Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 2, pp. 235-278.

We continue our study of the dynamics of mappings with small topological degree on projective complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic “equilibrium” measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points are equidistributed towards this measure. This generalizes results that were known in the invertible case and adds to the small number of situations in which a natural invariant measure for a non-invertible dynamical system is well-understood.

Nous poursuivons notre étude de la dynamique des applications rationnelles de petit degré topologique sur les surfaces complexes projectives. Dans un travail précédent nous avons construit une mesure ergodique naturelle, dite « d'équilibre », sous des hypothèses très générales. Nous étudions maintenant en détail les propriétés dynamiques de cette mesure : nous donnons des bornes optimales pour ses exposants de Lyapounov, montrons qu'elle est d'entropie maximale et qu'elle a une structure produit dans l'extension naturelle. Sous une hypothèse supplémentaire naturelle, nous montrons que cette mesure décrit la répartition des points selles. Ceci généralise des résultats qui étaient auparavant connus dans le cas inversible et vient ainsi s'ajouter au petit nombre de situations où une mesure invariante naturelle pour un système dynamique non inversible est vraiment bien comprise.

DOI: 10.24033/asens.2120
Classification: 37F10,  32H50,  32U40,  37B40,  37D99
Keywords: dynamics of meromorphic mappings, laminar and woven currents, entropy, natural extension
@article{ASENS_2010_4_43_2_235_0,
     author = {Diller, Jeffrey and Dujardin, Romain and Guedj, Vincent},
     title = {Dynamics of meromorphic maps with small topological degree {III:} geometric currents and ergodic theory},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {235--278},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 43},
     number = {2},
     year = {2010},
     doi = {10.24033/asens.2120},
     zbl = {1197.37059},
     mrnumber = {2662665},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/asens.2120/}
}
TY  - JOUR
AU  - Diller, Jeffrey
AU  - Dujardin, Romain
AU  - Guedj, Vincent
TI  - Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2010
DA  - 2010///
SP  - 235
EP  - 278
VL  - Ser. 4, 43
IS  - 2
PB  - Société mathématique de France
UR  - http://archive.numdam.org/articles/10.24033/asens.2120/
UR  - https://zbmath.org/?q=an%3A1197.37059
UR  - https://www.ams.org/mathscinet-getitem?mr=2662665
UR  - https://doi.org/10.24033/asens.2120
DO  - 10.24033/asens.2120
LA  - en
ID  - ASENS_2010_4_43_2_235_0
ER  - 
%0 Journal Article
%A Diller, Jeffrey
%A Dujardin, Romain
%A Guedj, Vincent
%T Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory
%J Annales scientifiques de l'École Normale Supérieure
%D 2010
%P 235-278
%V Ser. 4, 43
%N 2
%I Société mathématique de France
%U https://doi.org/10.24033/asens.2120
%R 10.24033/asens.2120
%G en
%F ASENS_2010_4_43_2_235_0
Diller, Jeffrey; Dujardin, Romain; Guedj, Vincent. Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 2, pp. 235-278. doi : 10.24033/asens.2120. http://archive.numdam.org/articles/10.24033/asens.2120/

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

[2] E. Bedford, M. Lyubich & J. Smillie, Distribution of periodic points of polynomial diffeomorphisms of 𝐂 2 , Invent. Math. 114 (1993), 277-288. | MR | Zbl

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

[4] J.-Y. Briend, Propriété de Bernoulli pour les extensions naturelles des endomorphismes de P k , Ergodic Theory Dynam. Systems 21 (2001), 1001-1007. | MR | Zbl

[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), 145-159. | Numdam | MR | Zbl

[6] S. Cantat, Dynamique des automorphismes des surfaces K3, Acta Math. 187 (2001), 1-57. | MR | Zbl

[7] J. Diller, R. Dujardin & V. Guedj, Dynamics of meromorphic maps with small topological degree I: from cohomology to currents, to appear in Indiana Univ. Math. J. | MR | Zbl

[8] J. Diller, R. Dujardin & V. Guedj, Dynamics of meromorphic maps with small topological degree II: energy and invariant measure, to appear in Comment. Math. Helvet. | MR

[9] J. Diller & C. Favre, Dynamics of bimeromorphic maps of surfaces, Amer. J. Math. 123 (2001), 1135-1169. | MR | Zbl

[10] T.-C. Dinh, Suites d'applications méromorphes multivaluées et courants laminaires, J. Geom. Anal. 15 (2005), 207-227. | MR | Zbl

