Communications in Mathematical Physics

, Volume 273, Issue 3, pp 755–783 | Cite as

The Parameter Planes of λz m exp(z) for m ≥ 2*

  • Núria FagellaEmail author
  • Antonio Garijo


We consider the families of entire transcendental maps given by F λ,m (z) = λz m exp(z), where m ≥ 2. All functions F λ,m have a superattracting fixed point at z = 0, and a critical point at z = −m. In the parameter planes we focus on the capture zones, i.e., λ values for which the critical point belongs to the basin of attraction of z = 0, denoted by A(0). In particular, we study the main capture zone (parameter values for which the critical point lies in the immediate basin, A *(0)) and prove that is bounded, connected and simply connected. All other capture zones are unbounded and simply connected. For each parameter λ in the main capture zone, A(0) consists of a single connected component with non-locally connected boundary. For all remaining values of λ, A *(0) is a quasidisk. On a different approach, we introduce some families of holomorphic maps of \(\mathbb {C}^*\) which serve as a model for F λ,m , in the sense that they are related by means of quasiconformal surgery to F λ,m .


Entire Function Quasiconformal Mapping Parameter Plane Capture Zone Dynamical Plane 
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  1. Ah.
    Ahlfors L. (1966). Lectures on quasiconformal mappings. Wadswoth & Brooks/Cole Mathematics Series, New York zbMATHGoogle Scholar
  2. A.
    Arnold V. (1965). Small denominators I, on the mappings of the circumference into itself. Amer. Math. Soc. Transl. (2) 46: 213–284 Google Scholar
  3. B.
    Baker I.N. (1975). The domains of normality of an entire function. Ann. Acad. Sci. Fenn. Ser. A I Math. 1: 277–283 MathSciNetzbMATHGoogle Scholar
  4. B1.
    Baker I.N. (1976). An entire function which has wandering domains. J. Austral. Math. Soc. 22: 173–176 zbMATHGoogle Scholar
  5. B2.
    Baker I.N. (1984). Wandering Domains in the Iteration of Entire Functions. Proc. London Math. Soc. 49: 563–576 zbMATHCrossRefMathSciNetGoogle Scholar
  6. BD.
    Baker I.N., Dominguez P. (2000). Some connectedness properties of Julia sets. Complex variables Theory Appl. 41: 371–389 zbMATHMathSciNetGoogle Scholar
  7. BA.
    Beurling A., Ahlfors L.V. (1956). The boundary correspondence under quasiconformal mappings. Acta Math. 96: 125–142 zbMATHCrossRefMathSciNetGoogle Scholar
  8. Be.
    Bergweiler W. (1995). Invariant domains and singularities. Math. Proc. Camb. Phil. Soc. 117: 525–532 zbMATHMathSciNetCrossRefGoogle Scholar
  9. Bo.
    Böttcher L.E. (1904). The principal laws of convergence of iterates and their application to analysis (Russian). Izv. Kazan. Fiz.-Mat. Obshch. 14: 155–234 Google Scholar
  10. BH1.
    Branner B., Hubbard J.H. (1992). The iteration of cubic polynomials Part I: The global toology of parameter space. Acta Math. 169: 143–206 CrossRefMathSciNetGoogle Scholar
  11. BH2.
    Branner B., Hubbard J.H. (1992). The iteration of cubic polynomials Part II: Patterns and parapatterns. Acta Math. 169: 229–325 zbMATHCrossRefMathSciNetGoogle Scholar
  12. BuHe.
    Buff X., Henriksen C. (2001). Julia sets in parameter spaces. Commun. Math. Phys. 220: 333–375 zbMATHCrossRefMathSciNetADSGoogle Scholar
  13. CG.
    Carleson L., Gamelin Th. (1993). Complex Dynamics. Springer, Berlin-Heidelberg-New York zbMATHGoogle Scholar
  14. DE.
    Douady A., Earle C.J. (1986). Conformally natural extension of homeomorphism of the circle. Acta Math. 157: 23–48 zbMATHCrossRefMathSciNetGoogle Scholar
  15. DT.
    Devaney R.L., Tangerman F. (1986). Dynamics of entire functions near the essential singularity. Ergodic Theory Dynam. Systems 6: 489–503 zbMATHMathSciNetGoogle Scholar
  16. dMvS.
    Melo W., Strien S. (1993). One-Dimensional dynamics. Springer-Verlag, Berlin-Heidelberg-New York zbMATHGoogle Scholar
  17. DH1.
    Douady, A., Hubbard, J.H.: \’Etude dynamique des polynômes complexes. Part I. Publ. math. d’Orsay, 1984Google Scholar
  18. DH2.
