Checkerboard Julia sets for rational maps

Paul Blanchard*, Figen Çilingir, Daniel Cuzzocreo, Robert L. Devaney, Daniel M. Look, Elizabeth D. Russell

*Corresponding author for this work

Research output: Contribution to journalReview article

6 Scopus citations

Abstract

In this paper, we consider the family of rational maps Fλ(z) = zn + λ/zd, where n ≥ 2, d ≥ 1, and λ ∈ ℂ. We consider the case where λ lies in the main cardioid of one of the n - 1 principal Mandelbrot sets in these families. We show that the Julia sets of these maps are always homeomorphic. However, two such maps F λ and Fμ are conjugate on these Julia sets only if the parameters at the centers of the given cardioids satisfy μ = νj(d+1)λ or μ = νj(d+1)λ where j ∈ ℤ and ν is an (n - 1)th root of unity. We define a dynamical invariant, which we call the minimal rotation number. It determines which of these maps are conjugate on their Julia sets, and we obtain an exact count of the number of distinct conjugacy classes of maps drawn from these main cardioids.

Original languageEnglish (US)
Article number1330004
JournalInternational Journal of Bifurcation and Chaos
Volume23
Issue number2
DOIs
StatePublished - Feb 2013

Keywords

  • Julia set
  • Mandelbrot set
  • symbolic dynamics

ASJC Scopus subject areas

  • Modeling and Simulation
  • Engineering (miscellaneous)
  • General
  • Applied Mathematics

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    Blanchard, P., Çilingir, F., Cuzzocreo, D., Devaney, R. L., Look, D. M., & Russell, E. D. (2013). Checkerboard Julia sets for rational maps. International Journal of Bifurcation and Chaos, 23(2), [1330004]. https://doi.org/10.1142/S0218127413300048