Advertisement

Foliations and polynomial diffeomorphisms of \({\mathbb{R}^{3}}\)

  • Carlos Gutierrez
  • Carlos MaqueraEmail author
Article

Abstract

Let \({Y=(f,g,h){:} \mathbb{R}^{3} \to \mathbb{R}^{3}}\) be a C 2 map and let Spec(Y) denote the set of eigenvalues of the derivative DY p , when p varies in \({\mathbb{R}^3}\) . We begin proving that if, for some ϵ > 0, \({Spec(Y)\cap (-\epsilon,\epsilon)=\emptyset,}\) then the foliation \({\mathcal{F}(k),}\) with \({k\in \{f,g,h\},}\) made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek’s Jacobian Conjecture for polynomial maps of \({\mathbb{R}^n.}\)

Keywords

Three dimensional vector field Global injectivity Foliation 

Mathematics Subject Classification (2000)

37C85 57R30 

References

  1. 1.
    Alexandrov V.A.: Remarks on Efimov’s theorem about differential tests of homeomorphism. Rev. Roumanie Math. Pures Appl. 36(3–4), 101–105 (1991)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bass H., Connell E., Wright D.: The Jacobian conjecture: reduction of degree and formal expansion of the inverse. Bull. Am. Math. Soc. 7, 287–330 (1982)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Białynicki-Birula A., Rosenlicht M.: Injective morphisms of real algebraic varieties. Proc. Am. Math. Soc. 13, 200–204 (1962)CrossRefzbMATHGoogle Scholar
  4. 4.
    Camacho, C.: Alcides Lins Neto geometric theory of foliations. Birkhäuser Boston Inc., (1985)Google Scholar
  5. 5.
    Campbell L.A.: Unipotent Jacobian matrices and univalent maps. Contemp. Math. 264, 157–177 (2000)Google Scholar
  6. 6.
    Černavskii A.V.: Finite-to-one open mappings of manifolds. Mat. Sbornik 65, 357–369 (1964)Google Scholar
  7. 7.
    Černavskii A.V.: Addendum to the paper Finite-to-one open mappings of manifolds. Mat. Sbornik 66, 471–472 (1965)Google Scholar
  8. 8.
    Chamberland M.: Characterizing two-dimensional maps whose Jacobians have constant eigenvalues. Can. Math. Bull. 46(3), 323–331 (2003)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cobo M., Gutierrez C., Llibre J.: On the injectivity of C 1 maps of the real plane. Can. J. Math. 54(6), 1187–1201 (2002)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chau and Nga. A remark on Yu’s theorem (1998, preprint)Google Scholar
  11. 11.
    Druskowski L.: An effective approach to Keller’s Jacobian conjecture. Math. Ann. 264, 303–313 (1983)CrossRefMathSciNetGoogle Scholar
  12. 12.
    van den Essen, A.: Polynomial automorphisms and the Jacobian conjecture. In: Progress in Mathematics, vol. 190. Birkhauser, Basel (2000)Google Scholar
  13. 13.
    Fernandes A., Gutierrez C., Rabanal R.: Global asymptotic stability for differentiable vector fields of \({ \mathbb{R}^2}\) . J. Differ. Equ. 206, 470–482 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Fernandes, A., Gutierrez, C., Rabanal, R.: On local diffeomorphisms of \({\mathbb{R}^n}\) that are injective. In: Qualitative Theory of Dynamical Systems, vol. 5, article no. 63, pp. 129–136, (2004)Google Scholar
  15. 15.
    Godbillon, C.: Feuilletages: Études géométriques. In: Progress in Math. vol. 14, Birkhäuser, Basel (1991)Google Scholar
  16. 16.
    Golubitsky, O., Guillemin, V.: Stable mappings and their singularities. In: Grad. Texts in Math. vol. 14. Springer, Heidelberg (1973)Google Scholar
  17. 17.
    Gutierrez, C.: A solution to the bidimensional global asymptotic stability conjecture. In: Ann. Inst. H. Poincaré. Analyse non Lineaire, vol. 12, No. 6, pp. 627–671 (1995)Google Scholar
  18. 18.
    Gutierrez E C., Rabanal R.: Injectivity of differentiable maps \({\mathbb{R}^2\to \mathbb{R}^2}\) at infinity. Bull. Braz. Math. Soc. 37(2), 217–239 (2006)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Gutierrez E Nguyen Van Chau C.: Properness and the Jacobian conjecture in \({\mathbb{R}^2}\) . Vietnam J. Math. 31(4), 421–427 (2003)MathSciNetGoogle Scholar
  20. 20.
    Gutierrez E Nguyen Van Chau C.: On nonsingular polynomial maps of \({\mathbb{R}^2}\) . Annales Polonici Mathematici 88, 193–204 (2006)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Gutierrez E Nguyen Van Chau, C.: A remark on an eigenvalue condition for the global injectivity of differentiable maps of \({\mathbb{R}^2}\) . In: Discrete & Continuous Dynamical Systems, Series A, vol. 17, No. 2, pp. 397–402Google Scholar
  22. 22.
    Jelonek Z.: Geometry of real polynomial mappings. Math. Zeitschrift 239, 321–333 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Jelonek Z: The set of points at which a polynomial map is not proper. Ann. Polon. Math. 58, 259–266 (1993)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Jelonek Z.: Testing sets for properness of polynomial mappings. Math. Ann. 315, 1–35 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Nollet, S., Xavier, F.: Global inversion via the Palais-smale condition. In: Discret and Continuous Dynamical Systems Ser. A, vol. 8, pp. 17–28 (2002)Google Scholar
  26. 26.
    Palmeira C.F.B.: Open manifolds foliated by planes. Ann. Math. 107, 109–131 (1978)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Peretz Nguyen Van Chau, R., Campbell L.A., Gutierrez, C.: Iterated images and the plane Jacobian conjecture. In: Discret and Continuous Dynamical Systems, vol. 16, No. 2, pp. 455–461 (2006)Google Scholar
  28. 28.
    Pinchuck S.: A counterexample to the strong Jacobian conjecture. Math. Z. 217, 1–4 (1994)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Rosenberg H.: Foliations by planes. Topology 7, 131–138 (1968)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Smyth B., Xavier F.: Injectivity of local diffeomorphisms from nearly spectral conditions. J. Diff. Equ. 130, 406–414 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Yagzhev A.V.: On Keller’s problem. Siberian Math. J. 21, 747–754 (1980)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Departamento de Matemática, Instituto de Ciências Matemáticas e de ComputaçãoUniversidade de São Paulo, São CarlosSão CarlosBrazil

Personalised recommendations