Checkerboard Julia Sets for Rational Maps

In this paper, we consider the family of rational maps Fλ(z) = z n + 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 Fλ and Fμ are conjugate on these Julia sets only if the parameters at the centers of the given cardioids satisfy μ = νλ or μ = νλ where j ∈ Z and ν is an n − 1 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. ∗The second author would like to thank the Department of Mathematics and Statistics at Boston University for its hospitality while this work was in progress. In addition, she would also like to thank TUBİTAK for their support while this research was in progress. 1 ar X iv :1 10 3. 38 03 v3 [ m at h. D S] 2 0 Ju n 20 11 In recent years there have been many papers dealing with the family of rational maps given by Fλ(z) = z n + λ zd , where n ≥ 2, d ≥ 1, and λ ∈ C [9]. For many parameter values, the Julia sets for these maps are Sierpiński curves, i.e., planar sets that are homeomorphic to the well-known Sierpiński carpet fractal. One distinguishing property of Sierpiński curve Julia sets is that the Fatou set consists of infinitely many open disks, each bounded by a simple closed curve, but no two of these bounding curves intersect. There are many different ways in which these Sierpiński curves arise as Julia sets in these families. For example, the Julia set is a Sierpiński curve if λ is a parameter for which 1. the critical orbits enter the immediate basin of attraction of ∞ after two or more iterations [5]; 2. the parameter lies in the main cardioid of a “buried” baby Mandelbrot set [3]; or 3. the parameter lies on a buried point in a Cantor necklace in the parameter plane [7]. The parameter planes for these maps in the cases where n = d = 3 and n = d = 4 are shown in Figure 1. The red disks not centered at the origin are regions where the first case above occurs. These disks are called Sierpiński holes. Many Mandelbrot sets are visible in Figure 1. The ones that touch the external red region are not “buried,” so their main cardioids do not contain Sierpiński curve Julia sets. Only the ones that do not meet this boundary contain parameters from case 2. Finally, numerous Cantor necklaces, i.e., sets homeomorphic to the Cantor middle-thirds set with the removed open intervals replaced by open disks, appear in these figures. The buried points in the Cantor set portion of the necklace are the parameters for which case 3 occurs.