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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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On the dynamics of the McMullen family $R(z)=z^m +\lambda /z^{\ell }$
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by Norbert Steinmetz PDF
Conform. Geom. Dyn. 10 (2006), 159-183 Request permission

Abstract:

In this note we discuss the parameter plane and the dynamics of the rational family $R(z)=z^m+\lambda /z^{\ell }$, with $m\ge 2$, $\ell \ge 1$, and $0<|\lambda |<\infty$.
References
  • Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR 1128089, DOI 10.1007/978-1-4612-4422-6
  • BDLSS P. Blanchard, R. L. Devaney, D. M. Look, P. Seal, and Y. Shapiro, Sierpiński curve Julia sets and singular perturbations of complex polynomials, preprint 2003. BDLMRSSU P. Blanchard, R. L. Devaney, D. M. Look, M. Morena Rocha, P. Seal, S. Siegmund, and D. Uminsky, Sierpiński carpets and gaskets as Julia sets of rational maps, preprint 2003.
  • Bodil Branner, The Mandelbrot set, Chaos and fractals (Providence, RI, 1988) Proc. Sympos. Appl. Math., vol. 39, Amer. Math. Soc., Providence, RI, 1989, pp. 75–105. MR 1010237, DOI 10.1090/psapm/039/1010237
  • Busse N. Busse, Dynamische Eigenschaften rekursiv definierter Polynomfolgen, Dissertation Dortmund 1992.
  • C. Carathéodory, Conformal representation, 2nd ed., Dover Publications, Inc., Mineola, NY, 1998. MR 1614918
  • Robert L. Devaney, Structure of the McMullen domain in the parameter planes for rational maps, Fund. Math. 185 (2005), no. 3, 267–285. MR 2161407, DOI 10.4064/fm185-3-5
  • Robert L. Devaney, Baby Mandelbrot sets adorned with halos in families of rational maps, Complex dynamics, Contemp. Math., vol. 396, Amer. Math. Soc., Providence, RI, 2006, pp. 37–50. MR 2209085, DOI 10.1090/conm/396/07392
  • Devaney3 R. L. Devaney, The McMullen domain: Satellite Mandelbrot Sets and Sierpinsky holes, preprint 2005.
  • Robert L. Devaney, Daniel M. Look, and David Uminsky, The escape trichotomy for singularly perturbed rational maps, Indiana Univ. Math. J. 54 (2005), no. 6, 1621–1634. MR 2189680, DOI 10.1512/iumj.2005.54.2615
  • DHLRU R. L. Devaney, M. Holzer, D. M. Look, M. Morena Rocha, and D. Uminsky, Singular perturbations of $z^n$, preprint 2004.
  • Robert L. Devaney and Daniel M. Look, Symbolic dynamics for a Sierpinski curve Julia set, J. Difference Equ. Appl. 11 (2005), no. 7, 581–596. American Mathematical Society Special Session on Difference Equations and Discrete Dynamics. MR 2173245, DOI 10.1080/10236190412331334473
  • Robert L. Devaney and Daniel M. Look, Buried Sierpinski curve Julia sets, Discrete Contin. Dyn. Syst. 13 (2005), no. 4, 1035–1046. MR 2166716, DOI 10.3934/dcds.2005.13.1035
  • DL3 R. L. Devaney and D. M. Look, A criterion for Sierpiński curve Julia sets for rational maps, preprint 2005. DMRS R. L. Devaney, M. Morena Rocha, and S. Siegmund, Rational maps with generalized Sierpiński gasket Julia sets, preprint 2004. DM R. L. Devaney and S. M. Moretta, The McMullen domain: rings around the boundary, preprint 2005.
  • Adrien Douady and John Hamal Hubbard, Itération des polynômes quadratiques complexes, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 3, 123–126 (French, with English summary). MR 651802
  • Adrien Douady and John Hamal Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 287–343. MR 816367, DOI 10.24033/asens.1491
  • R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217. MR 732343, DOI 10.24033/asens.1446
  • Mattler Ch. Mattler, Juliamengen und lokaler Zusammenhang, Dissertation Dortmund 1996.
  • Curt McMullen, Automorphisms of rational maps, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 10, Springer, New York, 1988, pp. 31–60. MR 955807, DOI 10.1007/978-1-4613-9602-4_{3}
  • Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
  • Curtis T. McMullen, The Mandelbrot set is universal, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 1–17. MR 1765082
  • John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240
  • Carsten Lunde Petersen and Gustav Ryd, Convergence of rational rays in parameter spaces, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 161–172. MR 1765088
  • Roesch P. Roesch, On captures for the family $f_\lambda (z)=z^2+\lambda /z^2$, to appear.
  • Norbert Steinmetz, Rational iteration, De Gruyter Studies in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1993. Complex analytic dynamical systems. MR 1224235, DOI 10.1515/9783110889314
  • Steinmetz2 N. Steinmetz, Sierpiński Curve Julia Sets of Rational Maps, Computational Methods and Function Theory 6 (2006), 317–327.
  • Lei Tan and Yongcheng Yin, Local connectivity of the Julia set for geometrically finite rational maps, Sci. China Ser. A 39 (1996), no. 1, 39–47. MR 1397233
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Additional Information
  • Norbert Steinmetz
  • Affiliation: Fachbereich Mathematik, Universität Dortmund, D-44221 Dortmund, Germany
  • Email: stein@math.uni-dortmund.de
  • Received by editor(s): January 31, 2006
  • Published electronically: August 22, 2006
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Conform. Geom. Dyn. 10 (2006), 159-183
  • MSC (2000): Primary 37F10, 37F15, 37F45
  • DOI: https://doi.org/10.1090/S1088-4173-06-00149-4
  • MathSciNet review: 2261046