Generalized Rings Around Sierpiński Holes

Abstract

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 [Formula: see text] The dividing ring [Formula: see text] is a simple closed curve passing through [Formula: see text] superstable parameters and the same number of escape parameters. Ring [Formula: see text] is a simple closed curve lying outside ring [Formula: see text] and meeting with [Formula: see text] at [Formula: see text] points and passing [Formula: see text] superstable parameters and [Formula: see text] escape parameters. For [Formula: see text], there exist [Formula: see text] rings [Formula: see text] and each ring passes through [Formula: see text] superstable parameters and the same number of escape parameters. Each ring [Formula: see text] is characterized by a unique symbolic sequence of length [Formula: see text] representing the property of critical orbit corresponding to a parameter lying on it. [Formula: see text] meets with one [Formula: see text]. There are [Formula: see text] rings [Formula: see text] meeting with [Formula: see text] and of those, [Formula: see text] and [Formula: see text] rings lie outside and inside ring [Formula: see text] meeting with [Formula: see text], respectively.