Baby Mandelbrot Sets and Spines in Some One-Dimensional Subspaces of the Parameter Space for Generalized McMullen Maps

• Suzanne Boyd ^[2] ; Matthew Hoeppner ^[1]
  1. [1] Hope College

     Hope College

     City of Holland, Estados Unidos

     
  2. [2] University of Wisconsin Milwaukee
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 4, 2025
• Idioma: inglés
• Enlaces
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• Resumen

  • For the family of complex rational functions of the form Rn,a,c(z) = zn + a zn + c, known as “Generalized McMullen maps”, for a = 0 and n ≥ 3 fixed, we study the boundedness locus in some one-dimensional slices of the (a, c)-parameter space, by fixing a parameter or imposing a relation. First, if we fix c with |c| ≥ 6 while allowing a to vary, assuming a modest lower bound on n in terms of |c|, we establish the location in the a-plane of n “baby" Mandelbrot sets, that is, homeomorphic copies of the original Mandelbrot set. We use polynomial-like maps, introduced by Douady and Hubbard ([9]) and applied for the subfamily Rn,a,0 by Devaney ([4]). Second, for slices in which c = ta, we again observe what look like baby Mandelbrot sets within these slices, and begin the study of this subfamily by establishing a neighborhood containing the boundedness locus.

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