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[Author: Backes, Lucas] AND [Keyword: Continuity] (1)	13 Mar 2025	Run
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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.
articleNo Access
An Infinite Sequence of Bitransitive Sierpiński Carpets for $z \mapsto λ (z + \frac{1}{z})$ $z \mapsto λ (z + \frac{1}{z})$
- Daniel M. Look
International Journal of Bifurcation and Chaos10 Mar 2022
Preview Abstract
We prove that there is a sequence of purely imaginary parameter values $λ_{n}$ $λ_{n}$ converging to $0$ $0$ such that the Julia set for $z \mapsto λ_{n} (z + 1 / z)$ $z \mapsto λ_{n} (z + 1 / z)$ is homeomorphic to the Sierpiński carpet fractal; however, for any distinct pair of such parameter values, the dynamics of the map restricted to the Julia set are not conjugate.