Equidistribution problems in complex dynamics of higher dimension. (English) Zbl 1386.32019

The paper under review provide an interesting survey on recent results concerning equidistribution of subvarieties, periodic points and orbits of points in higher-dimensional complex dynamics.
Let \(X\) be a compact Kähler manifold and \(f\) be a meromorphic self-map or a multi-valued map on \(X\). One first question is to estimate the cardinality of the set \(Q_n\) of isolated periodic points of period \(n\). Under some assumptions on \(f\), the first author et al. [Bull. Lond. Math. Soc. 49, No. 6, 947–964 (2017; Zbl 1401.37054)] recently showed that the cardinality of \(Q_n\) grows at most exponentially fast with \(n\). More precisely, \(\# Q_n= O(e^{n\lambda})\) for any constant \(\lambda>h_a(f)\) where \(h_a(f)\) is the algebraic entropy of \(f\).
Another key problem in this survey is the distribution of the points of \(Q_n\), that is, whether the sequence of probability measures \(\mu_n:= \frac{1}{\# Q_n}\sum_{a\in Q_n}\delta_a\) converges weakly to an \(f\)-invariant probability measure \(\mu\). If \(f\) is algebraically expanding, that is, its last dynamical degree \(d_k(f)\) is strictly bigger than the other dynamical degrees, the first author et al. [Indiana Univ. Math. J. 64, No. 6, 1805–1828 (2015; Zbl 1343.37030)] showed that \(\# Q_n\leq d_k(f)^n + o(d_k(f))^n\) and \(\frac{1}{d_k(f)^n} \sum_{a\in Q_n}\delta_a\) converges to an invariant probability measure \(\mu\). The authors obtain an analogous result for the equidistribution of the orbits of points.
Let \(V\) be a generic hypersurface of degree \(\mathrm{deg}(V)\) in \(\mathbb{P}^k\). Equidistribution of the backward orbit of \(V\) can be considered as the weak convergence of the sequence of pullbacks \((f^n)^*[V]\) of the \((1,1)\)-current of integration \([V]\) on \(V\). The authors [Ann. Sci. Éc. Norm. Supér. (4) 41, No. 2, 307–336 (2008; Zbl 1160.32029)] and C. Favre and M. Jonsson [Ann. Inst. Fourier 53, No. 5, 1461–1501 (2003; Zbl 1113.32005)] showed that, denoting by \(d_1(f)\) the first dynamical degree of \(f\), the sequence \(\mathrm{deg}(V)^{-1}d_1(f)^{-n}f^n)^*[V]\) converges to the Green \((1,1)\)-current \(T\) of \(f\). In the case of an analytic set \(V\) of arbitrary codimension \(p\), the authors show that \(\mathrm{deg}(V)^{-1}d_1(f)^{-pn}f^n)^*[V]\) converges to \(T^p\).
The authors provide also a brief discussion of super-potentials and other tools of pluripotential theory, and in the last section they give some open problems related to equidistribution and possible ways to attack these problems.


