Advertisement

Webs invariant by rational maps on surfaces

Article

Abstract

We prove that under mild hypothesis rational maps on a surface preserving webs are Lattès-like. We classify endomorphisms of \(\mathbb {P}^2\) preserving webs, extending former results of Dabija-Jonsson.

Keywords

Rational maps Webs Lattès-like maps 

Mathematics Subject Classification

Primary 37F75 Secondary 14E05 32S65 

Notes

Acknowledgments

We warmly thank the referee for his careful reading, and for his help in improving the readability of the paper.

References

  1. 1.
    Berteloot, F., Dupont, C.: Une caractérisation des endomorphismes de Lattès par leur mesure de Green. Comment. Math. Helv. 80, 433–454 (2005)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Boucksom, S., Favre, C., Jonsson, M.: Degree growth of meromorphic surface maps. Duke Math. J. 141(3), 519–538 (2008)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Beauville, A.: Finite subgroups of \({{\rm PGL}}2(K)\). Vector bundles and complex geometry. Contemp. Math. Amer. Math. Soc. Providence RI, 522, 23–29 (2010)Google Scholar
  4. 4.
    Cantat, S., Favre, C.: Symétries birationnelles des surfaces feuilletées. J. Reine Angew. Math. 561, 199–235 (2003) [Corrigendum à l’article “Symétries birationnelles des surfaces feuilletées”. J. Reine Angew. Math. 582, 229–231 (2005)]Google Scholar
  5. 5.
    Carleson, L., Gamelin, T.W.: Complex dynamics, Universitext: Tracts in Mathematics. Springer, New York (1993)CrossRefGoogle Scholar
  6. 6.
    Cerveau, D., Lins-Neto, A.: Hypersurfaces exceptionnelles des endomorphismes de \(\mathbb{C}{\rm P}(n)\). Bol. Soc. Brasil. Mat. 31(2), 155–161 (2000)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Dabija, M.: Algebraic and geometric dynamics in several complex variables. Ph.D. thesis. University of Michigan (2000)Google Scholar
  8. 8.
    Dabija, M., Jonsson, M.: Endomorphisms of the plane preserving a pencil of curves. Int. J. Math. 19(2), 217–221 (2008)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Dabija, M., Jonsson, M.: Algebraic webs invariant under endomorphisms. Publ. Mat. 54, 137–148 (2010)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Dinh, T.C.: Sur les endomorphismes polynomiaux permutables de \(\mathbb{C}^2\). Ann. Inst. Fourier 51, 431–459 (2001)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Dinh, T.C., Nguyen, V.A.: Comparison of dynamical degrees for semi-conjugate meromorphic maps. Commentarii Math. Helv. 86(4), 817–840 (2011)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Dinh, T.C., Sibony, N.: Sur les endomorphismes permutables de \(\mathbb{P}^k\). Math. Ann. 324, 33–70 (2002)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Dinh, T.C., Sibony, N.: Une borne supérieure pour l’entropie topologique d’une application rationnelle. Ann. Math. 161(3), 1637–1644 (2005)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Diller, J., Favre, C.: Dynamics of bimeromorphic maps of surfaces. Amer. J. Math. 123(6), 1135–1169 (2001)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Dupont, C.: Exemples de Lattès et domaines faiblement sphériques de \(\mathbb{C}^n\). Manuscripta Math. 111(3), 357–378 (2003)MATHMathSciNetGoogle Scholar
  16. 16.
    Favre, C., Pereira, J.V.: Foliations invariant by rational maps. Math. Zeitshrift 268, 753–770 (2011)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Fornaess, J.E., Sibony, N.: Complex analytic methods in dynamical systems (Rio de Janeiro, 1992). Astérisque 222(5), 201–231 (1994)Google Scholar
  18. 18.
    Gizatullin, M.H.: Rational \(G\)-surfaces. Izv. Akad. Nauk SSSR Ser. Mat. 44(1), 110–144 (1980)MATHMathSciNetGoogle Scholar
  19. 19.
    Gromov, M.: On the entropy of holomorphic maps. Enseign. Math. 49(3–4), 217–235 (2003)MATHMathSciNetGoogle Scholar
  20. 20.
    Harthorne, R.: Algebraic geometry. Graduate texts in Math., vol. 52. Springer (1977)Google Scholar
  21. 21.
    Hillman, J.A.: Four-manifolds, geometries and knots. Geometry & Topology Monographs, 5. Geometry & Topology Publications, Coventry (2002)Google Scholar
  22. 22.
    Kaneko, J., Tokunaga, S., Yoshida, M.: Complex crystallographic groups II. J. Math. Soc. Japan 34(4), 595–605 (1982)MATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Lang, S.: Algebra. Revised third edition. Graduate Texts in Mathematics, 211. Springer, New York (2002). xvi+914 ppGoogle Scholar
  24. 24.
    Marín, D., Pereira, J.V.: Rigid flat webs on the projective plane. Asian J. Math. 17(1), 163–192 (2013)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Milnor, J.: On Lattès maps. Dynamics on the Riemann sphere, Eur. Math. Soc. Zürich , 9–43 (2006)Google Scholar
  26. 26.
    Oda, T.: Convex bodies and algebraic geometry. An introduction to the theory of toric varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 15. Springer, Berlin (1988) [viii+212 pp] Google Scholar
  27. 27.
    Pereira, J.V., Pirio, L.: An invitation to web geometry, 27\(^o\) Colóquio Brasileiro de Matemática. Instituto Nacional de Matemática Pura e Aplicada (IMPA). Rio de Janeiro (2009) [xii+245 pp] Google Scholar
  28. 28.
    Schlomiuk, D., Vulpe, N.: Planar quadratic vector fields with invariant lines of total multiplicity at least five. Qual. Theory Dyn. Syst. 5(1), 135–194 (2004)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Shiffman, B., Shishikura, M., Ueda, T.: On totally invariant varieties of holomorphic mappings of \(\mathbb{P}^n\) (Preprint) Google Scholar
  30. 30.
    Tokunaga, S., Yoshida, M.: Complex crystallograhic groups I. J. Math. Soc. Japan 34(4), 581–593 (1982)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Ueda, T.: Fatou set in complex dynamics in projective spaces. J. Math. Soc. Japan 46(3), 545–555 (1994)MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Yoshihara, H.: Quotients of abelian surfaces. Publ. Res. Inst. Math. Sci. 31(1), 135–143 (1995)MATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Zariski, O., Samuel, P.: Commutative algebra. Vol. II. The University Series in Higher Mathematics. D. Van Nostrand Co., Inc, Princeton (1960). x+414 ppGoogle Scholar
  34. 34.
    Zdunik, A.: Parabolic orbifolds and the dimension of the maximal measure for rational maps. Invent. Math. 99(3), 627–649 (1990)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2015

Authors and Affiliations

  1. 1.CNRS and Centre de Mathématiques Laurent Schwartz, École PolytechniquePalaiseau CedexFrance
  2. 2.IMPARio de JaneiroBrazil

Personalised recommendations