Rabbits, Basilicas, and Other Julia Sets Wrapped in Sierpinski Carpets

0 lies on a periodic orbit. We then perturb F0 by adding a pole at the origin. Our goal is to investigate the structure of the Julia set of Fλ, which we denote by J(Fλ), when λ is nonzero. For these maps, the point at ∞ is always a superattracting fixed point, so we have an immediate basin of attraction of ∞ that we denote by Bλ. As a consequence, we may also define the filled Julia set for these maps to be the set of points whose orbits remain bounded. We denote this set by K(Fλ). In the case where c is chosen so that the map has a superattracting cycle of period 1, the structure of J(Fλ) has been well-studied [1], [3], [4], [5]. In this case, c = 0 and the map is z+λ/z. This map has four “free” critical points at the points cλ = λ , but there is essentially only one critical orbit, since one checks easily that F 2 λ (cλ) = 4λ + 1/4. Hence all four of the critical orbits land on the same point after two iterations. Then the following result is proved in [1].