32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
32U40 Currents
37F99 Dynamical systems over complex numbers
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
Full Text: DOI arXiv


[1] Ahn, T., Equidistribution in higher codimension for holomorphic endomorphisms of \(\Bbb P^k\), Trans. Amer. Math. Soc.368(5) (2016) 3359-3388. · Zbl 1418.37075
[2] Amerik, E. and Campana, F., Exceptional points of an endomorphism of the projective plane, Math. Z.249(4) (2005) 741-754. · Zbl 1075.14010
[3] Bedford, E., Lyubich, M. and Smillie, J., Distribution of periodic points of polynomial diffeomorphisms of \(C^2\), Invent. Math.114(2) (1993) 277-288. · Zbl 0799.58039
[4] Bedford, E., Lyubich, M. and Smillie, J., Polynomial diffeomorphisms of \(\mathbb{C}^2\). IV. The measure of maximal entropy and laminar currents, Invent. Math.112(1) (1993) 77-125. · Zbl 0792.58034
[5] F. Berteloot, F. Bianchi and C. Dupont, Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of \(\Bbb P^k\), preprint (2014), arXiv:1403.7603. · Zbl 1454.32011
[6] Boucksom, S., Favre, C. and Jonsson, M., Degree growth of meromorphic surface maps, Duke Math. J.141(3) (2008) 519-538. · Zbl 1185.32009
[7] Briend, J.-Y. and Duval, J., Exposants de Liapounoff et distribution des points périodiques d’un endomorphisme de \(\mathbb{C} \Bbb P^k\), Acta Math.182(2) (1999) 143-157. · Zbl 1144.37436
[8] J.-Y. Briend and J. Duval, Deux caractérisations de la mesure d’équilibre d’un endomorphisme de \(\text{P}^k(C)\), Publ. Math. Inst. Hautes Études Sci.93 (2001) 145-159, Erratum 109 (2009) 295-296. · Zbl 1185.37105
[9] Brolin, H., Invariant sets under iteration of rational functions, Ark. Mat.6 (1965) 103-144. · Zbl 0127.03401
[10] Cantat, S., Dynamique des automorphismes des surfaces K3, Acta Math.187 (2001) 1-57. · Zbl 1045.37007
[11] Cerveau, D. and Neto, A. Lins, Hypersurfaces exceptionnelles des endomorphismes de \(\mathbb{C} \Bbb P(n)\), Bol. Soc. Brasil. Mat. (N.S.)31(2) (2000) 155-161. · Zbl 0967.32022
[12] Clozel, L. and Otal, J.-P., Unique ergodicité des correspondances modulaires, in Essays on Geometry and Related Topics, Vols. 1 and 2, , Vol. 38 (Kundig, 2001), pp. 205-216. · Zbl 1048.37007
[13] Clozel, L. and Ullmo, E., Correspondances modulaires et mesures invariantes, J. Reine Angew. Math.558 (2003) 47-83. · Zbl 1042.11027
[14] J.-P. Demailly, Complex analytic and differential geometry, www.fourier.ujf-grenoble.fr/\( \sim\) demailly.
[15] De Thélin, H., Sur la construction de mesures selles, Ann. Inst. Fourier56(2) (2006) 337-372. · Zbl 1100.37029
[16] De Thélin, H., Sur les exposants de Lyapounov des applications méromorphes, Invent. Math.172(1) (2008) 89-116. · Zbl 1139.37037
[17] Diller, J., Dujardin, R. and Guedj, V., Dynamics of meromorphic maps with small topological degree III: Geometric currents and ergodic theory, Ann. Sci. École. Norm. Supér. (4)43(2) (2010) 235-278. · Zbl 1197.37059
[18] Dinh, T.-C., Distribution des préimages et des points périodiques d’une correspondance polynomiale, Bull. Soc. Math. France133(3) (2005) 363-394. · Zbl 1090.37032
[19] Dinh, T.-C., Suites d’applications méromorphes multivaluées et courants laminaires, J. Geom. Anal.15 (2005) 207-227. · Zbl 1085.37039
[20] Dinh, T.-C., Attracting current and equilibrium measure for attractors on \(\Bbb P^k\), J. Geom. Anal.17(2) (2007) 227-244. · Zbl 1139.37032
[21] Dinh, T.-C., Equidistribution of periodic points for modular correspondences, J. Geom. Anal.23(3) (2013) 1189-1195. · Zbl 1338.37012
[22] Dinh, T.-C. and Nguyen, V.-A., The mixed Hodge-Riemann bilinear relations for compact Kähler manifolds, Geom. Funct. Anal.16(4) (2006) 838-849. · Zbl 1126.32018
[23] Dinh, T.-C., Nguyen, V.-A. and Sibony, N., Exponential estimates for plurisubharmonic functions and stochastic dynamics, J. Differential Geom.84(3) (2010) 465-488. · Zbl 1211.32021
[24] Dinh, T.-C., Nguyen, V.-A. and Truong, T. T., On the dynamical degrees of meromorphic maps preserving a fibration, Commun. Contemp. Math.14(6) (2012) 18 p., doi: 10.1142/S0219199712500423. · Zbl 1311.37034
[25] Dinh, T.-C., Nguyen, V.-A. and Truong, T. T., Equidistribution for meromorphic maps with dominant topological degree, Indiana J. Math.64(6) (2015) 1805-1828. · Zbl 1343.37030
[26] T.-C. Dinh, V.-A. Nguyen and T. T. Truong, Growth of the number of periodic points for meromorphic maps, preprint (2016), arXiv:1601.03910.
