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Analytic automorphisms of \({\mathbb{C}^2}\) generated by polynomial vector fields

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Abstract

We give the classification of the transcendental automorphisms of \({\mathbb{C}^{2}}\) of the form \({(x,y)\to(P e^{Q},R e^{S})}\) with P,Q,R, \({S\in \mathbb{C}[x,y]}\).

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References

  1. Andersén E.: Complete vector fields on \({(\mathbb{C}^{\ast})^{n}}\), Proc. Amer. Math. Soc. 128, 1079–1085 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brunella M.: Sur les courbes intégrales propres des champs de vecteurs polynomiaux. Topology 37, 1229–1246 (1998)

    Article  MathSciNet  Google Scholar 

  3. Brunella M.: Complete vector fields on the complex plane. Topology 43, 433–445 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Bustinduy, On the entire solutions of a polynomial vector field on \({\mathbb{C}^2}\), Indiana Univ. Math. J. 53 (2004), 647–666.

  5. A. Bustinduy, Complete vector fields on \({\mathbb{C}^2}\) whose underlying foliation is polynomial, Int. J. Math. 21 (2010), 333–347.

  6. Y. Nishimura, Analytic automorphisms of \({\mathbb{C}^{2}}\) which preserve the coordinate axes, In: The Madison Symposium on Complex Analysis (Madison, WI, 1991,) volume 137 of Contemp. Math., pages 351–365, Amer. Math. Soc., Providence, RI, 1992.

  7. Shiga H.: On the automorphism of \({\mathbb{C}^2}\) with invariant axes. J. Math. Soc. Japan 29, 529–535 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Suzuki, Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l’espace \({\mathbb{C}^{2}}\), J. Math. Soc. Japan 26 (1974), 241–257.

  9. M. Suzuki, Sur les opérations holomorphes du groupe additif complexe sur l’espace de deux variables complexes, Ann. Sci. École Norm. Sup. 10 (1977), 517–546.

  10. Varolin D.: The density property for complex manifolds and geometric structures. J. Geom. Anal. 11, 135–160 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. L. Vivas, Remarks on automorphisms of \({\mathbb{C}^{*} \times \mathbb{C}^{*}}\) and their basins, Complex Var. Elliptic Equ. 54 (2009), 401–407.

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  1. Departamento de Ingeniería Industrial, Escuela Politécnica Superior, Universidad Antonio de Nebrija, C/ Pirineos 55, 28040, Madrid, Spain

    Alvaro Bustinduy

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  1. Alvaro Bustinduy
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Correspondence to Alvaro Bustinduy.

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Supported by Spanish MICINN Project MTM2015-63612-P.

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Bustinduy, A. Analytic automorphisms of \({\mathbb{C}^2}\) generated by polynomial vector fields. Arch. Math. 107, 251–258 (2016). https://doi.org/10.1007/s00013-016-0948-5

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  • DOI: https://doi.org/10.1007/s00013-016-0948-5

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