La mesure d’équilibre d’un endomorphisme de ${\mathbb{P}}^k\left(ℂ\right)$

Buff, Xavier

Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 939, pp. 33-69

Soit $f$ un endomorphisme holomorphe de ${\mathbb{P}}^k\left(ℂ\right)$. Je présenterai une construction géométrique, due à Briend et Duval, d’une mesure de probabilité $\mu$ ayant les propriétés suivantes : $\mu$ 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 $\mu$ et $\mu$ est l’unique mesure d’entropie maximale de $f$.

Let $f$ be a holomorphic endomorphism of ${\mathbb{P}}^k\left(ℂ\right)$. I will present a geometric construction, due to Briend and Duval, of a probability measure $\mu$ having the following properties: $\mu$ reflects the distribution of preimages of points outside an algebraic exceptional set, repelling periodic points of $f$ equidistribute with respect to $\mu$ and $\mu$ is the unique measure of maximal entropy of $f$.

Classification : 32H50
Mots-clés : dynamique holomorphe, mesure d'équilibre, ensemble exceptionnel, entropie
Keywords: holomorphic dynamics, equilibrium measure, exceptional set, entropy

[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. | Zbl | MR | EuDML | DOI

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

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

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

[5] J.-Y. Briend, S. Cantat & M. Shishikura - “Linearity of the exceptional set for maps of ${\mathbb{P}}^k\left(ℂ\right)$”, Math. Ann. 330 (2004), p. 39-43. | Zbl | MR

[6] J.-Y. Briend & J. Duval - “Exposants de Liapounoff et distribution des points périodiques d’un endomorphisme de ${\mathrm{ℂ\mathbb{P}}}^k$”, Acta Math. 182 (1999), p. 143-157. | Zbl | MR | DOI

[7] -, “Deux caractérisations de la mesure d’équilibre d’un endomorphisme de ${\mathbb{P}}^k\left(ℂ\right)$”, Publ. Math. Inst. Hautes Études Sci. 93 (2001), p. 145-159. | Zbl | Numdam | EuDML

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

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

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

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

[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. | Zbl | MR

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

[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. | Zbl | MR

[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. | Zbl | MR

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

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

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

[19] J. H. Hubbard & P. Papadopol - “Supperattractive fixed points in ${ℂ}^n$”, Indiana Univ. Math. J. 43 (1994), p. 321-365. | Zbl | MR | DOI

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

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

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

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

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

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

[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. | Zbl | MR

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