[27] Dinh, T.-C. and Sibony, N., Dynamique des applications d’allure polynomiale, J. Math. Pures Appl. (9)82(4) (2003) 367-423. · Zbl 1033.37023
[28] Dinh, T.-C. and Sibony, N., Regularization of currents and entropy, Ann. Sci. École Norm. Sup.37 (2004) 959-971. · Zbl 1074.53058
[29] Dinh, T.-C. and Sibony, N., Groupes commutatifs d’automorphismes holomorphes d’une variété kählérienne compacte, Duke Math. J.123 (2004) 311-328. · Zbl 1065.32012
[30] Dinh, T.-C. and Sibony, N., Distribution des valeurs d’une suite de transformations méromorphes et applications, Comment. Math. Helv.81 (2006) 221-258. · Zbl 1094.32005
[31] Dinh, T.-C. and Sibony, N., Pull-back currents by holomorphic maps, Manuscripta Math.123(3) (2007) 357-371. · Zbl 1128.32020
[32] Dinh, T.-C. and Sibony, N., Upper bound for the topological entropy of a meromorphic correspondence, Israel J. Math.163 (2008) 29-44. · Zbl 1163.37011
[33] Dinh, T.-C. and Sibony, N., Equidistribution towards the Green current for holomorphic maps, Ann. Sci. École Norm. Sup.41 (2008) 307-336. · Zbl 1160.32029
[34] Dinh, T.-C. and Sibony, N., Super-potentials of positive closed currents, intersection theory and dynamics, Acta Math.203(1) (2009) 1-82. · Zbl 1227.32024
[35] Dinh, T.-C. and Sibony, N., Equidistribution speed for endomorphisms of projective spaces, Math. Ann.347(3) (2010) 613-626. · Zbl 1206.37023
[36] Dinh, T.-C. and Sibony, N., Dynamics in several complex variables: Endomorphisms of projective spaces and polynomial-like mappings, in Holomorphic Dynamical Systems, , Vol. 1998 (Springer, 2010), pp. 165-294. · Zbl 1218.37055
[37] Dinh, T.-C. and Sibony, N., Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms, J. Algebraic Geom.19(3) (2010) 473-529. · Zbl 1202.32033
[38] Dinh, T.-C. and Sibony, N., Rigidity of Julia sets for Hénon type maps, J. Mod. Dyn.8(3-4) (2014) 499-548. · Zbl 1370.37090
[39] T.-C. Dinh and N. Sibony, Density of positive closed currents, a theory of non-generic intersections, preprint (2015), arXiv:1203.5810.