    Douady, A., Hubbard, J.H.: \’Etude dynamique des polynômes complexes. Part II. Publ. math. d’Orsay, 1985Google Scholar
  19. DH3.
    Douady A., Hubbard J.H. (1985). On the dynamics of Polynomial-like Mappings. Ann. Scient. Ec. norm. Sup. 18: 287–343 zbMATHMathSciNetGoogle Scholar
  20. EL.
    Eremenko, A.E., Lyubich, M.Yu.: Iterates of entire functions. Soviet Math. Dokl. 30, 592–594 (1984); translation from Dokl. Akad. Nauk. SSSR 279, 25–27 (1984)Google Scholar
  21. EL1.
    Eremenko A.E., Lyubich M.Yu. (1990). The dynamics of analytic transforms. Leningrad. Math. J. 1: 563–634 zbMATHMathSciNetGoogle Scholar
  22. EL2.
    Eremenko A.E., Lyubich M.Yu. (1992). Dynamical properties of some classes of entire functions. Ann. Inst. Fourier 42: 989–1020 zbMATHMathSciNetGoogle Scholar
  23. FG.
    Fagella N., Garijo A. (2003). Capture zones of the family of functions F λ,m(z) = λ zm exp(z). Inter. J. of Bif., Chaos (3) 9: 2623–2640 CrossRefMathSciNetGoogle Scholar
  24. F.
    Fatou P. (1926). Sur l’iterátion des fonctions transcendentes entières. Acta Math. 47: 337–370 CrossRefMathSciNetzbMATHGoogle Scholar
  25. Fau.
    Faught, D.: Local connectivity in a family of cubic polynomials. Ph.D Thesis, Cornell University, 1992Google Scholar
  26. G.
    Geyer L. (2001). Siegel discs, Herman rings and the Arnold Family. Trans. Amer. Math. Soc. 353: 3661–3683 zbMATHCrossRefMathSciNetGoogle Scholar
  27. GK.
    Goldberg L.R., Keen L. (1986). A finiteness theorem for a dynamical class of entire functions. Ergodic Th. Dynam. Sys. 6: 183–192 zbMATHMathSciNetGoogle Scholar
  28. Ke.
    Keen, L.: Dynamics of holomorphic self-maps of \(\mathbb {C}^*\). Proc. Workshop of Holomorphic Functions and Moduli, Berlin-Heidelberg-New York: Springer-Verlag 1988, pp. 9–30Google Scholar
  29. Ko1.
    Kotus, J.: Iterated holomorphic maps on the punctered plane. In: Dynamical Systems, Kurzhanski, A.B., Sigmund, K.J. eds. 287, Berline-Heidelber-New York: Springer Verlag, 1987, pp. 10–29Google Scholar
  30. Ko2.
    Kotus J. (1990). Ann. Acad. Sci. Fenn. (Ser. A, I. Math) 15: 329–340 zbMATHMathSciNetGoogle Scholar
  31. L.
    Lei, T.: The Mandelbrot set, theme and variations. London Math. Soc. Lecture Note Ser. 274, Cambridge: Cambridge Univ. Press, 2000Google Scholar
  32. LV.
    Letho O., Virtanen K.I. (1973). Quasiconformal mappings in the plane. Springer-Verlag, Berlin-Heidelberg-New York Google Scholar
  33. Mak.
    Makienko, P.: Iteration of analytic functions of \(\mathbb{C}^*\) (Russian). Dokl. Akad. Nauk. SSRR 297, 35–37 (1987); Translation in Sov. Math. Dokl 36, 418–420 (1988)Google Scholar
  34. MSS.
    Mañé R., Sad P., Sullivan D. (1983). On the dynamics of rational maps. Ann. Sci. École Norm. Sup. 16: 193–217 zbMATHGoogle Scholar
  35. M.
    Milnor, J.: On cubic polynomials with periodic critical point. Stony Brook Institute for Mathematical Sciences., 1991Google Scholar
  36. M1.
    Milnor J. (1999). Dynamics in one complex variable: Introductory lectures. Vieweg, Weshaden zbMATHGoogle Scholar
  37. MR.
    Misiurewicz, M., Rodrigues, A.: Double standard maps. Preprint. Scholar
  38. Pom.
    Pommrenke Ch. (1991). Boundary Behavior of Conformal Maps. Springer-Verlag, Berlin-Heidelberg-New York Google Scholar
  39. R.
    Roesch P. (1999). Puzzles de Yoccoz pour les applications à allure rationnelle. Enseign. Math. (2) 45(1–2): 133–168 zbMATHMathSciNetGoogle Scholar
  40. SS.
    Shub M., Sullivan D. (1985). Expanding endomorphims of the circle revisited. Ergodic Theory Dynamical Systems 5: 285–289 zbMATHMathSciNetGoogle Scholar
  41. S1.
    Słodkowski Z. (1991). Holomorphic motions and polynomials hulls. Proc. Amer. Math. Soc. 111: 347–355 CrossRefMathSciNetzbMATHGoogle Scholar
  42. Z.
    Zakeri S. (1999). Dynamics of cubic Siegel polynomials. Commun. Math. Physics. 206: 185–233 zbMATHCrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Dep. de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain
  2. 2.Dep. d’Eng. Informàtica i MatemàtiquesUniversitat Rovira i VirgiliTarragonaSpain

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