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[Author: Blanchard, Paul] AND [Keyword: Julia Set] (2)	13 Mar 2025	Run
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articleNo Access
A GENERALIZED VERSION OF THE MCMULLEN DOMAIN
International Journal of Bifurcation and Chaos01 Aug 2008
Preview Abstract
We study the family of complex maps given by F_λ(z) = zⁿ + λ/zⁿ + c where n ≥ 3 is an integer, λ is an arbitrarily small complex parameter, and c is chosen to be the center of a hyperbolic component of the corresponding Multibrot set. We focus on the structure of the Julia set for a map of this form generalizing a result of McMullen. We prove that it consists of a countable collection of Cantor sets of closed curves and an uncountable number of point components.
articleNo Access
CHECKERBOARD JULIA SETS FOR RATIONAL MAPS
International Journal of Bifurcation and Chaos01 Feb 2013
Preview Abstract
In this paper, we consider the family of rational maps
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 F_μ are conjugate on these Julia sets only if the parameters at the centers of the given cardioids satisfy μ = ν^j(d+1)λ or 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.