[40] Dinh, T.-C. and Sibony, N., Equidistribution of saddle periodic points for Hénon-type automorphisms of \(\mathbb{C}^k\), Math. Ann.366(3-4) (2016) 1207-1251. · Zbl 1361.32042
[41] Dujardin, R., Laminar currents and birational dynamics, Duke Math. J.131(2) (2006) 219-247. · Zbl 1099.37037
[42] Esnault, H., Oguiso, K. and Yu, X., Automorphisms of elliptic K3 surfaces and Salem numbers of maximal degree, Algebr. Geom.3(4) (2016) 496-507. · Zbl 1388.14112
[43] Esnault, H. and Srinivas, V., Algebraic versus topological entropy for surfaces over finite fields, Osaka J. Math.50(3) (2013) 827-846. · Zbl 1299.14024
[44] Favre, C., Points périodiques d’applications birationnelles de \(\Bbb P^2\). Ann. Inst. Fourier (Grenoble)48(4) (1998) 999-1023. · Zbl 0924.58083
[45] Favre, C. and Jonsson, M., Brolin’s theorem for curves in two complex dimensions, Ann. Inst. Fourier53(5) (2003) 1461-1501. · Zbl 1113.32005
[46] Favre, C. and Wulcan, E., Degree growth of monomial maps and McMullen’s polytope algebra, Indiana Univ. Math. J.61(2) (2012) 493-524. · Zbl 1291.37058
[47] Fornæss, J.-E. and Sibony, N., Complex Hénon mappings in \(\mathbb{C}^2\) and Fatou-Bieberbach domains, Duke Math. J.65(2) (1992) 345-380. · Zbl 0761.32015
[48] Fornæss, J.-E. and Sibony, N., Complex dynamics in higher dimension. I, in Complex Analytic Methods in Dynamical Systems, , Vol. 222 (Société Mathématique de France, 1994), pp. 201-231. · Zbl 0813.58030
[49] Fornæss, J.-E. and Sibony, N., Complex dynamics in higher dimensions. Notes partially written by Estela A. Gavosto, in Complex Potential Theory, , Vol. 439 (Kluwer Academic Publishers, Dordrecht, 1994), pp. 131-186. · Zbl 0811.32019
[50] Fornæss, J.-E. and Sibony, N., Complex dynamics in higher dimension. II, in Modern Methods in Complex Analysis, , Vol. 137 (Princeton University Press, Princeton, NJ, 1995), pp. 135-182. · Zbl 0847.58059
[51] Fornæss, J.-E. and Sibony, N., Dynamics of \(\Bbb P^2\) (examples), in Laminations and Foliations in Dynamics, Geometry and Topology, , Vol. 269 (American Mathematical Society, Providence, RI, 2001), pp. 47-85. · Zbl 1006.37025
[52] Freire, A., Lopes, A. and Mañé, R., An invariant measure for rational maps, Bol. Soc. Brasil. Mat.14(1) (1983) 45-62. · Zbl 0568.58027
[53] Friedland, S. and Milnor, J., Dynamical properties of plane polynomial automorphisms, Ergodic Theory Dynam. Systems9(1) (1989) 67-99. · Zbl 0651.58027
[54] Gromov, M., Convex sets and Kähler manifolds, in Advances in Differential Geometry and Topology (World Scientific Publishing, Teaneck, NJ, 1998), pp. 1-38. · Zbl 0770.53042
[55] Gromov, M., On the entropy of holomorphic maps, Enseignement Math.49 (2003) 217-235. · Zbl 1080.37051
[56] Guedj, V., Ergodic properties of rational mappings with large topological degree, Ann. of Math. (2)161(3) (2005) 1589-1607. · Zbl 1088.37020
[57] Hörmander, L., An Introduction to Complex Analysis in Several Variables, 3rd edn., , Vol. 7 (North-Holland, Amsterdam, 1990). · Zbl 0685.32001
[58] Huang, X. and Yuan, Y., Holomorphic isometry from a Kähler manifold into a product of complex projective manifolds, Geom. Funct. Anal.24(3) (2014) 854-886. · Zbl 1302.53078
[59] Hwang, J.-M. and Nakayama, N., On endomorphisms of Fano manifolds of Picard number one, Pure Appl. Math. Q.7(4) (2011), Special Issue: In memory of Eckart Viehweg, 1407-1426. · Zbl 1316.14081
[60] Iwasaki, K. and Uehara, T., Periodic points for area-preserving birational maps of surfaces, Math. Z.266(2) (2010) 289-318. · Zbl 1206.37010
[61] M. Jonsson and J. Reschke, On the complex dynamics of birational surface maps defined over number fields, Crelle’s Journal, to appear (2015), arXiv:1505.03559. · Zbl 1405.37098
[62] Kaloshin, V. Yu., Generic diffeomorphisms with superexponential growth of number of periodic orbits, Commun. Math. Phys.211(1) (2000) 253-271. · Zbl 0956.37017
