Generalized baby Mandelbrot sets adorned with halos in families of rational maps

Generalized Rings Around the McMullen Domain

“…In this paper we discuss some of the properties of the parameter plane for these maps. The following result can be found in [6].…”

“…In the second part, Proposition 3.4, we transfer this curve to the parameter λ−plane, using the Schwarz Lemma, to show the existence of S 1 . This new curve is also a Jordan curve that surrounds the origin and contains the centers of the main cardioids of (n−1) principal baby Mandelbrot sets and the same number of centers of Sierpinski holes in the case n = d. Indeed, the existence of these (n − 1) principal baby Mandelbrot set was proved in [6].…”

“…Note, however, that the cycles involved may be different depending on j and, indeed, they may even have different periods. Nonetheless, all points lying on these attracting cycles are of the form ν j z 0 for some z 0 ∈ C. Theorem 2.1 (Symmetries in dynamical plane, see [6]) Let n, d ∈ N with 1/n + 1/d < 1 and let ν = e i 2π m , then there exist r, q ∈ N such that for all k ≥ r,…”

“…The condition 1/n + 1/d < 1 is equivalent to nd > m and this defines the set of values n, d ≥ 2 with m ≥ 5. The dynamics of some of these maps, the topological structures of their Julia sets and the structure of the parameter λ−planes have been widely studied by several authors, see for example [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. A recent survey of results involving some of the maps in this family is given in [19].…”

“…The ideas behind the proof are similar to the ones used in [3]. However, some important differences are the fact that in the general case there is no involution of the form z → λ 1/n /z that conjugates F λ to itself as in the case n = d. Also, the proof in the more general case requires the use of generalized symmetries and ideas that where developed in [6,33].…”