93 (2001), p. 145-159. | Numdam | Zbl 1010.37004

[8] M. Brin & A. Katok - “On local entropy”, in Geometric dynamics, Lect. Notes in Math., vol. 1007, Springer-Verlag, 1983, p. 30-38. | MR 730261 | Zbl 0533.58020

[9] H. Brolin - “Invariant sets under iteration of rational functions”, Ark. Mat. 6 (1065), p. 103-144. | MR 194595 | Zbl 0127.03401

[10] T.-C. Dinh & N. Sibony - “Dynamique des applications d'allure polynomiale”, J. Math. Pures Appl. 82 (2003), p. 367-423. | Article | MR 1992375 | Zbl 1033.37023

[11] -, “Distribution des valeurs de transformations méromorphes et applications”, Comment. Math. Helv. 81 (2006), no. 1, p. 221-258. | MR 2208805 | Zbl 1094.32005

[12] J. E. Fornæss & N. Sibony - “Complex dynamics in higher dimension”, in Complex potential theory (Montreal, PQ, 1993), NATO Adv. Inst. Ser. C Math. Phys. Sci., vol. 439, Kluwer Acad. Press, Dordrecht, 1994, Notes partially written by Estela A. Gavosto, p. 131-186. | MR 1332961 | Zbl 0811.32019

[13] -, “Complex dynamics in higher dimension I”, in Complex analytic methods in dynamical systems (IMPA, janvier 1992), Astérisque, vol. 222, 1994, p. 201-231. | Numdam | MR 1285389 | Zbl 0813.58030

[14] -, “Complex dynamics in higher dimension, II”, in Modern Methods in Complex Analysis (Princeton, NJ, 1992), Ann. Math. Studies, vol. 137, Princeton University Press, Princeton, NJ, 1995, p. 135-187. | MR 1369137 | Zbl 0847.58059

[15] -, “Dynamics of 𝐏 2 (Examples)”, in Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998), Contemp. Math., vol. 269, Providence, RI, 2001, p. 47-85. | MR 1810536 | Zbl 1006.37025

[16] A. Freire, A. Lopez & R. Mañe - “An invariant measure for rational maps”, Bol. Soc. Brasil. Mat. 14 (1983), p. 45-62. | Article | MR 736568 | Zbl 0568.58027

[17] M. Gromov - “On the entropy of holomorphic maps”, 49 (2003), p. 217-235, Manuscrit 1977. | MR 2026895 | Zbl 1080.37051

[18] V. Guedj - “Ergodic properties of rational mappings with large topological degree”, Ann. of Math. (2) 161 (2005), no. 3, p. 1589-1607. | MR 2179389 | Zbl 1088.37020

[19] J. H. Hubbard & P. Papadopol - “Supperattractive fixed points in n , Indiana Univ. Math. J. 43 (1994), p. 321-365. | Article | MR 1275463 | Zbl 0858.32023

[20] P. Lelong - “Propriétés métriques des variétés analytiques complexes définies par une équation”, 67 (1950), p. 393-419. | Numdam | MR 47789 | Zbl 0039.08804

[21] M. Lyubich - “Entropy properties of rational endomorphisms of the Riemann sphere”, Ergodic Theory Dynamical Systems 3 (1983), p. 351-385. | MR 741393 | Zbl 0537.58035

[22] R. Mañe - “On the uniqueness of the maximizing measure for rational maps”, Bol. Soc. Brasil. Mat. 14 (1983), p. 27-43. | Article | MR 736567 | Zbl 0568.58028

[23] M. Misiurewicz & F. Przytycki - “Topological entropy and degree of smooth mappings”, 25 (1977), p. 573-574. | MR 458501 | Zbl 0362.54037

[24] W. Parry - Entropy and generators in ergodic theory, Benjamin Press, 1969. | MR 262464 | Zbl 0175.34001

[25] N. Sibony - “Dynamique des applications rationnelles de 𝐏 k , in Dynamique et géométrie complexes (Lyon, 1997), vol. 8, Paris, 1999, p. 97-185. | MR 1760844 | Zbl 1020.37026

[26] P. Tortrat - “Aspects potentialistes de l'itération des polynômes”, in Differential geometry and differential equations (C. Gu, M. Berger & R.L. Bryant, éds.), Lect. Notes in Math., vol. 1255, Springer, 1987, p. 195-209. | MR 1052425 | Zbl 0672.31003

[27] T. Ueda - “Fatou sets in complex dynamics on projective spaces”, J. Math. Soc. Japan 46 (1994), no. 3, p. 545-555. | MR 1276837 | Zbl 0829.58025