×

Sequences of multivalued meromorphic mappings and laminar currents. (Suites d’applications méromorphes multivaluées et Courants laminaires.) (French) Zbl 1085.37039

Let \((X_1,\omega_1)\) and \((X_2,\omega_2)\) be the compact Kählerian varieties of dimensions \(k_1\) and \(k_2\), respectively. Let \(F_n: X_1\to X_2\) be the meromorphic (multi-valued) mappings. For \(F_n\), denote by \(d_n\) the topological degree and \(\lambda_n\) the intermediate degree of the order \(k_2-1\). Suppose that the series \(\sum \lambda_n d_n^{-1}\) converges. For the case of the rational mappings between projective spaces, Sodin-Russakovskii-Shiffman showed that the preimages of \(F_n\) are equidistributed when \(n\) tends to infinity. More precisely, \(d_n^{-1}F_n^\ast(\delta_z)-d_n^{-1}F_n^\ast(\delta_{z'})\) tends weakly to 0 for \(z, z'\in X_2\) out of a pluripolar set \(\mathcal E\), where \(\delta_z\) denotes the Dirac mass in \(z\). If \(\{F_n\}=\{f^n\}\) is the sequence of the iterations of a meromorphic (multivalued) mapping \(f\) whose topological degree is bigger than the other dynamic degrees, \(\mathcal E\) is contained in a countable union of analytic ensembles. Besides, \(d_n^{-1}F_n^\ast(\delta_z)\) converges to the equilibrium measure \(\mu\) of \(f\). This measure also reflects the distribution of the repelling fixed-points of \(F_n\).
The author looks for the more general conditions on the sequence \(\{F_n\}\) in order that \(\mathcal E\) be contained in a countable union of analytic ensembles. It is shown that it is the case when the sequence of postcritical currents \(S_n\) of \(F_n\) converges to a current \(S_\infty\). Then the ensemble \(\mathcal E\) is contained essentially in \(\{\nu(S_\infty, z)> 0\}\), where \(\nu(S_\infty, z)\) denotes the Lelong number of \(S_\infty\) in \(z\). The distribution of the repelling fixed-points in the case when \(X_1\) and \(X_2\) are a compact Riemann surface \(X\) is also studied. Suppose that for all \(z\notin\mathcal E\), \(d_n^{-1}F_n^\ast(\delta_z)\) weakly tends to a measure \(\mu\) that is singular with respect to \(S_\infty\). Then the measure, equidistributed in the repelling stationary fixed-points of \(F_n\), tends weakly to \(\mu\). This result reflects the connection between the postcritical ensembles and the equidistribution of the repelling fixed-points of \(F_n(=f^n)\).
Giving an example, it is shown that the hypothesis of the connection between \(\mu\) and \(S_\infty\) is necessary. The author constructs the local inverse mappings of \(F_n\)-branches, which are defined on small balls and admit, as images, the ensembles whose diameters tend to \(0\) when \(n\to\infty\). This imply the equidistribution of preimages. Then the author studies the laminarity of certain positive closed currents of any dimension. Let \(\{Z_n\}\) be a sequence of images of \(\mathbb P^s\) in a projective variety \(X\) of dimension \(k>s\). Suppose that the sequence of integration currents on \(Z_n\), normalized by the appropriate way, tends to a current \(T\). Then \(T\) is tissued (geometric) and is laminar if \(s=k-1\) and the singularities of \(Z_n\) are sufficient. The construction of inverses branches is used to prove this result. More precisely, it is considered the projections \(F_n\) of \(Z_n\) on a projective space \(\mathbb P^s\). The inverse \(F_n\)-branches of the open sets of \(\mathbb P^s\), with controlled size, form the normal families of complex varieties in \(X\). Passing to the limit, one gets the complex varieties that form the current \(T\).
As an application, it is shown that the Green currents of some bidimensions, of a regular polynomial automorphism, are either laminar or tissued. The laminarity of Green currents is also valid for the automorphisms of a projective variety. To this end the author studies the laminarity of the limits \(T\) of \(\{Z_n\}\) when these analytic ensembles are not the images of \(\mathbb P^s\). Here it is used a notion of dual variety (curvature variety) \({\widehat Z}_n\) that is well adapted to the dynamic problems. In a certain sense it permits to work with the derivatives in the form of geometric objects. When the volume of \({\widehat Z}_n\) verifies vol\(({\widehat Z}_n) = O(\text{vol}(Z_n))\), then the current limits \(T\) are also tissued.

MSC:

