A Mandelpinski maze for rational maps of the form zn+λ/zd

Abstract

In this paper we identify a new type of structure that lies in the parameter plane of the family of maps z n + λ/z d where n ≥ 2 is even but d ≥ 3 is odd. We call this structure a Mandelbrot-Sierpinski maze. Basically, the maze consists at the first level of an infinite string of alternating Mandelbrot sets and Sierpinski holes that lie along an arc in the parameter plane for this family. At the next level, there are infinitely many smaller Mandelbrot sets and Sierpinski holes that alternate on the arc between each Mandelbrot set and Sierpinski hole on the previous level, and then finitely many other Mandelbrot sets and Sierpinski holes that extend away from the given Mandelbrot set in a pair of different directions. And then this structure repeats inductively to produce the "Mandelpinski" maze.

On Some Advances in Dynamics of One Variable Complex Functions

“…The structure MandelbrotSierpinski maze (MS-maze) explains by Devanay in [22] for family of rational maps z 2 + λ/z 3 . It is a newest structure that belongs to family of these functions in the parameter plane.…”

Generalized Rings Around the McMullen Domain

“…Following [3], we can consider that there is a Mandelbrot set whose central spine lies along the interval [λ − , λ + ], where λ − and λ + correspond to the interval [−2, 1/4] of the actual Mandelbrot set, contained in R + . The existence of these and other copies of Mandelbrot sets in the parameter plane of F λ is given in [6,20,21,22]. Proposition 3.7 (Superstable parameters for λ ∈ R + ) There is a decreasing sequence of parameters in R + , such that λ 1 > λ 2 • • • converging to λ − , and for λ = λ k , the critical point c 0 is periodic with period k, and the critical orbit in R + has the special form when k ∈ N and k ≥ 2 :…”

“…A recent survey of results involving some of the maps in this family is given in [19]. The structure of the parameter planes of these maps is furthered studied in [20,21,22].…”

“…However, as shown in [2], there exist Sierpiński holes corresponding to each escape time k ≥ 3, and these have the property that if λ 1 and λ 2 lie in Sierpiński holes with different escape times, then F λ 1 and F λ 2 are not topologically conjugate on their Julia sets. Much more information about the λ−planes for the case n = d ≥ 3 is given in [20,21,22].…”