Algebraic webs invariant under endomorphisms. (English) Zbl 1180.37057

Summary: We classify noninvertible, holomorphic selfmaps of the projective plane that preserve an algebraic web. In doing so, we obtain interesting examples of critically finite maps.


37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
14C21 Pencils, nets, webs in algebraic geometry
Full Text: DOI arXiv Euclid


[1] W. Blaschke and G. Bol, “Geometrie der Gewebe. Topologische Fragen der Differentialgeometrie” , J. W. Edwards, Ann Arbor, Michigan, 1944. · JFM 64.0727.03
[2] A. Bonifant and M. Dabija, Self-maps of \({\mathbb P}^ 2\) with invariant elliptic curves, in: “Complex manifolds and hyperbolic geometry” (Guanajuato, 2001), Contemp. Math. 311 , Amer. Math. Soc., Providence, RI, 2002, pp. 1\Ndash25. · Zbl 1023.32013
[3] S. Cantat and C. Favre, Symétries birationnelles des surfaces feuilletées, J. Reine Angew. Math. 561 (2003), 199\Ndash235. · Zbl 1070.32022
[4] L. Carleson and T. W. Gamelin, “Complex dynamics” , Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. · Zbl 0782.30022
[5] D. Cerveau and A. Lins Neto, Hypersurfaces exceptionnelles des endomorphismes de \(\mathbb{C}\mathbb{P}(n)\), Bol. Soc. Brasil. Mat. (N.S.) 31(2) (2000), 155\Ndash161. · Zbl 0967.32022
[6] M. Dabija, Algebraic and geometric dynamics in several complex variables, Thesis, University of Michigan (2000).
[7] M. Dabija and M. Jonsson, Endomorphisms of the plane preserving a pencil of curves, Internat. J. Math. 19(2) (2008), 217\Ndash221. · Zbl 1159.32009
[8] J. Diller, D. Jackson, and A. Sommese, Invariant curves for birational surface maps, Trans. Amer. Math. Soc. 359(6) (2007), 2793\Ndash2991 (electronic). · Zbl 1115.14007
[9] C. Favre and J. V. Pereira, Foliations invariant by rational maps, Preprint (2009), arxiv.org/abs/0907.1367. · Zbl 1237.37039
[10] J. E. Fornæss and N. Sibony, Critically finite rational maps on \(\mathbb{P}^ 2\), in: “The Madison Symposium on Complex Analysis” (Madison, WI, 1991), Contemp. Math. 137 , Amer. Math. Soc., Providence, RI, 1992, pp. 245\Ndash260. · Zbl 0772.58024
[11] J. E. Fornæss and N. Sibony, Complex dynamics in higher dimension. I. Complex analytic methods in dynamical systems (Rio de Janeiro, 1992), Astérisque 222 (1994), 5, 201\Ndash231. · Zbl 0813.58030
[12] P. Griffiths and J. Harris, “Principles of algebraic geometry” , Reprint of the 1978 original, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. · Zbl 0836.14001
[13] J. Grifone and E. Salem (eds.), “Web theory and related topics” , Papers from the Conference on Webs held in Toulouse, December 1996, World Scientific Publishing Co., Inc., River Edge, NJ, 2001.
[14] A. Hénaut, Sur la linéarisation des tissus de \(\mathbb{C}^ 2\), Topology 32(3) (1993), 531\Ndash542. · Zbl 0799.32010
[15] M. Homma, Plane curves whose tangent lines at collinear points are concurrent, Geom. Dedicata 53(3) (1994), 287\Ndash296. · Zbl 0851.14014
[16] M. Jonsson, Some properties of \(2\)-critically finite holomorphic maps of \(\mathbf{P}^ 2\), Ergodic Theory Dynam. Systems 18(1) (1998), 171\Ndash187. · Zbl 0915.58080
[17] M. Jonsson, Dynamics of polynomial skew products on \(\mathbf{C}^ 2\), Math. Ann. 314(3) (1999), 403\Ndash447. · Zbl 0940.37018
[18] M. Jonsson, Ergodic properties of fibered rational maps, Ark. Mat. 38(2) (2000), 281\Ndash317. · Zbl 1021.37019
[19] S. Koch, Teichmüller theory and endomorphisms of \(\mathbf{P}^n\), in preparation.
[20] J. V. Pereira, Algebraization of codimension one webs [after Tré-preau, Hénaut, Pirio, Robert,…], Séminaire Bourbaki, Exp. no. 974, Vol. 2006/2007, Astérisque 317 (2008), viii, 243\Ndash268. · Zbl 1184.14014
[21] L. Pirio, Équations fonctionnelles abéliennes et géométrie des tissus, Thèse de doctorat, Université Paris VI (2004).
[22] L. Pirio, Sur la linéarisation des tissus, Preprint (2008), arxiv.org/pdf/0811.1810.
[23] F. Rong, The Fatou set for critically finite maps, Proc. Amer. Math. Soc. 136(10) (2008), 3621\Ndash3625. · Zbl 1151.37045
[24] B. Shiffman, M. Shishikura, and T. Ueda, On totally invariant varieties of holomorphic mappings of \(\mathbb{P}^n\), Preprint (2000).
[25] N. Sibony, Dynamique des applications rationnelles de \(\mathbf P^ k\), in: “Dynamique et géométrie complexes” (Lyon, 1997), Panor. Synthèses 8 , Soc. Math. France, Paris, 1999, pp. ix\Ndashx, xi\Ndashxii, 97\Ndash185. · Zbl 1020.37026
[26] T. Ueda, Complex dynamics on projective spaces-index formula for fixed points, in: “Dynamical systems and chaos” , Vol. 1 (Hachioji, 1994), World Sci. Publ., River Edge, NJ, 1995, pp. 252\Ndash259. · Zbl 0991.32504
[27] T. Ueda, Critical orbits of holomorphic maps on projective spaces, J. Geom. Anal. 8(2) (1998), 319\Ndash334. · Zbl 0957.32009
[28] T. Ueda, Critically finite maps on projective spaces, (Japanese) Research on complex dynamical systems: current state and prospects (Japanese) (Kyoto, 1998), Sūrikaisekikenkyūsho Kōkyūroku 1087 (1999), 132\Ndash138. · Zbl 0951.32507
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.