[63] L. Kaufmann, A Skoda-type integrability theorem for singular Monge-Ampère measures, to appear in Michigan Math. J. · Zbl 1386.32032
[64] Lin, J.-L., Pulling back cohomology classes and dynamical degrees of monomial maps, Bull. Soc. Math. France140(4) (2012) 533-549. · Zbl 1333.37031
[65] Lyubich, M. J., Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dyn. Syst.3(3) (1983) 351-385. · Zbl 0537.58035
[66] Mok, N., Local holomorphic isometric embeddings arising from correspondences in the rank-1 case, in Contemporary Trends in Algebraic Geometry and Algebraic Topology, , Vol. 5 (World Scientific, River Edge, NJ, 2002), pp. 155-165. · Zbl 1083.32019
[67] Mok, N., Extension of germs of holomorphic isometries up to normalizing constants with respect to the Bergman metric, J. Eur. Math. Soc.14(5) (2012) 1617-1656. · Zbl 1266.32014
[68] Mok, N. and Ng, S.-C., Germs of measure-preserving holomorphic maps from bounded symmetric domains to their Cartesian products, J. Reine Angew. Math.669 (2012) 47-73. · Zbl 1254.32033
[69] Nguyen, V.-A., Green currents for quasi-algebraically stable meromorphic self-maps of \(\Bbb P^k\), Publ. Mat.56(1) (2012) 127-146. · Zbl 1297.37023
[70] K. Oguiso, Pisot units, Salem numbers and higher dimensional projective manifolds with primitive automorphisms of positive entropy, preprint (2016), arXiv:1608.03122. · Zbl 1432.37010
[71] Oguiso, K. and Truong, T. T., Explicit examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy, J. Math. Sci. Univ. Tokyo22 (2015) 361-385. · Zbl 1349.14055
[72] Russakovskii, A. and Shiffman, B., Value distribution for sequences of rational mappings and complex dynamics, Indiana Univ. Math. J.46(3) (1997) 897-932. · Zbl 0901.58023
[73] Saito, S., General fixed point formula for an algebraic surface and the theory of Swan representations for two-dimensional local rings, Amer. J. Math.109(6) (1987) 1009-1042. · Zbl 0647.14026
[74] Sibony, N., Dynamique des applications rationnelles de \(\Bbb P^k\), Panoramas et Synthèses8 (1999) 97-185. · Zbl 1020.37026
[75] Siu, Y. T., Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math.27 (1974) 53-156. · Zbl 0289.32003
[76] Taflin, J., Equidistribution speed towards the Green current for endomorphisms of \(\Bbb P^k\), Adv. Math.227(5) (2011) 2059-2081. · Zbl 1258.32004
[77] Taflin, J., Speed of convergence towards attracting sets for endomorphisms of \(\Bbb P^k\), Indiana Univ. Math. J.62(1) (2013) 33-44. · Zbl 1295.32031
[78] T. T. Truong, (Relative) dynamical degrees of rational maps over an algebraic closed field, preprint (2015), arXiv:1501.01523.
[79] T. T. Truong, Relative dynamical degrees of correspondences over a field of arbitrary characteristic, preprint (2016), arXiv:1605.05049.
[80] T. T. Truong, Relations between dynamical degrees, Weil’s Riemann hypothesis and the standard conjectures, preprint (2016), arXiv:1611.01124.
[81] C. Voisin, Théorie de Hodge et Géométrie Algébrique Complexe, Cours Spécialisés, Vol. 10 (Société Mathématique de France, Paris, 2002). · Zbl 1032.14001
[82] Vu, D.-V., Intersection of positive closed currents of higher bidegree, Michigan Math. J.65(4) (2016) 863-872. · Zbl 1367.32026
[83] D.-V. Vu, Complex Monge-Ampère equation for measures supported on real submanifolds, preprint (2016), arXiv:1608.02794.
[84] Xie, J., Periodic points of birational transformations on projective surfaces, Duke Math. J.164(5) (2015) 903-932. · Zbl 1392.37120
[85] Yomdin, Y., Volume growth and entropy, Israel J. Math.57(3) (1987) 285-300. · Zbl 0641.54036
[86] Zhang, D.-Q., Invariant hypersurfaces of endomorphisms of projective varieties, Adv. Math.252 (2014) 185-203. · Zbl 1291.14029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.