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

[12] R. Dujardin, Hénon-like mappings in 2 , Amer. J. Math. 126 (2004), 439-472. | MR | Zbl

[13] R. Dujardin, Sur l'intersection des courants laminaires, Publ. Mat. 48 (2004), 107-125. | MR | Zbl

[14] R. Dujardin, Structure properties of laminar currents on 2 , J. Geom. Anal. 15 (2005), 25-47. | MR | Zbl

[15] R. Dujardin, Laminar currents and birational dynamics, Duke Math. J. 131 (2006), 219-247. | MR | Zbl

[16] J. Duval, Singularités des courants d'Ahlfors, Ann. Sci. École Norm. Sup. 39 (2006), 527-533. | Numdam | MR | Zbl

[17] C. Favre & M. Jonsson, Dynamical compactifications of 𝐂 2 , à paraître aux Ann. Math. | Zbl

[18] J. E. Fornaess & N. Sibony, Complex dynamics in higher dimension. II, in Modern methods in complex analysis (Princeton, NJ, 1992), Ann. of Math. Stud. 137, Princeton Univ. Press, 1995, 135-182. | MR | Zbl

[19] A. Freire, A. Lopes & R. Mañé, An invariant measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), 45-62. | MR | Zbl

[20] W. Fulton, Intersection theory, second éd., Ergebn. Math. Grenzg. 2, Springer, 1998. | MR | Zbl

[21] É. Ghys, Laminations par surfaces de Riemann, in Dynamique et géométrie complexes (Lyon, 1997), Panor. Synthèses 8, Soc. Math. France, 1999, 49-95. | MR | Zbl

[22] M. Gromov, On the entropy of holomorphic maps, Enseign. Math. 49 (2003), 217-235. | MR | Zbl

[23] V. Guedj, Entropie topologique des applications méromorphes, Ergodic Theory Dynam. Systems 25 (2005), 1847-1855. | MR | Zbl

[24] V. Guedj, Ergodic properties of rational mappings with large topological degree, Ann. of Math. 161 (2005), 1589-1607. | MR | Zbl

[25] F. Ledrappier, Some properties of absolutely continuous invariant measures on an interval, Ergodic Theory Dynam. Systems 1 (1981), 77-93. | MR | Zbl

[26] F. Ledrappier & J.-M. Strelcyn, A proof of the estimation from below in Pesin's entropy formula, Ergodic Theory Dynam. Systems 2 (1982), 203-219. | MR | Zbl

[27] P. Lelong, Propriétés métriques des variétés analytiques complexes définies par une équation, Ann. Sci. École Norm. Sup. 67 (1950), 393-419. | Numdam | MR | Zbl

[28] M. J. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), 351-385. | MR | Zbl

[29] E. Mihailescu & M. Urbański, Holomorphic maps for which the unstable manifolds depend on prehistories, Discrete Contin. Dyn. Syst. 9 (2003), 443-450. | MR | Zbl

[30] C. C. Moore & C. Schochet, Global analysis on foliated spaces, Mathematical Sciences Research Institute Publications 9, Springer, 1988. | MR | Zbl

[31] D. Ornstein & B. Weiss, On the Bernoulli nature of systems with some hyperbolic structure, Ergodic Theory Dynam. Systems 18 (1998), 441-456. | MR | Zbl

[32] F. Przytycki, Anosov endomorphisms, Studia Math. 58 (1976), 249-285. | MR | Zbl

[33] F. Przytycki & M. Urbański, Conformal fractals: Ergodic theory methods, London Math. Soc. Lecture Note Series 371, 2010. | MR | Zbl

[34] M. Qian & S. Zhu, SRB measures and Pesin's entropy formula for endomorphisms, Trans. Amer. Math. Soc. 354 (2002), 1453-1471. | MR | Zbl

[35] V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measure, Uspehi Mat. Nauk 22 (1967), 3-56 ; English translation : Russian Math. Surveys 22 (1967), 1-52. | MR | Zbl

[36] H. De Thélin, Sur la construction de mesures selles, Ann. Inst. Fourier (Grenoble) 56 (2006), 337-372. | Numdam | MR | Zbl

[37] H. De Thélin, Sur les exposants de Lyapounov des applications méromorphes, Invent. Math. 172 (2008), 89-116. | MR | Zbl

[38] H. De Thélin & G. Vigny, Entropy of meromorphic maps and dynamics of birational maps, to appear in Mémoires de la SMF. | Numdam | Zbl

[39] Y. Yomdin, Volume growth and entropy, Israel J. Math. 57 (1987), 285-300. | MR | Zbl

Cited by Sources: