Generalized Rings Around Sierpiåƒski Holes

In this paper, we prove the existence and arrangement structure of rings passing through superstable parameters and escape parameters and lying outside dividing ring in the parameter plane of the McMullen family Fλ(z) = zn + λ zd 1 n + 1 d < 1,z ∈ℂ̄,λ ∈ ℂ∖{0}. The dividing ring S1 is a simple closed curve passing through n − 1 superstable parameters and the same number of escape parameters. Ring S2 is a simple closed curve lying outside ring S1 and meeting with S1 at n − 1 points and passing n(n − 1) superstable parameters and n2 − 1 escape parameters. For k ≥ 3, there exist (n − 1)2(n + d − 2)k−3 rings Sk and each ring passes through n + d superstable parameters and the same number of escape parameters. Each ring Sk is characterized by a unique symbolic sequence of length k − 1 representing the property of critical orbit corresponding to a parameter lying on it. Sk meets with one Sk−1. There are n + d − 2 rings Sk+1 meeting with Sk and of those, n − 1 and d − 1 rings lie outside and inside ring Sk−1 meeting with Sk, respectively.