We consider the family of rational maps given by F λ (z) = z n +λ/z d where n, d ∈ N with 1/n + 1/d < 1, the variable z ∈ C and the parameter λ ∈ C. It is known that when n = d ≥ 3 there are infinitely many rings S k with k ∈ N, around the McMullen domain. The Mc-Mullen domain is a region centered at the origin in the parameter λ-plane where the Julia sets of F λ are Cantor sets of simple closed curves. The rings S k converge to the boundary of the McMullen domain as k → ∞ and contain parameter values that lie at the center of Sierpiński holes, i.e., open simply connected subsets of the parameter space for which the Julia sets of F λ are Sierpiński curves. The rings also contain the same number of superstable parameter values, i.e., parameter values for which one of the critical points is periodic and correspond to the centers of the main cardioids of copies of Mandelbrot sets. In this paper we generalize the existence of these rings to the case when 1/n+1/d < 1 where n is not necessarily equal to d. The number of Sierpiński holes and superstable parameters on S 1 is τ n,d 1 = n − 1, and on S k for k > 1 is given by τ n,d k = dn k−2 (n − 1) − n k−1 + 1.
For the family of rational maps [Formula: see text] where [Formula: see text] it is known that there are infinitely many Mandelpiński necklaces [Formula: see text] with [Formula: see text] around the McMullen domain surrounding the origin in the parameter [Formula: see text]-plane. In this paper, we prove the existence of infinitely many these rings with a number of [Formula: see text] for fixed [Formula: see text] outside the Mandelpiński necklace [Formula: see text]. The ring [Formula: see text] is a simple closed curve meeting [Formula: see text] at [Formula: see text] points, such that it passes through exactly [Formula: see text] centers of Sierpinski holes and [Formula: see text] superstable parameter values. For each [Formula: see text], [Formula: see text] passes through precisely alternating [Formula: see text] superstable parameter values and the same number of centers of Sierpiński holes. There exist [Formula: see text] disjoint rings [Formula: see text] not meeting [Formula: see text] and surrounding the centers of Sierpiński holes lying on [Formula: see text], in the exterior and interior of a curve [Formula: see text], respectively. The number of such rings [Formula: see text] for fixed [Formula: see text] is [Formula: see text].
In this paper we discuss the dynamical structure of the rational family (f t ) given byEach map f t has two super-attracting immediate basins and two free critical points. We prove that for 0 < |t| ≤ 1 and |t| ≥ 1, either of these basins is completely invariant and at least one of the free critical points is inactive. Based on this separation we draw a detailed picture the structure of the dynamical and the parameter plane
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