Publicacions Matemàtiques

Joining Polynomial and Exponential Combinatorics for Some Entire Maps

Antonio Garijo, Xavier Jarque, and Mónica Moreno Rocha

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Abstract

We consider families of entire transcendental maps given by $F_{\lambda,m} (z) = \lambda z^m \exp(z) $ where $m \ge 2$. All these maps have a superattracting fixed point at $z=0$ and a free critical point at $z=-m$. In parameter planes we focus on the capture zones, i.e., we consider $\lambda$ values for which the free critical point belongs to the basin of attraction of $z=0$. We explain the connection between the dynamics near zero and the dynamics near infinity at the boundary of the immediate basin of attraction of the origin, thus, joining together exponential and polynomial behaviors in the same dynamical plane.

Article information

Source
Publ. Mat., Volume 54, Number 1 (2010), 113-136.

Dates
First available in Project Euclid: 8 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.pm/1262962135

Mathematical Reviews number (MathSciNet)
MR2603591

Zentralblatt MATH identifier
1180.37064

Subjects
Primary: 37F20: Combinatorics and topology 30D20: Entire functions, general theory

Keywords
Julia sets polynomial-like maps combinatorial dynamics

Citation

Garijo, Antonio; Jarque, Xavier; Moreno Rocha, Mónica. Joining Polynomial and Exponential Combinatorics for Some Entire Maps. Publ. Mat. 54 (2010), no. 1, 113--136. https://projecteuclid.org/euclid.pm/1262962135


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References

  • J. M. Aarts and L. G. Oversteegen, The geometry of Julia sets, Trans. Amer. Math. Soc. 338(2) (1993), 897\Ndash918.
  • I. N. Baker, J. Kotus, and L. Yinian, Iterates of meromorphic functions. IV. Critically finite functions, Results Math. 22(3\Ndash4) (1992), 651\Ndash656.
  • K. Barański, Trees and hairs for some hyperbolic entire maps of finite order, Math. Z. 257(1) (2007), 33\Ndash59.
  • W. Bergweiler, Invariant domains and singularities, Math. Proc. Cambridge Philos. Soc. 117(3) (1995), 525\Ndash532.
  • R. Bhattacharjee and R. L. Devaney, Tying hairs for structurally stable exponentials, Ergodic Theory Dynam. Systems 20(6) (2000), 1603\Ndash1617.
  • R. Bhattacharjee, R. L. Devaney, R. E. L. Deville, K. Josić, and M. Moreno Rocha, Accessible points in the Julia sets of stable exponentials, Discrete Contin. Dyn. Syst. Ser. B 1(3) (2001), 299\Ndash318.
  • C. Bodelón, R. L. Devaney, M. Hayes, G. Roberts, L. R. Goldberg, and J. H. Hubbard, Hairs for the complex exponential family, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9(8) (1999), 1517\Ndash1534.
  • R. L. Devaney and L. R. Goldberg, Uniformization of attracting basins for exponential maps, Duke Math. J. 55(2) (1987), 253\Ndash266.
  • R. L. Devaney and X. Jarque, Misiurewicz points for complex exponentials, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 7(7) (1997), 1599\Ndash1615.
  • R. L. Devaney, X. Jarque, and M. Moreno Rocha, Indecomposable continua and Misiurewicz points in exponential dynamics, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15(10) (2005), 3281\Ndash3293.
  • R. L. Devaney and M. Krych, Dynamics of $\exp(z)$, Ergodic Theory Dynam. Systems 4(1) (1984), 35\Ndash52.
  • R. L. Devaney and F. Tangerman, Dynamics of entire functions near the essential singularity, Ergodic Theory Dynam. Systems 6(4) (1986), 489\Ndash503.
  • A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18(2) (1985), 287\Ndash343.
  • A. È. Erëmenko and M. Yu. Lyubich, Iterations of entire functions, (Russian), Dokl. Akad. Nauk SSSR 279(1) (1984), 25\Ndash27; English translation in: Soviet Math. Dokl. 30(3) (1984), 592\Ndash594.
  • A. È. Erëmenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42(4) (1992), 989\Ndash1020.
  • N. Fagella and A. Garijo, Capture zones of the family of functions $\lambda z\sp m\exp(z)$, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13(9) (2003), 2623\Ndash2640.
  • L. R. Goldberg and L. Keen, A finiteness theorem for a dynamical class of entire functions, Ergodic Theory Dynam. Systems 6(2) (1986), 183\Ndash192.
  • C. McMullen, Automorphisms of rational maps, in: “Holomorphic functions and moduli”, Vol. I (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ. 10, Springer, New York, 1988, pp. 31\Ndash60.
  • J. W. Milnor, On cubic polynomial maps with periodic critical point, Part I, Preprint, Stony Brook Institute for Mathematical Sciences (2008), available at: http://www.math.sunysb.edu/$\sim$jack/.
  • L. Rempe, Rigidity of escaping dynamics for transcendental entire maps, Acta Math. (to appear), arXiv:math 0605058v3 (2009).
  • P. Roesch, Puzzles de Yoccoz pour les applications à allure rationnelle, Enseign. Math. (2) 45(1\Ndash2) (1999), 133\Ndash168.
  • D. Schleicher and J. Zimmer, Escaping points of exponential maps, J. London Math. Soc. (2) 67(2) (2003), 380\Ndash400.