Advertisement

Computational Methods and Function Theory

, Volume 19, Issue 4, pp 687–716 | Cite as

On the Multipliers at Fixed Points of Quadratic Self-Maps of the Projective Plane with an Invariant Line

  • Adolfo GuillotEmail author
  • Valente Ramírez
Article
  • 32 Downloads

Abstract

This paper deals with holomorphic self-maps of the complex projective plane and the algebraic relations among the eigenvalues of the derivatives at the fixed points. These eigenvalues are constrained by certain index theorems such as the holomorphic Lefschetz fixed-point theorem. A simple dimensional argument suggests there must exist even more algebraic relations that the ones currently known. In this work we analyze the case of quadratic self-maps having an invariant line and obtain all such relations. We also prove that a generic quadratic self-map with an invariant line is completely determined, up to linear equivalence, by the collection of these eigenvalues. Under the natural correspondence between quadratic rational maps of \({\mathbb {P}}^2\) and quadratic homogeneous vector fields on \({\mathbb {C}}^3\), the algebraic relations among multipliers translate to algebraic relations among the Kowalevski exponents of a vector field. As an application of our results, we describe the sets of integers that appear as the Kowalevski exponents of a class of quadratic homogeneous vector fields on \({\mathbb {C}}^3\) having exclusively single-valued solutions.

Keywords

Fixed point Self-map Projective space Multiplier Kowalevski exponent Painlevé test 

Mathematics Subject Classification

Primary 37C25 58C30 34M35 Secondary 37F10 14Q99 14N99 

Notes

Acknowledgements

The authors thank Jawad Snoussi and Javier Elizondo for helpful conversations, and Marco Abate and Masayo Fujimura for providing useful references.