37F05 Dynamical systems involving relations and correspondences in one complex variable
32C30 Integration on analytic sets and spaces, currents
32V40 Real submanifolds in complex manifolds
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
32H04 Meromorphic mappings in several complex variables
32U40 Currents
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alexander, H., Projective capacity, Ann. Math. Studies, 100, 3-27 (1981) · Zbl 0494.32001
[2] Bedford, E.; Lyubich, M.; Smillie, J., Polynomial diffeomorphisms of ℂ^2, IV. The measure of maximal entropy and laminar current, Invent. Math., 112, 77-125 (1993) · Zbl 0792.58034 · doi:10.1007/BF01232426
[3] Briend, J. Y.; Duval, J., Exposants de Liapounoff et distribution des points périodiques d’un endomorphisme de ℂℙ^k, Acta Math., 182, 143-157 (1999) · Zbl 1144.37436 · doi:10.1007/BF02392572
[4] Briend, J. Y.; Duval, J., Deux caractérisations de la mesure d’équilibre d’un endomorphisme de ℙ^k(ℂ), IHES Publ. Math., 93, 145-159 (2001) · Zbl 1010.37004 · doi:10.1007/s10240-001-8190-4
[5] Cantat, S., Dynamique des automorphismes des surfaces K3, Acta Math., 187, 1, 1-57 (2001) · Zbl 1045.37007 · doi:10.1007/BF02392831
[6] de Thélin, H., Sur la laminanté de certains courants, Ann. Sci. Ecole Norm. Sup. (4), 37, 2, 304-311 (2004) · Zbl 1061.32005
[7] Diller, J.; Favre, C., Dynamics of bimeromorphic maps of surfaces, Amer. J. Math., 123, 1135-1169 (2001) · Zbl 1112.37308 · doi:10.1353/ajm.2001.0038
[8] Dinh, T. C. Distribution des préimages et des points périodiques d’une correspondance polynomiale, à paraître au,Bull. Soc. Math. France. · Zbl 1090.37032
[9] Dinh, T. C.; Sibony, N., Dynamique des applications d’allure polynomials, J. Math. Pures et Appl., 82, 367-423 (2003) · Zbl 1033.37023
[10] Dinh, T. C. and Sibony, N. Distribution des valeurs de transformations méromorphes et applications, prépublication, (2003), arXiv:math.DS/0306095
[11] Dinh, T. C.; Sibony, N., Green currents for holomorphic automorphisms of compact Kähler manifolds, J. Amer. Math. Soc., 18, 291-312 (2005) · Zbl 1066.32024 · doi:10.1090/S0894-0347-04-00474-6
[12] Dujardin, R., Laminar currents in ℙ^2, Math. Ann., 325, 745-765 (2003) · Zbl 1021.37018 · doi:10.1007/s00208-002-0402-9
[13] Dujardin, R., Sur l’intersection des courants laminaires, Publicacions Mathematiques, 48, 107-125 (2004) · Zbl 1048.32021
[14] Dujardin, R. Laminar currents and entropy properties of surface birational maps, prépublication, (2003).
[15] Duval, J.; Sibony, N., Polynomial convexity, rational convexity, and currents, Duke Math. J., 79, 487-513 (1995) · Zbl 0838.32006 · doi:10.1215/S0012-7094-95-07912-5
[16] Federer, H., Geometric Measure Theory (1969), New York, NY: Springer-Verlag Inc., New York, NY · Zbl 0176.00801
[17] Fornæss, J. E.; Sibony, N., Dynamics ofP^2 (examples). Laminations and foliations in dynamics, geometry and topology [Stony Brook, NY, 47-85, (1998)], Contemp. Math. (2001), Providence, RI: Amer. Math. Soc., Providence, RI · Zbl 1006.37025
[18] Freire, A.; Lopes, A.; Mañé, R., An invariant measure for rational maps, Bol. Soc. Brasil. Mat., 14, 45-62 (1983) · Zbl 0568.58027 · doi:10.1007/BF02584744
[19] Fulton, W.Intersection Theory, Springer-Verlag, (1984). · Zbl 0541.14005
[20] Griffiths, P.; Harris, J., Principles of Algebraic Geometry (1994), New York: Wiley Classics Library, John Wiley & Sons, Inc., New York · Zbl 0836.14001
[21] Gromov, M., Convex sets and Kahler manifolds, Advances in Differential Geometry and Topology, 1-38 (1998), Teaneck, NJ: Word Sci. Publishing, Teaneck, NJ · Zbl 0770.53042
[22] Guedj, V. Ergodic properties of rational mappings with large topological degree,Ann. of Math., to appear. · Zbl 1088.37020
[23] Jonsson, M.; Weickert, B., A nonalgebraic attractor inP^2, Proc. Amer. Math. Soc., 128, 10, 2999-3002 (2000) · Zbl 0974.37030 · doi:10.1090/S0002-9939-00-05529-5
[24] Kirwan, F., Complex Algebraic Curves, London Mathematical Society Student Texts,23 (1992), Cambridge: Cambridge University Press, Cambridge · Zbl 0744.14018
[25] Lyubich, M. J., Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems, 3, 351-385 (1983) · Zbl 0537.58035
[26] McMullen, C. T., Dynamics on K3 surfaces: Salem numbers and Siegel disks, J. Reine Angew. Math., 545, 201-233 (2002) · Zbl 1054.37026
[27] Méo, M., Image inverse d’un courant positif fermé par une application surjective, C.R.A.S., 322, 1141-1144 (1996) · Zbl 0858.32012
[28] Russakovskii, A.; Shiffman, B., Value distribution for sequences of rational mappings and complex dynamics, Indiana Univ. Math. J., 46, 897-932 (1997) · Zbl 0901.58023 · doi:10.1512/iumj.1997.46.1441
[29] Sibony, N., Dynamique des applications rationnelles de ℙ^k, Panor. Synthèses, 8, 97-185 (1999) · Zbl 1020.37026
[30] Sibony, N.; Wong, P. M., Some results on global analytic sets, Séminaire Lelong-Skoda, L.N., 822, 221-237 (1980) · Zbl 0444.32006
[31] Siu, Y. T., Analyticity of sets associated to Lelong numbers and the extension of closed positive currents, Invent. Math., 27, 53-156 (1974) · Zbl 0289.32003 · doi:10.1007/BF01389965
[32] Skoda, H., Prolongement des courants positifs, fermés de masse finie, Invent. Math., 66, 361-376 (1982) · Zbl 0488.58002 · doi:10.1007/BF01389217
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.