References

  1. 1.
    Abate, M.: Index theorems for meromorphic self-maps of the projective space. In: Frontiers in Complex Dynamics, Princeton Math. Ser., vol. 51, pp. 451–462. Princeton Univ. Press, Princeton, NJ (2014).  https://doi.org/10.1515/9781400851317-018 CrossRefGoogle Scholar
  2. 2.
    Atiyah, M.F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes. II. Applications. Ann. Math. 2(88), 451–491 (1968).  https://doi.org/10.2307/1970721 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Audin, M.: Fatou, Julia, Montel. Lecture Notes in Mathematics, vol. 2014. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  4. 4.
    Baum, P.F., Bott, R.: On the zeros of meromorphic vector-fields. In: Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), pp. 29–47. Springer, New York (1970).  https://doi.org/10.1007/978-3-642-49197-9_4 CrossRefGoogle Scholar
  5. 5.
    Bonifant, A., Dabija, M., Milnor, J.: Elliptic curves as attractors in \(\mathbb{P}^2\). I. Dynamics. Experiment. Math. 16(4), 385–420 (2007).  https://doi.org/10.1080/10586458.2007.10129016 MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bonifant, A.M., Dabija, M.: Self-maps of \({\mathbb{P}}^2\) with invariant elliptic curves. In: Complex Manifolds and Hyperbolic Geometry (Guanajuato, 2001), Contemp. Math., vol. 311, pp. 1–25. Am. Math. Soc., Providence, RI (2002). https://doi.org/10.1090/conm/311/05444
  7. 7.
    Camacho, C., Sad, P.: Invariant varieties through singularities of holomorphic vector fields. Ann. Math. (2) 115(3), 579–595 (1982).  https://doi.org/10.2307/2007013 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chazy, J.: Sur les équations différentielles du troisième ordre et d’ordre supérieur dont l’intégrale générale a ses points critiques fixes. Acta Math. 34(1), 317–385 (1911).  https://doi.org/10.1007/BF02393131 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cosgrove, C.M.: Higher-order Painlevé equations in the polynomial class. I. Bureau symbol \(\rm P2\). Stud. Appl. Math 104(1), 1–65 (2000).  https://doi.org/10.1111/1467-9590.00130 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fatou, P.: Sur les substitutions rationnelles. C. R. Acad. Sci., Paris 165, 992–995 (1917). https://gallica.bnf.fr/ark:/12148/bpt6k3118k.f992
  11. 11.
    Fatou, P.: Sur les équations fonctionnelles. Bull. Soc. Math. France 47, 161–271 (1919). https://doi.org/10.24033/bsmf.998 MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fujimura, M.: The moduli space of rational maps and surjectivity of multiplier representation. Comput. Methods Funct. Theory 7(2), 345–360 (2007).  https://doi.org/10.1007/BF03321649 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, resultants, and multidimensional determinants. Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA (1994).  https://doi.org/10.1007/978-0-8176-4771-1 CrossRefGoogle Scholar
  14. 14.
    Ghys, E., Rebelo, J.C.: Singularités des flots holomorphes. II. Ann. Inst. Fourier (Grenoble) 47(4), 1117–1174 (1997).  https://doi.org/10.5802/aif.1594 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Goriely, A.: A brief history of Kovalevskaya exponents and modern developments. Regul. Chaotic Dyn. 5(1), 3–15 (2000).  https://doi.org/10.1070/rd2000v005n01ABEH000120. (Sophia Kovalevskaya to the 150th anniversary) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Grayson, D., Stillman, M.: Macaulay2, a software system for research in algebraic geometry (Version 1.11) (2018). http://www.math.uiuc.edu/Macaulay2/
  17. 17.
    Griffiths, P., Harris, J.: Principles of algebraic geometry. Wiley Classics Library. Wiley, New York (1994).  https://doi.org/10.1002/9781118032527. Reprint of the 1978 originalCrossRefGoogle Scholar
  18. 18.
    Guillot, A.: Un théorème de point fixe pour les endomorphismes de l’espace projectif avec des applications aux feuilletages algébriques. Bull. Braz. Math. Soc. (N.S.) 35(3), 345–362 (2004).  https://doi.org/10.1007/s00574-004-0018-7 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Guillot, A.: Semicompleteness of homogeneous quadratic vector fields. Ann. Inst. Fourier (Grenoble) 56(5), 1583–1615 (2006).  https://doi.org/10.5802/aif.2221 MathSciNetCrossRefGoogle Scholar
  20. 20.
    Guillot, A.: The geometry of Chazy’s homogeneous third-order differential equations. Funkcial. Ekvac. 55(1), 67–87 (2012).  https://doi.org/10.1619/fesi.55.67 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Guillot, A.: Quadratic differential equations in three variables without multivalued solutions: Part I. SIGMA Symmetry Integrability Geom. Methods Appl. 14, 122, 46 (2018).  https://doi.org/10.3842/SIGMA.2018.122
  22. 22.
    Guillot, A., Ramirez, V.: Multipliers-of-self-maps-on-P2. GitHub repository (2018). https://github.com/valentermz/Multipliers-of-self-maps-on-P2
  23. 23.
    Guillot, A., Rebelo, J.: Semicomplete meromorphic vector fields on complex surfaces. J. Reine Angew. Math. 667, 27–65 (2012).  https://doi.org/10.1515/CRELLE.2011.127 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ince, E.L.: Ordinary Differential Equations. Dover Publications, New York (1944)zbMATHGoogle Scholar
  25. 25.
    Julia, G.: Sur les substitutions rationnelles. C. R. Acad. Sci., Paris 165, 1098–1100 (1917). http://gallica.bnf.fr/ark:/12148/bpt6k3118k.f1098
  26. 26.
    Kudryashov, Y., Ramírez, V.: Spectra of quadratic vector fields on \({\mathbb{C}}^2\): The missing relation (2018). http://arxiv.org/abs/1705.06340. Preprint arXiv:1705.06340 [math.CV] (to appear in Mosc. Math. J.)
  27. 27.
    Lins Neto, A.: Fibers of the Baum-Bott map for foliations of degree two on \(\mathbb{P}^2\). Bull. Braz. Math. Soc. (N.S.) 43(1), 129–169 (2012).  https://doi.org/10.1007/s00574-012-0008-0 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Mattuck, A.: The field of multisymmetric functions. Proc. Am. Math. Soc. 19, 764–765 (1968).  https://doi.org/10.2307/2035879 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Milnor, J.: Remarks on iterated cubic maps. Experiment. Math. 1(1), 5–24 (1992). http://projecteuclid.org/euclid.em/1048709112
  30. 30.
    Milnor, J.: Geometry and dynamics of quadratic rational maps. Experiment. Math. 2(1), 37–83 (1993).  https://doi.org/10.1080/10586458.1993.10504267. With an appendix by the author and Lei TanMathSciNetCrossRefGoogle Scholar
  31. 31.
    Milnor, J.: Dynamics in one complex variable, Annals of Mathematics Studies, vol. 160, third edn. Princeton University Press, Princeton (2006). https://www.degruyter.com/view/product/451239
  32. 32.
    O’Brian, N.R.: Zeroes of holomorphic vector fields and Grothendieck duality theory. Trans. Am. Math. Soc. 229, 289–306 (1977).  https://doi.org/10.2307/1998512 MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Painlevé, P.: Mémoire sur les équations différentielles dont l’intégrale générale est uniforme. Bull. Soc. Math. France 28, 201–261 (1900).  https://doi.org/10.24033/bsmf.633 CrossRefGoogle Scholar
  34. 34.
    Ramírez, V.: The Woods Hole trace formula and indices for vector fields and foliations on \({\mathbb{C }}^{2}\) (2016). http://arxiv.org/abs/1608.05321. Preprint arXiv:1608.05321 [math.CV]
  35. 35.
    Ramírez, V.: Twin vector fields and independence of spectra for quadratic vector fields. J. Dyn. Control Syst. 23(3), 623–633 (2017).  https://doi.org/10.1007/s10883-016-9344-5 MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    The Sage Developers: The Sage Mathematics Software System (Version 8.1) (2017). http://www.sagemath.org
  37. 37.
    Ueda, T.: Complex dynamics on projective spaces—index formula for fixed points. In: Dynamical systems and chaos, Vol. 1 (Hachioji, 1994), pp. 252–259. World Sci. Publ., River Edge, NJ (1995).  https://doi.org/10.1142/9789814536165

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Instituto de MatematicasUniversidad Nacional Autónoma de México, Ciudad UniversitariaMexico CityMexico
  2. 2.Institut de Recherche Mathématique de RennesUniversité de Rennes 1, UMR 6625RennesFrance

Personalised recommendations